THE GROUP-THEORETICAL METHOD IN THE QUANTUM PHYSICS OF SOLIDS*)

(Point Symmetry)

A. V. Sokolov and V. P. Shirokovskii

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
2. Symmetry of physical systems and its connection with group theory . . . . . 620
3. Elements of group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
4. Fundamentals of representation theory . . . . . . . . . . . . . . . . . . . 628
5. Calculation of characters. Reduction of representations . . . . . . . . . . 632
6. Representations of the rotation group . . . . . . . . . . . . . . . . . . . . 637
7. Infinitesimal transformations . . . . . . . . . . . . . . . . . . . . . . . . 644
8. The octahedral group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
9. Application of group theory in quantum mechanics . . . . . . . . . . . . . 655
10. Conservation laws and a complete set of physical quantities . . . . . . . 658
11. The hydrogen-atom problem . . . . . . . . . . . . . . . . . . . . . . . . . 663
12. States of an electron in a crystalline field . . . . . . . . . . . . . . . . 667
13. Splitting of atomic terms in a crystal without spin taken into account . . 675
14. Splitting of atomic terms in a crystal with spin taken into account . . . . 679
15. Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683
16. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687

The aim of the present review is to set forth the basic ideas of the application of the method of group theory to the quantum physics of solids in a form accessible to a broad circle of physicists. Until now there has not been, in the Soviet literature, a sufficiently systematic exposition of this question. The article covers the theoretical material on point symmetry.

*) In preparing the review the authors used both textbook and scientific literature devoted, to one degree or another, to questions of the application of group theory to quantum physics¹, as well as special literature on group theory and the theory of symmetry². As a rule, however, references to these monographs are not given in the text of the review; only journal sources are indicated.

A. V. SOKOLOV AND V. P. SHIROKOVSKII

1. INTRODUCTION

Already at the dawn of the development of modern quantum theory it became clear that the method of group theory and, in particular, the theory of representations and characters acquire enormous importance in quantum physics. It may be said that at the present time one cannot engage in physics without making use, to one degree or another, of the concept of a group.

However, because group theory and, in particular, the theory of representations and characters of the permutation group, first applied in quantum mechanics by Wigner and Neumann3, proved very difficult even for specialists, a reaction arose directed against the “group plague” in quantum mechanics. Physicists (for example, Dirac, Slater) met the new methods with hostility. Therefore there existed, and still continues to exist, a widespread opinion that the apparatus of group theory is too complicated and cumbersome for broad application, and that all results can be obtained by simpler methods. At present it is already clear that such an opinion is unfounded and harmful.

In order to extend the results obtained by Heisenberg for the helium atom to more complicated many-electron systems, Wigner and Neumann explicitly used group theory. In their investigations, permutations of electrons and exchange energy played an essential role. The true and sole reason for the success of the theory was that the Schrödinger equation remains invariant under permutation of electrons, i.e., that it admits the group of permutations of identical particles. This was the beginning of a series of remarkable works that made it possible fully to classify atomic spectra and to approach the study of molecules and chemical bonds. Since then, a large part of the results found with the help of group theory has been obtained by other methods, but these methods are nothing other than implicit applications of group theory (symmetry theory), simplified by the Pauli principle.

The second series of works, which we owe chiefly to Wigner and Weyl, showed that not only the permutation group plays an essential role in quantum physics. In the works of these authors it was established that group theory has fundamental significance for the understanding of the basic questions of quantum physics. The use of group theory makes it possible to formulate the fundamental propositions of quantum theory in the most general form, without resorting to any model representations.

The application of group theory in quantum mechanics and quantum theory of the solid state makes it possible to deal with such questions as the problem of degeneracy, the splitting of terms in a crystal, to examine zone theory rigorously, and other questions which, by means of ordinary methods of investigation, are either almost impossible,

because of their cumbersomeness, or cannot be given with such completeness as group theory makes possible.

Group theory makes it possible to separate those properties of a system that have a geometrical or kinematic origin from its purely dynamical properties, which depend on the nature of the interaction between the particles of the system, on the form of the potential energy. Only these latter properties require a complete solution of the wave equation; for the remaining properties a general treatment is sufficient, in which group theory plays approximately the same role as in crystallography.

The reluctance of many physicists to apply a method that is essentially simple can be explained only by the unfamiliarity of the apparatus and by the fact that all existing monographs devoted to the application of group-theoretical methods to quantum physics are written so ponderously that they are beyond even theoretical physicists.

The theory of symmetry has special significance in the physics of the solid state. If in atomic physics it is still possible, to a certain degree, to get by without the strict and consistently developed mathematical apparatus of group theory, then ignoring the latter in the study of molecules and crystals must inevitably lead to a lowering of the general level of scientific research in these fields. It is known that the successes of certain aspects of the theory of the solid state depended on knowledge of group theory. For example, without solving the wave equation of the many-electron system of a crystal, one can write down the general form of the wave function of the system and, with its aid, obtain a number of important conclusions of fundamental significance in the theory of the solid state.

The application of group theory to the quantum physics of the solid state began essentially with the fundamental work of Bethe[^4]. In this work a theoretical investigation was made of the splitting by the field of a crystal of levels degenerate in a free atom (molecule). The next very important stage is the series of papers by Seitz[^5] devoted to the theory of space groups. In his papers the foundations were given for finding the eigenfunctions corresponding to the symmetry of a given space group, using the Born—Karman boundary conditions. Seitz showed that these functions will always have the form of Bloch functions. The aforementioned papers of Seitz and the fundamental work of Bouckaert, Smoluchowski, and Wigner[^6] initiated a whole series of works devoted to the band theory of the solid state. Bouckaert, Smoluchowski, and Wigner, using the concept of the group of the wave vector, obtained tables of characters of irreducible representations of the symmetry groups of simple lattices; and in the works of Herring and of Döring and Zehler[^7] this method was extended and applied to more complicated lattices. It turns out that consideration of band theory using the rigorous apparatus of group theory makes it possible to elucidate some-

620  A. V. Sokolov and V. P. Shirokovsky

subtle questions that escape view under ordinary methods of consideration. For example, taking into account not only the translational but also the rotational symmetry of the crystal lattice leads to the joining of energy bands. Such joining may exert a substantial influence, for example, on the density of electronic states, the intensity of X-ray absorption and emission spectra in a metal, and other phenomena.

It should be noted that in the foreign literature in recent years the number of works in which problems of solid-state physics are solved by methods of group theory has increased considerably. One may say that group theory has become a working apparatus for foreign physicists. Here we shall mention only the works of Dering, Zehner, Herring, Elliott, Parmenter, and Dresselhaus[^8]. In the studies of these authors the band theory of atomic semiconductors is considered and, in particular, the question of the influence of spin-orbit coupling on magnetic resonance in them.

In the present article the physical foundations of the application of the method of group theory in quantum mechanics and in solid-state physics are discussed. In particular, the question of conservation laws and the complete set of physical quantities is considered; a description of electronic states in a crystalline field is given. The very important problem of the splitting of atomic terms in crystals is considered in two separate sections. Finally, in the last section general selection rules are given.

2. SYMMETRY OF PHYSICAL SYSTEMS AND ITS CONNECTION WITH GROUP THEORY

Although the consideration of physical problems with the aid of the method of group theory is very rigorous and elegant, it has the drawback that it is abstract and difficult to survey unless it is based on clear geometric representations. Therefore, throughout the exposition, the abstract consideration of physical systems is accompanied by geometric representations.

All bodies existing in nature possess a certain symmetry. Isotropic bodies are distinguished—gases, liquids, and amorphous solids, in which the physical properties are the same in all directions—and anisotropic bodies, whose physical properties differ in different directions; these are crystals. In application to physics, group theory is essentially a theory of the symmetry properties of physical systems.

The symmetry of a system is determined by the totality of those displacements that bring the system into coincidence with itself; these displacements are referred to as operations (transformations) of symmetry.

If it is required to transfer a system $\tau$ from position $\tau_1$ to position $\tau_2$, this can be done in many ways. However, for

THE GROUP-THEORY METHOD IN QUANTUM PHYSICS OF THE SOLID STATE

for our consideration it is completely immaterial what path the system follows in passing from position $\tau_1$ to position $\tau_2$, and therefore all motions effecting this transition are regarded as equivalent, and the simplest is chosen from among them. Such simplest motions include translations, rotations, and screw motions. An arbitrary motion is equivalent to one or to a combination of several of these motions.

A rotation in space is characterized by a certain axis $a$ and by an angle of rotation $\varphi$ about this axis. If a body coincides with itself upon rotation about the axis $a$ through the angle $2\pi/n$, then this axis is called an axis of symmetry of order $n$.

Let us now consider, together with the system $\tau$, the system $\bar{\tau}$ obtained from the first by mirror reflection in some plane $\sigma$. The system and its mirror image cannot be brought into coincidence with one another by ordinary motions; this can be done only by invoking the operation of mirror reflection. If $\tau_1$ is some position of the system $\tau$ and $\bar{\tau}_2$ is some position of the system $\bar{\tau}$, then we can first reflect $\tau_1$ in some plane, and then bring the mirror image $\bar{\tau}_1$ thereby obtained into coincidence with $\bar{\tau}_2$ by a motion. Such a process is called an operation of the second kind, in contrast to ordinary motions—operations of the first kind. The simplest operations of the second kind include reflections, reflections in glide planes, and roto-reflection transformations.

A reflection is characterized by some plane $\sigma$ and is denoted by $\Sigma$. A roto-reflection transformation is characterized by the presence of an axis $a$ and a plane $\sigma$ perpendicular to it. This operation consists of a rotation about $a$ through an angle $\varphi$ and a subsequent reflection in the plane $\sigma$. One says that a physical system possesses a roto-reflection axis of order $n$ if it coincides with itself upon rotation about this axis through an angle $2\pi/n$ and subsequent reflection in the plane perpendicular to the axis. For a roto-reflection transformation the notation $S(\varphi)$ is introduced.

If the angle of rotation of a roto-reflection transformation is $\varphi=\pi$, then this operation leads to inversion with respect to the point of intersection of the axis $a$ and the plane $\sigma$.

The elements of symmetry are: a center of symmetry, an axis of symmetry, a plane of symmetry, etc. They correspond to symmetry operations: to a center of symmetry—inversion; to an axis of symmetry—rotation; to a plane of symmetry—reflection in this plane, etc.

Let us denote two arbitrary operations by $M$ and $L$. The result of their successive application is again some operation $N$, which we shall call the product $LM=N$. If $M$ and $L$ are both operations of the second kind, then the operation $M$ will carry the sy-

system from position $\tau_1$ to position $\overline{\tau}_2$, and then the operation $L$ transfers it to position $\tau_3$. But it is always possible to transfer the system directly from position $\tau_1$ to $\tau_3$ by means of a motion. Thus, one may assert that if $M$ and $L$ are both operations of the second kind, then their product will be an operation of the first kind; if, however, one of them is an operation of the first kind and the other an operation of the second kind, then their resultant will be an operation of the second kind.

Let us explain the definition of the product of operations using rotations as an example. We shall denote the operation of rotation about the axis $a$ through an angle $\varphi$ by $A(\varphi)$ and consider the rotations

\[ A(\varphi),\ A(2\varphi), \ldots,\ A(k\varphi), \ldots \tag{2,1} \]

It is convenient to regard the rotation $A(k\varphi)$ as the result of applying the rotation $A(\varphi)$ repeatedly $k$ times. Then the sequence of rotations (2,1) may be written symbolically in the form

\[ A,\ A^2,\ldots,\ A^k,\ldots, \tag{2,2} \]

putting $A(k\varphi)=A^k$. The following arguments convince us of the expediency of such notation. Let

\[ A(h\varphi)=A^h,\quad A(k\varphi)=A^k, \tag{2,3} \]

but then the equality

\[ A^h \cdot A^k=A^{h+k}, \tag{2,4} \]

must be satisfied, which is indeed true, since rotations performed one after another through the angles $k\varphi$ and $h\varphi$ are equivalent to a rotation through the angle $(h+k)\varphi$.

The angles used in crystallography are always rational parts of $2\pi$. Let, for example, $\varphi=2\pi/3$. Then $A^2$ is a rotation through $4\pi/3$, and $A^3$ is a rotation through $2\pi$. However, a full revolution is equivalent to a rotation through $0$, which therefore must be included in consideration. A rotation through $0$ is called the identity transformation and is denoted by $A(0)$, $A^0$, $E$.

If one performs a rotation about the axis $a$ through the angle $\varphi$ in the opposite direction, the operation obtained is the inverse of $A$, which it is expedient to denote by $A^{-1}$. Indeed, the application of a direct and an inverse rotation gives the identity transformation; the same follows also on the basis of operations with powers

\[ A \cdot A^{-1}=E \quad \text{and, in general,} \quad A^k \cdot A^{-k}=E. \tag{2,5} \]

If by $L^{-1}$ we always understand the operation inverse to $L$, then we have:

\[ L^{-1}\cdot L=L\cdot L^{-1}=E. \tag{2,6} \]

It is easy to verify that for symmetry operations the associative law holds:

\[ (L\cdot M)\cdot N=L\cdot(M\cdot N), \tag{2,7} \]

but, generally speaking, the commutative law does not hold:

\[ L\cdot M\ne M\cdot L. \tag{2,8} \]

Let us point out a number of other important properties of symmetry operations.

If we have planes \(\sigma\) and \(\sigma'\) forming an angle \(\alpha\) with each other, and \(a\) is the line of their intersection, then it is easy to verify that the following relation holds:

\[ \Sigma\cdot\Sigma'=A(2\alpha), \tag{2,9} \]

i.e. the product of two reflections is a rotation about the line of intersection of the reflecting planes, the angle of rotation being equal to twice the angle between the planes. Indeed, from Fig. 1 it is clear that every point of the straight line \(a\) remains fixed under both reflections, and therefore the product of the reflections must be a rotation about \(a\). If now \(l\) is any straight line perpendicular to \(a\) in the plane \(\sigma\), then under reflection in \(\sigma\) it will remain unchanged, while under reflection in \(\sigma'\) it will pass into some position \(l'\). The angle \(\widehat{(ll')}=2\alpha\) is the angle of rotation.

Fig. 1.

If we multiply (2,9) on the left by \(\Sigma\) or on the right by \(\Sigma'\), then, since

\[ \Sigma^{2}=E, \tag{2,10} \]

we obtain, respectively:

\[ \Sigma=A(2\alpha)\Sigma',\qquad \Sigma'=\Sigma A(2\alpha), \tag{2,9a} \]

i.e. the product of a rotation and a reflection whose plane contains the axis of rotation is again a reflection in a plane containing this axis and making with the first plane an angle equal to one half of the angle of rotation.

The product of two rotations whose axes intersect at a certain point \(O\) will again be a rotation about some axis passing through \(O\). Indeed, let the planes \(\sigma_1,\sigma_2,\sigma_3\) form a trihedral angle with vertex at the point \(O\) and edges \(a,b\), and \(c\). Denote the angles between the planes by \(\alpha/2,\beta/2,\gamma/2\), respectively. Then according to (2,10) we have:

\[ \Sigma_1\Sigma_3\Sigma_3\Sigma_2\Sigma_2\Sigma_1=E, \]

but by (2,9)

\[ \Sigma_1 \Sigma_3 = C(\gamma), \quad \Sigma_3 \Sigma_2 = B(\beta), \quad \Sigma_2 \Sigma_1 = A(\alpha) \]

and, consequently,

\[ C(\gamma) B(\beta) A(\alpha)=E. \]

Multiplying the last equality by \(C^{-1}(\gamma)\) and taking (2,6) into account, we find:

\[ B(\beta) A(\alpha)=C^{-1}(\gamma). \tag{2,11} \]

The following special case is important: the product of two rotations \(U\) and \(V\) through \(\pi\), whose axes \(u\) and \(v\) intersect at some point \(O\), will again be a rotation about some axis \(a\), passing through the point \(O\) perpendicular to the axes \(u\) and \(v\), and the angle of rotation will be equal to twice the angle between the axes. Indeed, from Fig. 2 it is seen that under a rotation about the axis \(u\) through \(\pi\) the axis \(a\) is reversed, and under a rotation about the axis \(v\) it again returns to its initial position. This means that the axis \(a\) remains fixed under the resulting rotation and, consequently, is the axis of rotation, i.e. the product of the operations is a rotation about the axis \(a\). Its magnitude is easy to determine by noting that under the operation \(U\) the axis \(u\) remains fixed, while under the operation \(V\) it passes into the position \(u'\). Consequently, finally we have:

\[ U \cdot V = A\bigl(2\widehat{uv}\bigr). \tag{2,12} \]

Fig. 2.

Operations \(M\) and \(L\) are called commuting if the equality

\[ LM = ML \]

holds.

In particular, the following operations commute:

1. Reflection and rotation about an axis perpendicular to the plane of reflection.

2. Rotation through \(\pi\) about axes intersecting at a right angle.

3. Reflections in two mutually perpendicular planes.

4. Inversion commutes with any operation.

The totality of all symmetry operations of a given physical system is called its symmetry group. The study of symmetry groups is rationally carried out with the aid of the general apparatus of group theory.

3. ELEMENTS OF GROUP THEORY

Let there be a finite or infinite collection \(\mathfrak{G}\) of elements \(g_1, g_2, \ldots g_k, \ldots\). This collection forms a group if the following conditions are satisfied:

1. The product \((g_i g_j)\) of any two elements of the collection, taken in a definite order, is an element of the same collection \((g_i g_j)=g_k\).

2. The collection \(\mathfrak{G}\) contains an element \(e\), satisfying the relation

\[ e g_k = g_k e = g_k, \]

called the identity element.

1. To every element \(g_k\) of the collection \(\mathfrak{G}\) there corresponds, in the same collection, another element \(g_k^{-1}\), defined by the relation

\[ g_k^{-1} g_k = g_k g_k^{-1} = e, \]

called the inverse.

1. The product of elements obeys the associative law:

\[ (g_i g_j) g_k = g_i (g_j g_k). \]

The commutative law, generally speaking, does not hold, i.e., in the general case

\[ g_i g_j \ne g_j g_i. \]

The element inverse to the product \(g_i g_j\) is equal to

\[ (g_i g_j)^{-1} = g_j^{-1} g_i^{-1}, \tag{3,1} \]

which is easily proved on the basis of the third condition.

If the number \(g\) of elements of the group \(\mathfrak{G}\) is finite, then the group is called finite and its order is said to be \(g\). If, for some element of the group \(\mathfrak{G}\), the equality \(g_k^n=e\) holds, and \(n\) is the smallest power satisfying this equality, then \(n\) is called the order of the element \(g_k\).

When all elements of a group are mutually permutable, the group is called commutative (abelian); otherwise it is noncommutative (nonabelian). A special case of abelian groups is formed by cyclic groups. A group is called cyclic if all its elements can be obtained by successive exponentiation of one of them, i.e.,

\[ a, a^2, \ldots a^n = e. \]

It can be shown that if \(\mathfrak{G}\) is a group, and \(g_k\) is one of its elements, then the relation \(g_k \mathfrak{G}=\mathfrak{G}\) holds.

A set \(\mathfrak{H}\), composed of an arbitrary number of elements of the group \(\mathfrak{G}\), is called a subgroup of the group \(\mathfrak{G}\) if this set is itself a group with respect to the operation defined in \(\mathfrak{G}\). The number of elements \(h\) of this subgroup is called the order of the subgroup \(\mathfrak{H}\). It can be shown that the order of a subgroup is a divisor of the order of the whole group

\[ g=hm. \tag{3.2} \]

The elements \(g_i\) and \(g_j\) of the group \(\mathfrak{G}\) are called conjugate in this group if in \(\mathfrak{G}\) there is at least one such element \(g_k\) that the equality

\[ g_j=g_k g_i g_k^{-1} \tag{3.3} \]

is satisfied.

Geometrically, the concept of conjugate elements may be approached as follows. Consider an operation \(A\) (a rotation) with symmetry element \(a\) (the axis of rotation), and an operation \(B\) with symmetry element \(b\), and also an operation \(Q\) which transforms the symmetry element \(a\) into \(b\). We shall show that the operation \(B\) is conjugate to \(A\), i.e.

\[ B=QAQ^{-1}. \tag{3.4} \]

Although this result is general, in the proof we shall confine ourselves to the case when \(A\) is a rotation, \(A=A(\varphi)\). We shall show that \(B\) is a rotation. To this end let us consider the action of the operation \(QAQ^{-1}\) on the element \(b\). Since \(Q\) transforms \(a\) into \(b\), \(Q^{-1}\) transforms \(b\) into \(a\). The action of the operation \(A\) on the result will leave it unchanged, and, finally, \(Q\) will transform it back into \(b\). Since the resulting rotation \(QAQ^{-1}\) leaves \(b\) unchanged, the symmetry element \(b\) is the axis of rotation. It is easy to understand that the angle of rotation about the axis \(b\) must also be equal to \(\varphi\). Thus two rotations through the same angle belong to the same class if among the elements of the group there is an operation by means of which one rotation axis can be brought into coincidence with the other. In exactly the same way, two reflections in different planes belong to the same class if some operation of the group carries one plane into the other. The axes and planes of symmetry themselves, whose directions can be made to coincide with one another, are said to be equivalent.

If the element \(g_j\) of the group \(\mathfrak{H}\) is conjugate to the element \(g_i\), then the element \(g_i\) is conjugate to the element \(g_j\). Indeed, from (3.3) it follows, if one multiplies on the right by \(g_k\), and on the left by \(g_k^{-1}\), that

\[ g_i=g_k^{-1}g_j g_k, \]

i.e. \(g_i\) is obtained from \(g_j\) by transforming with the element \(g_k^{-1}\). Every element \(g_k\) is conjugate to itself, since \(g_k=e g_k e^{-1}\). Finally, if

\[ g_j=g_k g_i g_k^{-1}\quad \text{and}\quad g_l=g_m g_j g_m^{-1}, \]

then

\[ g_l=(g_m g_k)g_i(g_m g_k)^{-1}, \]

i.e., the property of conjugacy of elements is transitive. Hence it follows that every group \(\mathfrak G\) decomposes into nonintersecting sets of conjugate elements or, as they say, into classes of conjugate elements. For Abelian groups each element constitutes a class by itself, since \(a_k a_i a_k^{-1}=a_i\) always holds. The identity element constitutes a class by itself in any group.

The product\(^*\) \(K_iK_j\) of two classes of conjugate elements \(K_i\) and \(K_j\) of the group \(\mathfrak G\) consists of several classes of conjugate elements. Indeed, if \(g_i\) belongs to \(K_i\) and \(g_j\) belongs to \(K_j\), then \(K_iK_j\) consists of elements \(g_ig_j\). Consider the elements conjugate to \(g_ig_j\):

\[ g_k(g_i g_j)g_k^{-1}=(g_k g_i g_k^{-1})(g_k g_j g_k^{-1}), \]

i.e., every element conjugate to the element \(g_ig_j\) from \(K_iK_j\) is already contained in this product. Thus, the product \(K_iK_j\), together with every element, also contains all elements conjugate to it, and consequently also the entire corresponding class. Therefore we finally have:

\[ K_iK_j=\sum_{k=1} c_{ijk}K_k . \tag{3,5} \]

This equality expresses the fact that the product of two classes of conjugate elements will consist of the collection of some number of classes of conjugate elements.

The subgroups \(\mathfrak H\) and \(g_k\mathfrak H g_k^{-1}\) of the group \(\mathfrak G\) shall be called conjugate subgroups. Just as a group decomposes into classes of conjugate elements, so the set of all subgroups of the group \(\mathfrak G\) decomposes into nonintersecting classes of conjugate subgroups.

A subgroup which coincides with its conjugate is called a normal divisor or invariant subgroup of the group \(\mathfrak G\). Consequently, if the subgroup \(\mathfrak N\) is a normal divisor, then for every element \(g_k\) of the group \(\mathfrak G\) the equalities

\[ g_k\mathfrak N g_k^{-1}=\mathfrak N \quad \text{or} \quad g_k\mathfrak N=\mathfrak N g_k \tag{3,6} \]

hold. Thus, a normal divisor commutes with any element of the group. If \(n_i\) belongs to the normal divisor \(\mathfrak N\), then every conjugate element \(g_kn_ig_k^{-1}\) also belongs to \(\mathfrak N\), i.e.

\(^*\) By the product \(K_iK_j\) of classes one should understand the result of multiplying each of the elements of \(K_i\) by each of the elements of \(K_j\).

a normal divisor contains the whole class corresponding to the element \(n_i\). Indeed, if \(n_i\) belongs to \(\mathfrak N\), then \(n_i\mathfrak N=\mathfrak N\). Transforming this equality and using (3.6), we find:

\[ g_k n_i \mathfrak N g_k^{-1}=g_k n_i g_k^{-1}\mathfrak N=\mathfrak N . \]

Consequently, the element \(g_k n_i g_k^{-1}\) belongs to \(\mathfrak N\). This condition is at the same time also sufficient for the subgroup to be a normal divisor.

Let us introduce the following definitions as well.

1. Let \(\mathfrak M\) be an arbitrary set consisting of some number of elements of the group \(\mathfrak G\). The subgroup consisting of all elements of the group equal to products of a finite number of powers of the elements of the set \(\mathfrak M\) is called the subgroup generated by the set \(\mathfrak M\), and is denoted by \(\{\mathfrak M\}\).

2. A group \(\mathfrak G\) is called the direct product of its subgroups \(\mathfrak H_1, \mathfrak H_2,\ldots,\mathfrak H_n\), if the following requirements are fulfilled: a) the elements of any two subgroups \(\mathfrak H_i\) and \(\mathfrak H_j\), for \(i\ne j\), commute with each other; b) every element \(g_k\) of \(\mathfrak G\) is represented uniquely in the form of a product

\[ g_k=h_1h_2\cdots h_n, \]

where \(h_i\) is an element of \(\mathfrak H_i\) for \(i=1,2,\ldots,n\). The direct product is usually denoted by

\[ \mathfrak G=\mathfrak H_1\times\mathfrak H_2\times\cdots\times\mathfrak H_n . \]

4. FOUNDATIONS OF THE THEORY OF REPRESENTATIONS

To the elements of a group one may assign certain concrete images, in particular matrices of orders \(1,2,\ldots\). For example, to the identity transformation, inversion, and reflection in the plane perpendicular to the \(z\)-axis there correspond the coordinate transformations

\[ \begin{pmatrix} x\\ y\\ z \end{pmatrix} \to \begin{pmatrix} x\\ y\\ z \end{pmatrix}, \qquad \begin{pmatrix} x\\ y\\ z \end{pmatrix} \to \begin{pmatrix} -x\\ -y\\ -z \end{pmatrix}, \qquad \begin{pmatrix} x\\ y\\ z \end{pmatrix} \to \begin{pmatrix} x\\ y\\ -z \end{pmatrix}, \]

which in matrix form are written as

\[ \begin{pmatrix} x\\ y\\ z \end{pmatrix} = \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{pmatrix} \begin{pmatrix} x\\ y\\ z \end{pmatrix}, \]

\[ \begin{pmatrix} -x\\ -y\\ -z \end{pmatrix} = \begin{pmatrix} -1&0&0\\ 0&-1&0\\ 0&0&-1 \end{pmatrix} \begin{pmatrix} x\\ y\\ z \end{pmatrix}, \qquad \begin{pmatrix} x\\ y\\ -z \end{pmatrix} = \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&-1 \end{pmatrix} \begin{pmatrix} x\\ y\\ z \end{pmatrix}. \]

Thus, one may say that the symmetry operations are represented by matrices of the third order:

\[ E= \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}, \quad I= \begin{pmatrix} -1&0&0\\ 0&-1&0\\ 0&0&-1 \end{pmatrix}, \quad \Sigma_z= \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&-1 \end{pmatrix}. \]

In the general case, elements of a group may be represented by matrices of higher order.

Linear substitutions in \(n\) variables are transformations of the form

\[ \begin{aligned} y_1&=a_{11}x_1+\ldots+a_{1n}x_n,\\ &\left. \begin{array}{c} \\[-1.2em] \vdots\\[-0.2em] \\[-1.2em] \end{array} \right\} \tag{4,1}\\[-0.2em] y_n&=a_{n1}x_1+\ldots+a_{nn}x_n, \end{aligned} \]

or, briefly,

\[ \mathbf y=A\mathbf x. \tag{4,1a} \]

The substitution is completely determined by the matrix of coefficients

\[ A= \begin{pmatrix} a_{11}\ldots a_{1n}\\ \cdots\\ \cdots\\ a_{n1}\ldots a_{nn} \end{pmatrix}. \tag{4,2} \]

Therefore, in speaking of linear substitutions, we are in fact speaking of their matrices, since it is quite immaterial what meaning is assigned to the quantities \(\mathbf x\).

By the product of two substitutions we shall understand the result of their successive application. If to the quantities \(\mathbf x\) one applies the substitution \(A\), which transforms them into \(\mathbf y\), and then to the resulting quantities \(\mathbf y\) the substitution \(B\), which transforms them into \(\mathbf z\), then the operation \(C\), which transforms \(\mathbf x\) into \(\mathbf z\), will be equal to \(BA\). This means that the product of substitutions defined in this way leads, in the matrix representation, to the multiplication of their matrices. Obviously, this operation obeys the associative law. Further, let the substitution \(A\) transform the quantities \(\mathbf x\) into \(\mathbf y\). Then the inverse substitution will be the substitution transforming \(\mathbf y\) into \(\mathbf x\), and its matrix is equal to \(A^{-1}\). This matrix exists only in the case when \(A\) is nonsingular, i.e. \(\operatorname{Det} A\ne0\). Finally, there always exists the identity substitution with the unit matrix. Thus, the totality of all nonsingular linear substitutions of a given dimension forms a group, called the group of linear substitutions \(\Gamma\).

If the matrices \(A\) and \(B\) are connected by the relation \(B=QAQ^{-1}\), then one says that \(B\) is obtained from \(A\) by the transformation \(Q\). A group of linear substitutions \(\Gamma'\), obtained from another group of linear substitutions \(\Gamma\) by transformation by some nonsingular matrix, will be called equivalent to \(\Gamma\). Equivalent groups will be regarded as equal, and only one of them will be considered.

If all matrices of a group of substitutions are unitary (orthogonal), then the group is called a unitary (orthogonal) group of substitutions. It can be shown that every finite (real) group of substitutions is equivalent to a unitary (orthogonal) group of substitutions.

A group of substitutions is called reducible if it can be transformed so that all its matrices have the form

\[ A=\begin{pmatrix} A_1 & 0\\ 0 & A_2 \end{pmatrix}. \tag{4,3} \]

Such a group is determined by its component parts:

\[ \Gamma_1=E_1,A_1,B_1,\ldots; \]

\[ \Gamma_2=E_2,A_2,B_2,\ldots . \]

The circumstance that \(\Gamma\) is transformed in the indicated way into a reducible group is expressed by the equality

\[ \Gamma=\Gamma_1+\Gamma_2. \]

In the general case, if \(\Gamma\) can be reduced in such a way that the component \(\Gamma_k\) occurs \(n_k\) times, then one writes:

\[ \Gamma=n_1\Gamma_1+\cdots+n_k\Gamma_k+\cdots . \tag{4,4} \]

The \(\Gamma_k\), in turn, may decompose further, but in the end one arrives at such component parts as do not admit any further reduction. Such groups of substitutions are called irreducible. The general result may be formulated as follows: every finite group of substitutions is either irreducible, or completely reducible to a sum of irreducible groups.

Let there be a group \(\mathfrak{G}\), and let to each of its elements \(g_k\) there be assigned, from some collection of matrices of degree \(n\), a matrix \(A(g_k)\), so that for any \(g_i\) and \(g_j\) from \(\mathfrak{G}\) the equality holds

\[ A(g_i g_j)=A(g_i)\cdot A(g_j). \tag{4,5} \]

Obviously, in this case the collection of matrices of degree \(n\) forms

group. We shall say that this group of linear substitutions forms an \(n\)-dimensional representation of the group \(\mathfrak G\), and the variables that transform according to the indicated matrices we shall call the basis of the representation. In the case where the group of linear substitutions is irreducible, the representation is called irreducible; in the opposite case—reducible. Let us note that the decomposition of a representation into irreducible parts means that the space of basis vectors is split into subspaces, each of which, under the transformations of the group, is transformed into itself. The obvious one-dimensional representation of a group is obtained by assigning to each element \(g_k\) the one-dimensional unit matrix, i.e. simply the number 1. This representation is called the identity representation.

Without derivation, let us write the orthogonality relations for the matrix elements of two irreducible representations, assuming—as is always possible—the representations to be unitary:

\[ \sum_{g_k} a_{\lambda \nu}(g_k)\, b_{\gamma \mu}^{*}(g_k) = \begin{cases} 0, & A(g_k) \ne B(g_k),\\[4pt] \dfrac{g}{n}\,\hat{\delta}_{\lambda\gamma}\hat{\delta}_{\nu\mu}, & A(g_k)=B(g_k). \end{cases} \tag{4,6} \]

The sum of the diagonal elements of the matrix \(A(g_k)\) in a certain irreducible representation is called the character of the element \(g_k\) in this representation and is denoted \(X(g_k)\). Conjugate elements of a group have identical characters. Consequently, the characters are functions of classes of conjugate elements, and therefore they are often denoted by \(X_j^{(m)}\), where the upper index indicates the number of the representation and the lower one the number of the class.

Between the systems of characters of irreducible representations there exist the following orthogonality relations:

\[ \sum_{g_k} X^{(m)}(g_k)X^{(l)}(g_k)=g\hat{\delta}_{ml}. \tag{4,7} \]

It is easily obtained from relation (4,6), if in it one sets \(x=\lambda\), \(\nu=\mu\) and sums over \(\lambda\) and \(\mu\).

On the basis of formulas (4,4) and (4,7) it can be shown that a necessary and sufficient condition for the equivalence of two irreducible representations consists in the equality of their systems of characters.

A reducible representation of the group \(\mathfrak G\) is decomposed in a unique way into a sum of irreducible representations. Indeed, let

\[ \Gamma = a_1\Gamma_1+\cdots+a_r\Gamma_r = b_1\Gamma_1+\cdots+b_r\Gamma_r. \]

Let us denote the character of \(\Gamma\) by \(X\); then for each element

groups the equality holds

\[ X=a_1X^{(1)}+\ldots+a_rX^{(r)}=b_1X^{(1)}+\ldots+b_rX^{(r)}. \]

Hence, by formula (4.7), we obtain:

\[ \sum_{g_k} X(g_k)\,X^{(m)}(g_k)=a_m g=b_m g \]

and, consequently, \(a_m=b_m\).

The source from which all irreducible representations of the group are obtained is the representation by means of the regular permutation group. Such a representation can be constructed if the elements of the group, arranged in a definite order, are taken as basis vectors, and to each element of the group there is assigned the matrix effecting the same permutation of elements as occurs when the group is multiplied by the given element. In this representation

\[ X(e)=g,\qquad X(g_k)=0. \tag{4.8} \]

Let us denote the regular representation by \(\Pi\) and decompose it into irreducible components:

\[ \Pi=n_1\Gamma_1+\ldots+n_r\Gamma_r. \]

Equating the characters of both sides of this equality and taking (4.8) into account, we obtain:

\[ \sum_{k=1}^{r} n_k X^{(k)}(g_i)=g\,\delta_i^{0} e. \]

Multiplying this equality by \(X^{(l)*}(g_i)\) and summing over all elements \(G\), according to (4.7), we find:

\[ n_l=X^{(l)}(e). \tag{4.9} \]

Thus, the regular representation of the group contains each irreducible representation as many times as its dimension.

It can be shown that the number of distinct irreducible representations of a group is equal to the number of classes of conjugate elements \(r\).

5. COMPUTATION OF CHARACTERS. PRODUCT OF REPRESENTATIONS

Knowledge of the structure of an abstract group makes it possible to compute all \(r\) systems of simple characters. For this one may use multiplication of classes. We already know that the relations have place—

solutions (3.5)

\[ K_iK_j=\sum_{k=1}^{r} c_{ijk}K_k. \]

Let us now consider, for some representation, the matrix equal to the sum of the matrices of all elements of the class \(K_i\), and denote it by \(M_i^{(m)}\). Then we shall have:

\[ M_i^{(m)}M_j^{(m)}=\sum_{k=1}^{r} c_{ijk}M_k^{(m)}. \tag{5.1} \]

Let us also note that the matrix \(M_i^{(m)}\) commutes with all elements of an irreducible representation and, consequently, must be scalar, i.e.

\[ M_i^{(m)}=x_i^{(m)}E_{n_m}. \]

If \(h_i\) is the order of \(K_i\), then, obviously,

\[ X\!\left(M_i^{(m)}\right)=h_iX_i^{(m)}. \]

On the other hand,

\[ X\!\left(M_i^{(m)}\right)=x_i^{(m)}n_m, \]

whence

\[ \frac{x_i^{(m)}}{h_i}=\frac{X_i^{(m)}}{n_m}. \tag{5.2} \]

Equality (5.1) is equivalent to the following:

\[ x_i^{(m)}x_j^{(m)}=\sum_{k=1}^{r} c_{ijk}x_k^{(m)}. \tag{5.3} \]

Then a system of \(r\) equations is obtained:

\[ \left. \begin{aligned} x_i^{(m)}x_1^{(m)}&=\sum_{k=1}^{r} c_{i1k}x_k^{(m)},\\ &\cdots\\ x_i^{(m)}x_r^{(m)}&=\sum_{k=1}^{r} c_{irk}x_k^{(m)}. \end{aligned} \right\} \tag{5.4} \]

Among the numbers \(x_i^{(m)}\), at least \(x_1^{(m)}\ne0\), but this system has

nonzero solutions with respect to \(x_1^{(m)} \ldots x_r^{(m)}\) only under the condition

\[ \left| \begin{array}{cccc} c_{i11}-x_i & c_{i12} & \ldots & c_{i1r} \\ c_{i21} & c_{i22}-x_i & \ldots & c_{i2r} \\ \ldots & \ldots & \ldots & \ldots \\ c_{ir1} & c_{ir2} & \ldots & c_{irr}-x_i \end{array} \right|=0. \tag{5,5} \]

The numbers \(c_{ijk}\) are all known if we know the structure of the group; consequently, this equation makes it possible to find all \(r\) values of the number \(x_i^{(m)}\). The orders of the classes \(h\) are also known from the structure of the group, so that, after solving equations (5,5), the numbers \(X_i^{(m)}/n_m\) have been found for all \(i\). Then, from formula (4,7) of character theory, which may be written in the form

\[ \sum_{k=1}^{r}\frac{X_k^{(m)}}{n_m}\frac{X_k^{(m)*}}{n_m}=\frac{g}{n_m^2}, \]

we determine \(n_m\), and, finally, completely find \(X_i^{(m)}\).

Let us also note that the dimensions of the irreducible representations \(n_m\) can often be found by a simpler method. From formulas (4,8) and (4,9) we obtain:

\[ \sum_{m=1}^{r} X^{(m)}(e)X^{(m)}(g_k)=g\delta_k^0 \]

or, recalling that \(X^{(m)}(e)=n_m\), we shall have:

\[ n_1^2+n_2^2+\ldots+n_r^2=g. \tag{5,6} \]

Consequently, if we can show that the order of the group \(g\) is decomposed in a unique way into a sum of a given number of squares of integers, then we in fact find the dimensions of the representations.

In considering certain problems of quantum physics (for example, addition of angular momenta, selection rules, and others), it is very important to know the properties of the product of representations.

Let us introduce the notion of the direct product of representations of the group \(\mathfrak{G}\). Consider two bases \(\mathbf{x}\) and \(\mathbf{y}\) of irreducible representations \(A\) and \(B\), with components \(x_1\ldots x_n\) and \(y_1\ldots y_m\), respectively. To these vectors let us put into correspondence another vector \(\mathbf{z}\), whose components are the products \(x_i y_j\) \((i=1\ldots n,\ j=1\ldots m)\). Then one obtains a system of \(nm\) new quantities, which can

serves as the basis for a certain new representation of dimension \(nm\). With the aid of the basis of this representation one can define a new transformation \(C\)

\[ z=Cz', \tag{5.7} \]

which is a consequence of the original transformations

\[ x=Ax',\qquad y=By'. \tag{5.8} \]

Since the transformations (5.8) in components can be represented in the form

\[ \begin{aligned} x_i&=\sum_{k=1}^{n} a_{ik}x'_k,\\ y_j&=\sum_{l=1}^{m} b_{jl}y'_l, \end{aligned} \tag{5.8a} \]

we have:

\[ x_i y_j=\sum_{k,l} a_{ik}b_{jl}x'_k y'_l \]

and, consequently, the coefficient of \(x'_k y'_l\)

\[ a_{ik}b_{jl}=c_{ij,kl} \tag{5.9} \]

will be an element of the matrix \(C\), situated at the intersection of row \(ij\) and column \(kl\). This matrix of order \(nm\) is called the direct (Kronecker) product of the original ones and is written in the form

\[ C=A\times B. \]

Therefore the representation whose basis is formed by the components \(z\) is called the direct product of the original representations.

The character of a representation that is the direct product of two others is equal to the product of the characters of both factors:

\[ X(C)=\sum_{ij} C_{ij,ij}=\sum_{ij} a_{ii}b_{jj}=\sum_i a_{ii}\sum_j b_{jj}=X(A)X(B). \tag{5.10} \]

The representation obtained will, generally speaking, be reducible; it is irreducible only in the case when one of the original representations is one-dimensional.

Both multiplied irreducible representations may, in particular, coincide. Then two cases must be distinguished: when the bases \(x\) and \(y\) are different and when they coincide.

Let us have two distinct bases \(x\) and \(y\), realizing one and the same representation \(\Gamma\). The direct product of the representation \(\Gamma\) by itself, \(\Gamma \times \Gamma\), has as a basis the \(n^2\) quantities \(x_i y_j\), and its characters are determined by the formula

\[ X(A \times A) = [X(A)]^2 . \]

This representation, as we have already indicated, will be reducible. A partial reduction can be carried out at once. One of the representations into which the given one decomposes will be realized by the \(n(n+1)/2\) quantities \(x_i y_j + x_j y_i\), and the other by the \(n(n-1)/2\) quantities of the form \(x_i y_j - x_j y_i\) (it is not difficult to understand that the quantities of each of these sets are transformed only into one another). The first product is called the symmetric square of the representation by itself, and its characters are denoted by \([X^2]\), while the second is called the antisymmetric square, and its characters are denoted by \(\{X^2\}\).

It is easy to determine the characters of the symmetric square. Indeed,

\[ \begin{aligned} A(x_i y_j + x_j y_i) &= \sum_k a_{ik} x_k \sum_l a_{jl} y_l + \sum_m a_{jm} x_m \sum_h a_{ih} y_h \\ &= \sum_{k,l} a_{ik}a_{jl} x_k y_l + \sum_{m,h} a_{jm}a_{ih} x_m y_h \\ &= \sum_{k,l} (a_{ik}a_{jl}+a_{jk}a_{il}) x_k y_l \\ &= \frac12 \sum_{k,l} (a_{ik}a_{jl}+a_{jk}a_{il})\,(x_k y_l+x_l y_k). \end{aligned} \]

Hence for the character we have:

\[ [X^2](A)=\frac12 \sum_{i,j}(a_{ii}a_{jj}+a_{ij}a_{ji}). \]

But since

\[ \sum_i a_{ii}=X(A), \qquad \sum_{i,j} a_{ij}a_{ji}=X(A^2), \]

we finally obtain the formula

\[ [X^2](A)=\frac12\{[X(A)]^2+X(A^2)\}, \tag{5.11} \]

which makes it possible to determine the characters of the symmetric square

representations onto itself by the characters of the original representation. In an entirely analogous way, for the characters of the antisymmetric product we find the formula

\[ \{\mathbf{X}^{2}\}(A)=\frac{1}{2}\{[\mathbf{X}(A)]^{2}-\mathbf{X}(A^{2})\}. \tag{5,12} \]

If, however, the sets \(\mathbf{x}\) and \(\mathbf{y}\) coincide, then only the symmetric product can be defined.

It is now possible to indicate one more method for obtaining irreducible representations and their characters, which often proves useful. Let the group \(\mathfrak{G}\) be the direct product of two groups \(\mathfrak{A}\) and \(\mathfrak{B}\):

\[ \mathfrak{G}=\mathfrak{A}\times \mathfrak{B}. \]

Let the irreducible representations of \(\mathfrak{A}\) be \(A_{1}, A_{2},\ldots,A_{p}\), and those of the group \(\mathfrak{B}\) be \(B_{1}, B_{2},\ldots,B_{q}\). Then we obtain all irreducible representations of the group \(\mathfrak{G}\) by directly multiplying each representation of \(\mathfrak{A}\) by each representation of \(\mathfrak{B}\), i.e. by forming the products

\[ \Gamma_{k}=A_{m}\times B_{n}. \tag{5,13} \]

The character of the element \(g_{ij}=a_i b_j\) is determined by the formula

\[ \mathbf{X}^{(k)}(g_{ij})=\mathbf{X}^{(m)}(a_i)\mathbf{X}^{(n)}(b_j). \tag{5,14} \]

6. REPRESENTATIONS OF THE ROTATION GROUP

Some physical systems, for example the hydrogen atom, coincide with themselves under a rotation through any angle about an axis passing through some fixed point in an arbitrary direction. In other words, their symmetry group is the group of rotations about a fixed center. The elements of the rotation group are determined by the values of three parameters, two of which specify the direction of the rotation axis, and the third—the angle of rotation about this axis. Therefore the rotation group is called three-parameter.

The determination of all representations of the rotation group reduces to bringing into correspondence with its transformations certain matrices of a unitary group depending on three parameters.

An example of a group representing the rotation group is the unimodular group of transformations \(\mathfrak{U}_{2}\) of two complex variables, whose operations have the form

\[ \Delta= \begin{cases} \xi'=\alpha \xi+\beta \eta,\\ \eta'=-\beta^{*}\xi+\alpha^{*}\eta. \end{cases} \tag{6,1} \]

It is obtained from the group of all possible mappings of the two-dimensional complex space onto itself:

\[ \begin{aligned} \xi'&=\alpha \xi+\beta \eta,\\ \eta'&=\gamma \xi+\delta \eta, \end{aligned} \qquad \Delta=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}, \tag{6,2} \]

if the condition of unimodularity is imposed on the transformation matrix, i.e., if one requires that \(\Delta\) be unitary and that

\[ \operatorname{Det}\Delta=1. \tag{6,3} \]

Indeed, in order that the transformation \(\Delta\) be unitary, the condition

\[ \Delta^{-1}=\widetilde{\Delta}^{*}, \tag{6,4} \]

must be satisfied, where \(\widetilde{\Delta}\) denotes the matrix transposed to \(\Delta\). Using (6,3) and (6,4), it is not difficult to show that \(\delta=\alpha^{*}\) and \(\gamma=-\beta^{*}\), i.e., the transformation \(\Delta\) can indeed be represented in the form

\[ \Delta=\begin{pmatrix} \alpha&\beta\\ -\beta^{*}&\alpha^{*} \end{pmatrix} \tag{6,5} \]

with the additional condition \(\alpha\alpha^{*}+\beta\beta^{*}=1\), following from (6,3) and reducing the number of independent variables from four to three.

With the aid of stereographic projection one can show how to associate with every rotation \(A\) (an element of the rotation group) a certain transformation \(\Delta\) with quite definite coefficients \(\alpha\) and \(\beta\).

Fig. 3.

Consider three-dimensional space with coordinate axes \(x, y, z\) and the sphere \(\mathfrak{S}\) with center at the origin and radius equal to unity (Fig. 3). Let the coordinates of the pole \(s\) be \(0,0,-1\), and let \(p\) be a variable point on the sphere. The straight line \(sp\) intersects the plane \(xy\) at some point \(p'\), and we thus have a perfectly definite correspondence between the points of the sphere \(\mathfrak{S}\) and the points of the plane \(xy\), with the point of the sphere \(s\) corresponding to the infinitely distant point of the plane. The established correspondence of points also gives us the stereographic projection of the sphere onto the plane.

Let us derive the formulas giving the stereographic projection. Let \(pq\) be the perpendicular from the point \(p\) to the \(z\)-axis. From the similarity of triangles we have \(qp/op'=qs/os=1+oq/1\), since \(os=1\), and therefore

\[ qp=(1+oq)op'. \]
Denoting by \((xyz)\) the coordinates of the point \(p\) and by \((x'y')\) the coordinates of the point \(p'\), we can write:
\[ qp=(1+z)op'. \]

Projecting the parallel segments \(op'\) and \(qp\) onto the \(x\)- and \(y\)-axes, we obtain:
\[ \begin{aligned} x&=(1+z)x',\\ y&=(1+z)y'. \end{aligned} \tag{6,6} \]

The equation of the sphere gives us a quadratic equation for \(z\), solving which we find:
\[ z=\frac{\pm 1-(x'^2+y'^2)}{1+(x'^2+y'^2)}. \tag{6,7} \]

But for all points \((x'y')\) at a finite distance we have \(z>-1\), and consequently in (6,7) the sign \(+\) must be taken. Using also (6,6), we obtain the expression of \((xyz)\) in terms of \((x'y')\):
\[ x=\frac{2x'}{1+x'^2+y'^2}, \quad y=\frac{2y'}{1+x'^2+y'^2}, \quad z=\frac{1-(x'^2+y'^2)}{1+x'^2+y'^2}. \tag{6,8} \]

Introduce the complex coordinate
\[ \zeta=x'+iy'. \]

Then (6,8) can be represented in the form
\[ x+iy=\frac{2\zeta}{1+\zeta\zeta^*}, \quad x-iy=\frac{2\zeta^*}{1+\zeta\zeta^*}, \quad z=\frac{1-\zeta\zeta^*}{1+\zeta\zeta^*}. \tag{6,9} \]

Finally, choosing homogeneous complex coordinates \(\xi\) and \(\eta\) such that \(\zeta=\eta/\xi\), and satisfying the condition
\[ \xi\xi^*+\eta\eta^*=1, \tag{6,10} \]
we obtain:
\[ x=\eta\xi^*+\xi\eta^*, \quad y=-i(\eta\xi^*-\xi\eta^*), \quad z=\xi\xi^*-\eta\eta^*. \tag{6,11} \]

To every pair of numbers \(\xi,\eta\) there corresponds a point on the sphere, since according to (6,11) the equality (6,10) is equivalent to \(x^2+y^2+z^2=1\). Under any unitary transformation we must have
\[ \xi'\xi'^*+\eta'\eta'^*=\xi\xi^*+\eta\eta^*, \]
and therefore a point of the sphere is transformed into a point of the sphere. It is easy to verify that, in this case, the angles between radius vectors drawn from the center \(o\) to various points of the sphere are preserved. Consequently, we are dealing with a rotation.

To every transformation \(\Delta\) of type (6,1) there corresponds, therefore, a rotation \(A\); the converse is not quite true, for if the signs of \(\alpha\) and \(\beta\) are changed, then \(\xi'\) and \(\eta'\) change sign (\(\xi'=-\xi,\ \eta'=-\eta\)), but \(x', y', z'\), according to (6,11), remain unchanged. As a consequence, to every rotation \(A\) there correspond two transformations of the unimodular group \(\mathfrak{U}_2\): \(\Delta\) and \(-\Delta\). Since the transformations are linear, the product \(A_2A_1\) of two successive rotations corresponds to the product \(\Delta_2\Delta_1\). Consequently, the group \(\mathfrak{U}_2\) is a two-dimensional two-valued representation of the rotation group. \(\alpha\) and \(\beta\) in (6,1) are called the Cayley–Klein parameters. In terms of the Euler angles they are expressed as follows:

\[ \begin{aligned} \alpha &= \exp\left[-\,\frac{i(\varphi+\psi)}{2}\right]\cos\frac{\theta}{2},\\ \beta &= -\,i\exp\left[-\,\frac{i(\varphi-\psi)}{2}\right]\sin\frac{\theta}{2}. \end{aligned} \tag{6,12} \]

Then the matrix \(\Delta\) in (6,5) takes the form

\[ \begin{pmatrix} \exp\left[-\,\frac{i(\varphi+\psi)}{2}\right]\cos\dfrac{\theta}{2} & -\,i\exp\left[-\,\frac{i(\varphi-\psi)}{2}\right]\sin\dfrac{\theta}{2} \\[6pt] -\,i\exp\left[\dfrac{i(\varphi-\psi)}{2}\right]\sin\dfrac{\theta}{2} & \exp\left[\dfrac{i(\varphi+\psi)}{2}\right]\cos\dfrac{\theta}{2} \end{pmatrix}. \tag{6,13} \]

It is obvious that under a rotation about the \(z\)-axis by \(2\pi\) the matrix (6,13) is multiplied by \(-1\). Thus, to the physically equivalent rotation operations by \(0\) and \(2\pi\), in the representation found, there correspond two distinct matrices. This is precisely the expression of the two-valuedness noted above. The meaning of this two-valuedness will be clarified later.

The method considered at once gives us an infinite set of representations of the rotation group, since all representations of the group \(\mathfrak{U}_2\), in view of (6,12), are also, obviously, representations of the rotation group, and they are easy to construct.

Let us form tensors of the unitary space \(\xi, \eta\). A tensor of rank \(n\) will have \(n+1\) components:

\[ \xi^n,\ \xi^{\,n-1}\eta,\ldots,\ \xi^{\,n-r}\eta^r,\ldots,\ \xi\eta^{\,n-1},\ \eta^n . \tag{6,14} \]

Let us take the component \(\xi^{\,n-r}\eta^r\) and carry out on \(\xi\) and \(\eta\) the transformation (6,1). Then we obtain:

\[ \xi'^{\,n-r}\eta'^{\,r} = (\alpha\xi+\beta\eta)^{\,n-r}(-\beta^*\xi+\alpha^*\eta)^r = \sum_{k=0}^{n} a^{(n)}_{rk}\xi^{\,n-k}\eta^k . \tag{6,15} \]

From this it is clear that the components of the tensor under the transformation (6.1) are linearly transformed into one another, defining the transformation matrix

\[ A^{(n)}=\bigl(a_{lk}^{(n)}\bigr). \]

The transformation (6.15) is not unitary, but becomes such if the tensor components are normalized by putting

\[ q_k=\frac{\xi^{\,n-k}\eta^k}{\sqrt{(n-k)!\,k!}}. \tag{6.16} \]

Let us note here that under the rotation \(A(\psi)\) with rotation angle \(\psi\) about the \(z\)-axis, under which, according to (6.13), \(\xi\) is multiplied by \(\exp\left(-\frac{i\psi}{2}\right)\), and \(\eta\) by \(\exp\left(\frac{i\psi}{2}\right)\), the component \(q_k\) is multiplied by

\[ \exp\left[-\frac{i}{2}(n-2k)\psi\right]. \]

Thus, to each rotation with Euler angles \(\varphi,\theta,\psi\) there corresponds a matrix \(A^{(n)}\). The matrices \(A^{(n)}\) are multiplied among themselves like the matrices \(\Delta\). Consequently, we have obtained an infinite set of representations of the group, generated by each tensor of rank \(n=0,1,\ldots\). In quantum notation we shall put \(n=2j\) and shall denote the representation by the symbol \(D_j\), understanding by this the \((2j+1)\)-dimensional representation of the rotation group. Moreover, we denote \(j-k=m\); then the expression for \(q_k\) will take the symmetric form:

\[ q_m^{(j)}=\frac{\xi^{\,j+m}\eta^{\,j-m}}{\sqrt{(j+m)!\,(j-m)!}},\quad m=j,\ j-1,\ldots,\ -j, \tag{6.17} \]

for \(k=j-m\) and \(n-k=2j-j+m=j+m\). Since the product of the representation matrices (6.1) corresponds to the product of the representation matrices \(\bigl(a_{lk}^{(n)}\bigr)\), we shall therefore have a linear representation of the group (6.1) of dimension \((2j+1)\), i.e. the unitary group will be represented by matrices of dimension \((2j+1)\).

Let us now proceed to the determination of the elements of the transformation matrices. Taking into account (6.17) and (6.1), we shall have:

\[ q_m^{(j)\prime} =\frac{\xi'^{\,j+m}\eta'^{\,j-m}}{\sqrt{(j+m)!\,(j-m)!}} = \frac{(\alpha\xi+\beta\eta)^{j+m}(-\beta^*\xi+\alpha^*\eta)^{j-m}}{\sqrt{(j+m)!\,(j-m)!}}, \]

and we must represent the right-hand side in the form of a linear combination

quantities \(q_m^{(j)}\). Elementary calculations give

\[ q_m^{(j)'}= \sum_{k=0}^{j+m}\sum_{k'=0}^{j-m} (-1)^{j-m-k'} \frac{\sqrt{(j+m)!(j-m)!}}{(j+m-k)!(j-m-k')!k!k'!} \times \]

\[ {}\times \alpha^{j+m-k}\alpha^{*k}\beta^k\beta^{*\,j-m-k'}\xi^{j-k-k'}\eta^{k+k'} . \tag{6,18} \]

If one sets \(p! = \infty\) when \(p\) is a negative integer, then in (6,18) the summation over \(k\) and \(k'\) may be carried out from \(-\infty\) to \(+\infty\), since the extraneous terms will contain in the denominator a factor equal to infinity and will vanish. Instead of \(k'\) let us introduce a new summation variable \(s=j-k-k'\), over which the summation may likewise be carried out from \(-\infty\) to \(+\infty\), over integral or half-integral values according as \(j\) is integral or half-integral. Thus we obtain:

\[ q_m^{(j)'}= \sum_{k,s}(-1)^{k+s-m} \frac{\sqrt{(j+m)!(j-m)!}}{(j+m-k)!(k+s-m)!k!(j-k-s)!} \times \]

\[ {}\times \alpha^{j+m-k}\alpha^{*\,j-k-s}\beta^k\beta^{*\,k+s-m}\xi^j+s\eta^{j-s}. \]

But according to (6,17) we have:

\[ \xi^{j+s}\eta^{j-s}=\sqrt{(j+s)!(j-s)!}\,q_s^{(j)} \]

and finally we obtain the required linear dependence in the form

\[ q_m^{(j)'}= \sum_{k,s}(-1)^{k+s-m} \frac{\sqrt{(j+m)!(j-m)!(j+s)!(j-s)!}}{(j+m-k)!(k+s-m)!k!(j-k-s)!} \times \]

\[ {}\times \alpha^{j+m-k}\alpha^{*\,j-k-s}\beta^k\beta^{*\,k+s-m}q_s^{(j)}, \tag{6,19} \]

whence for the elements of the matrices we have:

\[ d_{ms}^{(j)}= \sum_k(-1)^k \frac{\sqrt{(j+m)!(j-m)!(j+s)!(j-s)!}}{(j+m-k)!(k+s-m)!k!(j-k-s)!} \times \]

\[ {}\times \alpha^{j+m-k}\alpha^{*\,j-k-s}\beta^k\beta^{*\,k+s-m}. \tag{6,20} \]

Here the indices \(m, s\) run through the values from \(-j\) to \(+j\), while the summation over \(k\) is taken from the greatest of \(0, m-s\) to the smallest of \(j+m, j-s\). The constant factor \((-1)^{s-m}\) which appears may be discarded, since the representation is always determined only up to equivalence.

Let us note without proof that the matrices \(D_j\) obtained give irreducible linear representations of the unitary group. The repre-

giving \(j\) a series of values

\[ j=0,\ \frac{1}{2},\ 1,\ \frac{3}{2},\ldots, \]

we obtain an infinite set of these linear representations.

Let us also show how the direct product \(D_{j_1}\times D_{j_2}\) decomposes into irreducible representations. We begin with the calculation of the character. Since conjugate elements have equal characters, it is enough to calculate the character of a rotation about the \(z\)-axis, for in the rotation group any rotation is equivalent to a rotation through the same angle about the \(z\)-axis. The diagonal elements of the rotation matrix are obtained from formula (6.20) for \(m'=s\). Since, for a rotation about the \(z\)-axis, the angle \(\theta\) is equal to 0, in the sums (6.20) only terms with \(k=0\) remain, and we obtain:

\[ d_{mm}^{(j)}(\varphi,0)=e^{-im\varphi}. \]

Then for the expression of the character we find:

\[ X_j=\sum_{m=-j}^{j} e^{-im\varphi}, \]

or, after elementary calculations, we have:

\[ X_j=\frac{\sin\left(j+\frac{1}{2}\right)\varphi}{\sin\frac{1}{2}\varphi}. \tag{6,21} \]

The character of the direct product of two representations is equal to the product of the characters of the component parts:

\[ X_{j_1}X_{j_2} = \sum_{m_1=-j_1}^{+j_1} e^{im_1\varphi} \sum_{m_2=-j_2}^{+j_2} e^{im_2\varphi} = \]

\[ = \sum_{m_1=-j_1}^{+j_1} e^{im_1\varphi} \frac{e^{i(j_2+1)\varphi}-e^{-ij_2\varphi}}{e^{i\varphi}-1}. \]

We must represent this expression in the form of a sum of the characters of irreducible representations \(D_j\), i.e. in the form

\[ \sum_j \frac{e^{i(j+1)\varphi}-e^{ij\varphi}}{e^{i\varphi}-1}. \]

Equating the two expressions, canceling by \((e^{i\varphi}-1)\), and carrying out

multiplication, we obtain:

\[ e^{i(j_1+j_2+1)\varphi}+e^{i(j_1+j_2)\varphi}+\ldots-e^{-i(j_1-j_2)\varphi}-\ldots-e^{-i(j_1+j_2)\varphi} = \sum_j e^{i(j+1)\varphi}-e^{-ij\varphi}. \]

Combining in the left-hand side of the equality the positive and negative terms in pairs, we see that \(j\) must run through a series of values from \(j_1+j_2\) to \(j_1-j_2\), each once. Thus, we have:

\[ D_{j_1}\times D_{j_2}=D_{j_1+j_2}+\ldots+D_{j_1-j_2} = \sum_{j=j_1+j_2}^{j_1-j_2} D_j . \tag{6,22} \]

This is the desired decomposition.

7. INFINITESIMAL TRANSFORMATIONS

The rotation group of three-dimensional space is an example of an infinite group whose elements depend on parameters varying continuously. For the rotation group the role of parameters may be played, for example, by the Euler angles. The rotation group consists of linear transformations, and the dependence of the group on the parameters reduces to the fact that the elements of the matrices which determine the mentioned linear transformations depend on these parameters. For further consideration it is convenient for us to introduce other parameters. Every rotation can be represented by a vector issuing from the origin of coordinates, directed along the axis of rotation and of length equal to the angle of rotation. The projections of this vector on the coordinate axes will serve as our parameters. Then the parameter space is a sphere of radius \(\pi\), in which diametrically opposite points of the surface are identified. The product of two rotations \(A_\alpha(\alpha_x,\alpha_y,\alpha_z)\) and \(A_\beta(\beta_x,\beta_y,\beta_z)\) is the rotation \(A_\gamma=A_\beta A_\alpha\), where the parameters \(\gamma_s\), characterizing the element \(A_\gamma\), are single-valued functions of the parameters \(\alpha_s\) and \(\beta_s\): \(\gamma_s=f_s(\alpha,\beta)\).

We now introduce the so-called infinitesimal transformations of the group by the formula

\[ I_k=\left(\frac{\partial A_\alpha}{\partial \alpha_k}\right)_{\alpha_s=0}. \tag{7,1} \]

The symbol \(I_k\) denotes, obviously, a certain matrix of third order with numerical elements.

Let us carry out directly the computation of the matrices of infinitesimal transformations for the three-dimensional rotation group. In cho-

THE GROUP-THEORY METHOD IN THE QUANTUM PHYSICS OF SOLIDS

In calculating \(\hat I_k\), we may assume that \(\alpha_y=\alpha_z=0\). This means that we consider a rotation about the \(x\)-axis through an angle \(\alpha_x\), which leads to the transformation matrix

\[ \begin{pmatrix} 1 & 0 & 0\\ 0 & \cos\alpha_x & -\sin\alpha_x\\ 0 & \sin\alpha_x & \cos\alpha_x \end{pmatrix}. \tag{7,2} \]

Differentiating this matrix with respect to \(\alpha_x\) and setting \(\alpha_x=0\), we obtain:

\[ \hat I_x= \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & -1\\ 0 & 1 & 0 \end{pmatrix} \tag{7,3} \]

and analogously

\[ \hat I_y= \begin{pmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ -1 & 0 & 0 \end{pmatrix}, \qquad \hat I_z= \begin{pmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}. \tag{7,3a} \]

Elementary calculations lead to the three relations:

\[ \hat I_x\hat I_y-\hat I_y\hat I_x=\hat I_z,\qquad \hat I_y\hat I_z-\hat I_z\hat I_y=\hat I_x,\qquad \hat I_z\hat I_x-\hat I_x\hat I_z=\hat I_y. \tag{7,4} \]

Let the operations of the group now act on some three-dimensional vector

\[ \mathbf r'=A_\alpha \mathbf r. \tag{7,5} \]

We expand the right-hand side of (7,5) in a series in \(\alpha_s\) and restrict ourselves to first-order terms:

\[ A_\alpha \mathbf r= \left[ A_{\alpha_s=0} +\left(\frac{\partial A_\alpha}{\partial \alpha_x}\right)_{\alpha_s=0}\alpha_x +\left(\frac{\partial A_\alpha}{\partial \alpha_y}\right)_{\alpha_s=0}\alpha_y +\left(\frac{\partial A_\alpha}{\partial \alpha_z}\right)_{\alpha_s=0}\alpha_z \right]\mathbf r, \tag{7,6} \]

whence, taking into account definition (7,1), we obtain:

\[ \mathbf r'=\left[1+(\alpha_x\hat I_x+\alpha_y\hat I_y+\alpha_z\hat I_z)\right]\mathbf r. \tag{7,7} \]

Consequently, as a result of the indicated transformation the vector \(\mathbf r\) undergoes the following change:

\[ \delta \mathbf r=(\alpha_x\hat I_x+\alpha_y\hat I_y+\alpha_z\hat I_z)\mathbf r. \]

Each term on the right gives the change in \(\mathbf r\) under an infinitesimally small rotation about one of the coordinate axes. Thus, for example, we obtain the following changes of the components \((xyz)\) of the vector \(\mathbf r\) under a rotation through a small angle \(\alpha_x\) about the \(x\)-axis:

\[ \delta \mathbf r = \begin{pmatrix} \delta x\\ \delta y\\ \delta z \end{pmatrix} = \alpha_x \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & -1\\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} x\\ y\\ z \end{pmatrix}, \]

whence

\[ \delta x=0,\qquad \delta y=-\alpha_x z,\qquad \delta z=\alpha_x y. \tag{7,8} \]

From formula (7,7) it follows that an infinitesimally small rotation corresponds to the operator

\[ 1+\alpha_x \hat I_x+\alpha_y \hat I_y+\alpha_z \hat I_z . \tag{7,9} \]

Let now \(\psi(\mathbf r)\) be a function of the coordinates. Then, under a rotation about the \(x\)-axis through an angle \(\alpha_x\), its coordinates undergo the changes (7,8). To this rotation there corresponds the operator \(1+\alpha_x \hat I_x\), which transforms the function \(\psi(xyz)\) into \(\psi(x,\ y+\alpha_x z,\ z-\alpha_x y)\). Consequently, we have:

\[ (1+\alpha_x \hat I_x)\psi(\mathbf r) = \psi(\mathbf r) - \left( -\frac{\partial \psi}{\partial y}z + \frac{\partial \psi}{\partial z}y \right)\alpha_x, \]

whence

\[ \hat I_x = - \left( y\frac{\partial}{\partial z} - z\frac{\partial}{\partial y} \right) \tag{7,10} \]

and analogously

\[ \begin{aligned} \hat I_y &= - \left( z\frac{\partial}{\partial x} - x\frac{\partial}{\partial z} \right),\\ \hat I_z &= - \left( x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x} \right). \end{aligned} \tag{7,10a} \]

Formulas (7,10) in vector form may be written in the following way:

\[ \hat{\mathbf I}=-[\mathbf r \hat{\nabla}]. \tag{7,10б} \]

Let us now clarify in more detail the connection between what was said above about infinitesimal transformations and the representation of the rotation group. We shall denote the representation in a neighborhood of the identity transformation by matrices \(A(\alpha_x,\alpha_y,\alpha_z)\) of order \(n\), where the matrix elements are continuous and differentiable functions of the parameters \(\alpha_x,\alpha_y,\alpha_z\).

The whole problem reduces to finding infinitesimal transformations of the representation. Instead of the sought matrices \(I_x, I_y, I_z\), we introduce new matrices:

\[ \left. \begin{aligned} \hat L_x &= i\hat I_x,\quad \hat L_y=i\hat I_y,\quad \hat L_z=i\hat I_z,\\ \hat L_p &= \hat L_x+i\hat L_y,\quad \hat L_q=\hat L_x-i\hat L_y. \end{aligned} \right\} \tag{7,11} \]

The commutation relations will be

\[ \left. \begin{aligned} \hat L_z\hat L_p-\hat L_p\hat L_z&=\hat L_p,\\ \hat L_z\hat L_q-\hat L_q\hat L_z&=-\hat L_q,\\ \hat L_p\hat L_q-\hat L_q\hat L_p&=2\hat L_z. \end{aligned} \right\} \tag{7,12} \]

The representation by the matrices \(A(\alpha_x,\alpha_y,\alpha_z)\) must also include the representation of the Abelian subgroup of rotations about the axis \(z\), to whose elements correspond the matrices \(A(0,0,\alpha_z)\). For the vectors that will thereby play the role of a basis, the transformation will have the form

\[ A(0,0,\alpha_z)\mathbf v_m=e^{-im\alpha_z}\mathbf v_m. \tag{7,13} \]

Hence, on the basis of the definition of \(\hat I_z\), we obtain:

\[ \hat L_z\mathbf v_m=i\hat I_z\mathbf v_m =i\left(\frac{\partial}{\partial\alpha_z}e^{-im\alpha_z}\right)\mathbf v_m =m\mathbf v_m. \]

Consequently,

\[ \hat L_z\mathbf v_m=m\mathbf v_m, \tag{7,14} \]

i.e. \(\mathbf v_m\) is an eigenvector of the operator \(\hat L_z\), corresponding to the eigenvalue \(m\).

Lemma. If some vector \(\mathbf v_m\) is an eigenvector of the operator \(\hat L_z\), corresponding to the eigenvalue \(m\), then \(\hat L_p\mathbf v_m\) is also an eigenvector of \(\hat L_z\), corresponding to the eigenvalue \((m+1)\), and, analogously, \(\hat L_q\mathbf v_m\) is an eigenvector of \(\hat L_z\), corresponding to the eigenvalue \((m-1)\). By assumption,

\[ \hat L_z\mathbf v_m=m\mathbf v_m, \]

and by (7,12)

\[ \begin{aligned} \hat L_z\bigl(\hat L_p\mathbf v_m\bigr) &=\bigl(\hat L_p\hat L_z+\hat L_p\bigr)\mathbf v_m =\hat L_p\bigl(\hat L_z\mathbf v_m\bigr)+\hat L_p\mathbf v_m \\ &=\hat L_p(m\mathbf v_m)+\hat L_p\mathbf v_m =(m+1)\bigl(\hat L_p\mathbf v_m\bigr). \end{aligned} \tag{7,15} \]

and similarly for \(\hat L_q\),

\[ \hat L_z\left(\hat L_q \mathbf v_m\right)=(m-1)\left(\hat L_q \mathbf v_m\right). \tag{7.16} \]

The number of distinct eigenvalues for \(\hat L_z\) is not greater than \(n\). Denote the eigenvalue with the largest real part by \(j\), and let \(\mathbf v_j\) be the corresponding eigenvector. Then, according to the lemma, we shall have:

\[ \hat L_z\left(\hat L_p \mathbf v_j\right)=(j+1)\left(\hat L_p \mathbf v_j\right), \tag{7.17} \]

but \(\hat L_z\) has no such eigenvalue, and, consequently,

\[ \hat L_p \mathbf v_j=0. \tag{7.18} \]

By virtue of the lemma proved above, the vectors

\[ \mathbf v_{j-1}=\hat L_q \mathbf v_j,\quad \mathbf v_{j-2}=\hat L_q \mathbf v_{j-1},\ldots, \tag{7.19} \]

if they are nonzero, correspond to the eigenvalues \((j-1),(j-2),\ldots\) of the operator \(\hat L_z\). The sequence of vectors must, of course, lead to the zero vector, since the number of distinct eigenvalues of \(\hat L_z\) is not greater than \(n\). Let us show that for \(m=j,j-1,\ldots\) the formula

\[ \hat L_p \mathbf v_m=\rho_m \mathbf v_{m+1}, \tag{7.20} \]

holds, where \(\rho_m\) are integers. By (7.18) it is true for \(m=j\), with \(\rho_j=0\). Suppose now that the formula is true for some \(m=\mu\), and prove it for \(m=\mu-1\). According to (7.17), (7.19), and (7.20), we have:

\[ \begin{aligned} \hat L_p \mathbf v_{\mu-1} &=\hat L_p \hat L_q \mathbf v_\mu =\left(\hat L_q \hat L_p+2\hat L_z\right)\mathbf v_\mu =\hat L_q\left(\hat L_p \mathbf v_\mu\right)+2\hat L_z\mathbf v_\mu \\ &=\hat L_q\rho_\mu\mathbf v_{\mu+1}+2\mu\mathbf v_\mu =\left(\rho_\mu+2\mu\right)\mathbf v_\mu . \end{aligned} \tag{7.21} \]

Note that for \(\mu=j\) we do not use here the formula \(\hat L_q \mathbf v_{\mu+1}=\mathbf v_\mu\), since \(\rho_\mu=0\) for \(\mu=j\). This proves relation (7.20). According to (7.20),

\[ \hat L_p \mathbf v_{\mu-1}=\rho_{\mu-1}\mathbf v_\mu, \]

and, taking (7.21) into account, we have \(\rho_{\mu-1}=\rho_\mu+2\mu\), whence, performing successive calculations, we obtain:

\[ \rho_\mu=j(j+1)-\mu(\mu-1). \]

Then

\[ \hat L_p \mathbf v_m = [j(j+1)-m(m+1)]\mathbf v_{m+1};\quad m=j,\quad j-1,\ldots \tag{7,22} \]

Using this equality, let us determine the index \(s\) of the first of the vectors (7,19) equal to zero, i.e. \(\mathbf v_s=0\), but \(\mathbf v_{s+1}\ne 0\). Then from (7,22) it follows that \(\rho_s=0\), i.e.

\[ j(j+1)=s(s-1). \]

Solving the quadratic equation, we obtain:

\[ s=j, \]

\[ s=-(j+1). \]

The value \(s=j\) must be rejected, since the vector \(\mathbf v_j\ne 0\). Consequently, we have the sequence

\[ \mathbf v_j,\quad \mathbf v_{j-1},\ldots,\mathbf v_{-j+1},\quad \mathbf v_{-j} \tag{7,23} \]

and the number of vectors is \(2j+1\). Hence it is clear that \(j\) is either an integer or a half-integer.

If \(2j+1=n\), then the vectors (7,23) may be taken as unit vectors. This representation will be irreducible. In the case \(n>2j+1\) we would obtain that the representation \(A^{(n)}\) is reducible. But there exists only one, up to equivalence, irreducible representation of a given dimension. We have already constructed them with the aid of Euler angles. Consequently, representations of this kind exhaust all representations of the rotation group of the given dimension.

The vectors in relations (7,19) and (7,22) may be multiplied by arbitrary factors. The latter may be chosen so that the following final relations hold:

\[ \left. \begin{aligned} \hat L_p \mathbf v_m &= \sqrt{j(j+1)-m(m+1)}\,\mathbf v_{m+1},\\ \hat L_q \mathbf v_m &= \sqrt{j(j+1)-m(m-1)}\,\mathbf v_{m-1},\\ \hat L_z \mathbf v_m &= m\mathbf v_m. \end{aligned} \right\} \tag{7,24} \]

The subspace \(\mathbf v_j,\mathbf v_{j-1},\ldots,\mathbf v_{-j}\) of our vector space is transformed into itself by the operations \(\hat L_p,\hat L_q,\hat L_z\), and consequently also by the infinitesimal rotations \(\hat I_x,\hat I_y,\hat I_z\). Hence it follows that this subspace is transformed into itself also by all transformations of the rotation group, i.e. the vectors \(\mathbf v_j\ldots \mathbf v_{-j}\) define an invariant subspace \(\mathfrak R_{2j+1}\). The transformations of this sub-

spaces form a representation of the rotation group, completely determined by equations (7.24). In the space \(\mathfrak{R}_{2j+1}\) the operator \(\hat L_z\) has simple eigenvalues \(m=j,\, j-1,\ldots,-j\) with eigenvectors \(\mathbf v_m\). Let us also note that all vectors of the space \(\mathfrak{R}_{2j+1}\) are eigenvectors of the operator

\[ \hat L^2=\hat L_x^2+\hat L_y^2+\hat L_z^2 =\frac{1}{2}\left(\hat L_p\hat L_q+\hat L_q\hat L_p\right)+\hat L_z^2. \tag{7.25} \]

From (7.24), after simple computations, we obtain:

\[ \hat L^2\mathbf v_m=j(j+1)\mathbf v_m. \tag{7.26} \]

The representation of degree \(2j+1\) defined by formulas (7.24) is equivalent to the representation found in § 6 and denoted by \(D_j\). Indeed, in the space \(\ldots \xi^{\,n-r}\eta^r\ldots\) of the representation \(D_j\), the basis vectors \(\xi^{\,n-r}\eta^r\), under the rotation \((0,0,a_z)\), are multiplied by

\[ e^{-ima_z}=e^{-\frac{1}{2}i(n-2r)a_z}, \]

and consequently the values

\[ m=\frac{n}{2}-r \qquad \left(=\frac{n}{2},\ \frac{n}{2}-1,\ldots,-\frac{n}{2}\right) \]

occur once each. If now in this space one constructs the subspace \(\mathbf v_m\) of \(2j+1\) dimensions of the structure described above, then it coincides with the whole space (since both have the same number of dimensions). The quantities \(\mathbf v_m\) of the subspace of \((2j+1)\) dimensions must coincide with the products \(\xi^{\,j+m}\eta^{\,j-m}\) of the representation \(D_j\), up to a numerical factor. Introducing the numerical factor, we obtain:

\[ \mathbf v_m=\frac{\xi^{\,j+m}\eta^{\,j-m}}{\sqrt{(j+m)!(j-m)!}}. \tag{7.27} \]

These \(\mathbf v\) form at the same time, according to (6.17), a normalized orthogonal system.

In an analogous way one can prove that, when \(j\) is equal to an integer \(l\), the representation \(D_l\), defined by formula (7.24), coincides with the representation expressed in terms of spherical functions of order \(l\), \(Y_l^m\). Indeed, the number of the latter is \(2l+1\), and therefore the largest eigenvalue of the operator \(\hat L_z\) is the value \(m=l\). Consequently, the spherical functions \(Y_l^m\) transform according to the irreducible representation \(D_l\), i.e. we can choose the normalizing

multiplier of the spherical functions \(Y_l^m\) in such a way that the relations (7.24) are exactly satisfied for them. Hence it also follows that \(D_l\) is a single-valued representation.

8. THE OCTAHEDRON GROUP \(^{4,5,9}\)

Let us carry out a detailed study of the group \(\mathfrak{D}_h\). Such a consideration is intended partly to illustrate the general propositions set forth in §§ 3, 4, and 5, and partly is needed for the subsequent exposition, since the general methods of investigating quantum-mechanical systems by means of group theory are explained below precisely on the example of a system with cubic symmetry. We begin with a description of the group \(\mathfrak{D}\) (the octahedron group). The system of axes of this group is the system of axes of symmetry of the cube: three fourth-order axes pass through the centers of opposite faces, four third-order axes pass through opposite vertices, and six second-order axes pass through the midpoints of opposite edges (Fig. 4).

Before carrying out the division of the elements of the group \(\mathfrak{D}\) into classes of conjugate elements, we shall make several preliminary remarks. It was already indicated in § 3 that two symmetry operations will be conjugate if they are of the same order and if the corresponding symmetry elements are equivalent. An additional remark is required in the case when rotations about one and the same axis are conjugate.

Let \(A\) be a rotation about some axis \(a\), and let \(A^{-1}\) be the inverse transformation. It is obvious that a direct and an inverse transformation always have one and the same order. If now, among the elements of the group, there is a rotation through \(\pi\) about an axis perpendicular to the given one, then the operations \(A\) and \(A^{-1}\), according to the general rule (see § 3), turn out to be conjugate, since such a rotation brings the initial position of the axis \(a\) into the opposite one.

Reflection in the plane \(\sigma_h\), perpendicular to the axis \(a\), also changes the direction of the axis, but at the same time changes the direction of rotation. Consequently, the presence of \(\sigma_h\) does not make the elements \(A\) and \(A^{-1}\) conjugate. Reflection in the plane \(\sigma_v\), passing through the axis \(a\), does not change the direction of the axis, but changes the direction of rotation. Therefore, in the presence of \(\sigma_v\), the elements \(A\) and \(A^{-1}\) will be conjugate.

If rotations about some axis through one and the same angle in opposite directions are conjugate, then the axis is called two-sided.

It can be shown that in the group \(\mathfrak{O}\) all axes of the same order are equivalent and each of them is two-sided. Therefore the 24 elements of the group are distributed among the following five classes:

\[ \begin{aligned} K_1&\text{ — consisting of the identity transformation;}\\ K_2&\text{ — consisting of three rotations through }\pi\text{ about the coordinate axes;}\\ K_3&\text{ — consisting of three rotations through }\pi/2\text{ and three rotations}\\ &\qquad\text{through }3\pi/2\text{ about the coordinate axes;}\\ K_4&\text{ — consisting of six rotations through }\pi\text{ about axes passing}\\ &\qquad\text{through the midpoints of opposite edges;}\\ K_5&\text{ — consisting of four rotations through }2\pi/3\text{ and four}\\ &\qquad\text{rotations through }4\pi/3\text{ about the space diagonals of the cube.} \end{aligned} \]

Consequently, the octahedral group has five irreducible representations. The sum of the squares of the dimensions of all the representations must be equal to the number of elements of the group. The only decomposition of the number 24 into a sum of five squares is

\[ 3^2+3^2+2^2+1^2+1^2, \]

i.e., the octahedral group has two three-dimensional representations, one two-dimensional representation, and two one-dimensional representations. Let us find the class multiplication table:

\[ \begin{aligned} K_2K_2&=3K_1+2K_2, & K_3K_3&=K_4K_4=6K_1+2K_2+3K_5,\\ K_2K_3&=K_3+2K_4, & K_3K_4&=4K_2+3K_5,\\ K_2K_4&=2K_3+K_4, & K_3K_5&=K_4K_5=4K_3+4K_4,\\ K_2K_5&=3K_5, & K_5K_5&=8K_1+8K_2+4K_5. \end{aligned} \]

Taking these relations into account, we compose equations of the form (5.5):

\[ \left| \begin{array}{ccccc} -x_2 & 1 & 0 & 0 & 0\\ 3 & 2-x_2 & 0 & 0 & 0\\ 0 & 0 & 1-x_2 & 2 & 0\\ 0 & 0 & 2 & 1-x_2 & 0\\ 0 & 0 & 0 & 0 & 3-x_2 \end{array} \right|=0, \]

\[ \left| \begin{array}{ccccc} -x_3 & 0 & 1 & 0 & 0\\ 0 & -x_3 & 1 & 2 & 0\\ 6 & 2 & -x_3 & 0 & 3\\ 0 & 4 & 0 & -x_3 & 3\\ 0 & 0 & 4 & 4 & -x_3 \end{array} \right|=0, \]

$$ \left| \begin{array}{ccccc} -x_4 & 0 & 0 & 1 & 0\\ 0 & -x_4 & 2 & 1 & 0\\ 0 & 4 & -x_4 & 0 & 3\\ 6 & 2 & 0 & -x_4 & 3\\ 0 & 0 & 4 & 4 & -x_4 \end{array} \right|=0, $$

$$ \left| \begin{array}{ccccc} -x_5 & 0 & 0 & 0 & 1\\ 0 & -x_5 & 0 & 0 & 3\\ 0 & 0 & 4-x_5 & 4 & 0\\ 0 & 0 & 4 & 4-x_5 & 0\\ 8 & 8 & 0 & 0 & 4-x_5 \end{array} \right|=0. $$

Solving them, we obtain:

$$ \begin{aligned} x_2&=3,\ 3,\ 3,\ -1,\ -1, & x_3&=6,\ -6,\ 0,\ 2,\ -2,\\ x_4&=6,\ -6,\ 0,\ -2,\ 2, & x_5&=8,\ 8,\ -4,\ 0,\ 0 \end{aligned} $$

and, finally, according to the general theory, we find the following table of characters:

Table 1

0	$K_1$	$K_2$	$K_3$	$K_4$	$K_5$
$\Gamma_1$	1	1	1	1	1
$\Gamma_2$	1	1	−1	−1	1
$\Gamma_3$	2	2	0	0	−1
$\Gamma_4$	3	−1	−1	1	0
$\Gamma_5$	3	−1	1	−1	0

Without presenting the long and tedious calculations, let us write down explicitly the matrices of the two-dimensional and three-dimensional representations of our group:

$$ \Gamma_3 \qquad \left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right),\quad \left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right),\quad \left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right),\quad \left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right); $$

$$ \left(\begin{array}{cc} 1 & 1\\ 0 & -1 \end{array}\right) \left(\begin{array}{cc} 1 & 1\\ 0 & -1 \end{array}\right) \left(\begin{array}{cc} 0 & -1\\ -1 & 0 \end{array}\right),\quad \left(\begin{array}{cc} 0 & -1\\ -1 & 0 \end{array}\right),\quad \left(\begin{array}{cc} -1 & 0\\ 1 & 1 \end{array}\right),\quad \left(\begin{array}{cc} -1 & 0\\ 1 & 1 \end{array}\right); $$

$$ \left(\begin{array}{cc} -1 & 0\\ 1 & 1 \end{array}\right),\quad \left(\begin{array}{cc} -1 & 0\\ 1 & 1 \end{array}\right),\quad \left(\begin{array}{cc} 1 & 1\\ 0 & -1 \end{array}\right),\quad \left(\begin{array}{cc} 1 & 1\\ 0 & -1 \end{array}\right), $$

$$ \left(\begin{array}{cc} 0 & -1\\ -1 & 0 \end{array}\right) \left(\begin{array}{cc} 0 & -1\\ -1 & 0 \end{array}\right); $$

\[ \begin{pmatrix} 0 & 1\\ -1 & -1 \end{pmatrix}, \quad \begin{pmatrix} -1 & -1\\ 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} -1 & -1\\ 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 1\\ -1 & -1 \end{pmatrix}; \]

\[ \begin{pmatrix} 0 & 1\\ -1 & -1 \end{pmatrix}, \quad \begin{pmatrix} -1 & -1\\ 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} -1 & -1\\ 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 1\\ -1 & -1 \end{pmatrix}. \]

\[ \Gamma_4 \quad \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}, \quad \begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}, \quad \begin{pmatrix} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1 \end{pmatrix}, \quad \begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1 \end{pmatrix}; \]

\[ \begin{pmatrix} -1 & 0 & 0\\ 0 & 0 & -1\\ 0 & 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} -1 & 0 & 0\\ 0 & 0 & 1\\ 0 & -1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & -1 \end{pmatrix}, \]

\[ \begin{pmatrix} 0 & 1 & 0\\ -1 & 0 & 0\\ 0 & 0 & -1 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & -1\\ 0 & -1 & 0\\ 1 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & 1\\ 0 & -1 & 0\\ -1 & 0 & 1 \end{pmatrix}; \]

\[ \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & -1\\ 0 & -1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix}, \]

\[ \begin{pmatrix} 0 & -1 & 0\\ -1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & -1\\ 0 & 1 & 0\\ -1 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0 \end{pmatrix}; \]

\[ \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & -1\\ 1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & -1\\ -1 & 0 & 0\\ 0 & 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}; \]

\[ \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & -1\\ -1 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & -1 & 0\\ 0 & 0 & 1\\ -1 & 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & -1 & 0\\ 0 & 0 & -1\\ 1 & 0 & 0 \end{pmatrix}. \]

\[ \Gamma_5 \quad \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}, \quad \begin{pmatrix} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1 \end{pmatrix}, \quad \begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1 \end{pmatrix}, \quad \begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}; \]

$$ \begin{pmatrix} 0&0&1\\ 0&1&0\\ -1&0&0 \end{pmatrix}, \quad \begin{pmatrix} 0&0&-1\\ 0&1&0\\ 1&0&0 \end{pmatrix}, \quad \begin{pmatrix} 0&1&0\\ -1&0&0\\ 0&0&1 \end{pmatrix}, $$

$$ \begin{pmatrix} 0&-1&0\\ 1&0&0\\ 0&0&1 \end{pmatrix}, \quad \begin{pmatrix} 1&0&0\\ 0&0&1\\ 0&-1&0 \end{pmatrix}, \quad \begin{pmatrix} 1&0&0\\ 0&0&-1\\ 0&1&0 \end{pmatrix}; $$

$$ \begin{pmatrix} 0&1&0\\ 1&0&0\\ 0&0&-1 \end{pmatrix}, \quad \begin{pmatrix} 0&-1&0\\ -1&0&0\\ 0&0&-1 \end{pmatrix}, \quad \begin{pmatrix} -1&0&0\\ 0&0&-1\\ 0&-1&0 \end{pmatrix}, $$

$$ \begin{pmatrix} -1&0&0\\ 0&0&1\\ 0&1&0 \end{pmatrix}, \quad \begin{pmatrix} 0&0&1\\ 0&-1&0\\ 1&0&0 \end{pmatrix}, \quad \begin{pmatrix} 0&0&-1\\ 0&-1&0\\ -1&0&0 \end{pmatrix}; $$

$$ \begin{pmatrix} 0&0&-1\\ 1&0&0\\ 0&-1&0 \end{pmatrix}, \quad \begin{pmatrix} 0&0&1\\ 1&0&0\\ 0&1&0 \end{pmatrix}, \quad \begin{pmatrix} 0&0&-1\\ -1&0&0\\ 0&1&0 \end{pmatrix}, \quad \begin{pmatrix} 0&0&1\\ -1&0&0\\ 0&-1&0 \end{pmatrix}; $$

$$ \begin{pmatrix} 0&1&0\\ 0&0&-1\\ -1&0&0 \end{pmatrix}, \quad \begin{pmatrix} 0&1&0\\ 0&0&1\\ 1&0&0 \end{pmatrix}, \quad \begin{pmatrix} 0&-1&0\\ 0&0&1\\ -1&0&0 \end{pmatrix}, \quad \begin{pmatrix} 0&-1&0\\ 0&0&-1\\ 1&0&0 \end{pmatrix}. $$

In physical problems one most often has to deal with the group $\mathfrak{D}_h$. This group is obtained from the group $\mathfrak{D}$ by adding the center of inversion. It can be represented as the direct product of the group $\mathfrak{D}$ and the group $\mathfrak{C}$, consisting of only two elements: the identity $E$ and the inversion $I$. Therefore its representations and character table can be found by using the rule stated in § 5.

9. APPLICATION OF GROUP THEORY IN QUANTUM MECHANICS

The motion of any physical system takes place in a certain force field. The field is the material carrier of those forces which act on the physical system. The presence of precisely this force field determines the symmetry properties of the space in which the system moves. This is why the study of the symmetry properties of force fields acting on a physical system must precede the study of the system itself, since its behavior is determined precisely by these fields.

A general and well-developed apparatus that makes it possible to study the properties of physical systems that follow from symmetry is group theory. The application of group theory in quantum mechanics is based on the fact that the Schrödinger equation, which describes the state of a system (atom, molecule, crystal, etc.), remains invariant under the following symmetry transformations of the given system:

1) under permutations of the coordinates of different particles that play the same role in the system;

2) under translations, rotations, and reflections of space that do not change the force field acting in the system.

If the nucleus in an atom is regarded as a fixed center of forces, then one is dealing with rotations about this center and inversion with respect to it. For an atom in a homogeneous field, the rotation group is replaced by the subgroup of rotations about a fixed axis directed along the field, and by reflections in planes passing through this axis. In a metal, in the crystalline state, the wave equation remains invariant with respect to the operations of the space group, etc.

Along with this, one must also take into account the invariance of the Schrödinger equation (in the absence of a magnetic field) with respect to a change in the sign of time. The Schrödinger equation

\[ \hat H\left(x,-i\hbar\frac{\partial}{\partial x},t\right)\psi(x,t) = i\hbar\frac{\partial}{\partial t}\psi(x,t) \tag{9,1} \]

under the replacement of \(t\) by \(-t\) becomes

\[ \hat H\left(x,-i\hbar\frac{\partial}{\partial x},-t\right)\psi(x,-t) = -\,i\hbar\frac{\partial}{\partial t}\psi(x,-t). \tag{9,2} \]

We now form its complex conjugate,

\[ \hat H^{*}\left(x,-i\hbar\frac{\partial}{\partial x},-t\right)\psi^{*}(x,-t) = i\hbar\frac{\partial}{\partial t}\psi^{*}(x,-t). \tag{9,3} \]

Next we require that the relation

\[ \hat H^{*}\left(x,-i\hbar\frac{\partial}{\partial x},-t\right) \equiv \hat H\left(x,-i\hbar\frac{\partial}{\partial x},t\right) \tag{9,4} \]

hold. The latter requirement expresses only the fact that the energy operator is real and remains unchanged when the sign of time is changed. Then equalities (9,1) and (9,3) show that the functions \(\psi(x,t)\) and \(\psi^{*}(x,-t)\) satisfy one and the same equation.

If we restrict ourselves to the consideration of stationary states, i.e., assume the energy operator not to depend explicitly on time, then ...

the eigenfunctions in this case will be

\[ \psi(x,t)=\psi(x)e^{-\frac{iE}{\hbar}t} \tag{9,5} \]

and the Schrödinger equation will take the form

\[ \hat H\left(x,-i\hbar\frac{\partial}{\partial x}\right)\psi(x)=E\psi(x). \tag{9,6} \]

In this case our last assertion reduces to the statement that, in an electric field, to every energy level, together with the function \(\psi(x)\), there also belongs \(\psi^*(x)\). From this it follows immediately that the eigenfunctions of nondegenerate states automatically turn out to be real, while for degenerate states they can always be reduced to real ones by choosing the corresponding linear combinations.

The operation of forming the complex conjugate expression may be regarded as an operation that takes the wave function used by one observer to describe a certain state into the wave function used to describe the very same physical state by an observer in a coordinate system with the opposite direction of the time axis.

Symmetry transformations that leave the wave equation invariant form, in each case, a group. The operations corresponding to the elements of these groups at the same time give transformations of the wave functions \(\psi(x)\), if it is assumed that every transformation \(A\) that carries the coordinate system \(x\) into \(x'\) transforms the function \(\psi\) into \(\psi'\), with

\[ \psi'(x')=\psi(x). \tag{9,7} \]

Let us show that if \(\psi_k\) is an eigenfunction of the operator \(\hat\Omega\) and belongs to the eigenvalue \(\omega_k\), then the function \(A\psi_k\), where \(A\) is an arbitrary transformation of the symmetry group of the operator \(\hat\Omega\), will also be an eigenfunction of \(\hat\Omega\), corresponding to the same eigenvalue \(\omega_k\). Indeed, we may write:

\[ \hat\Omega\psi_k=\omega_k\psi_k. \tag{9,8} \]

Subjecting both sides of equation (9,8) to the transformation \(A\), we obtain:

\[ A\hat\Omega\psi_k=\hat\Omega A\psi_k=\omega_k A\psi_k. \tag{9,9} \]

It follows that the function \(A\psi_k\) is also a solution of equation (9,8) with eigenvalue \(\omega_k\).

Invariance of the operator $\hat{\Omega}$, i.e., its “insensitivity” to operations of the symmetry group, can be expressed symbolically as follows:

\[ A\hat{\Omega}=\hat{\Omega}A. \tag{9,10} \]

If we now apply the assertion proved above to the Hamiltonian operator $\hat{H}$, we obtain the well-known Wigner theorem. If $\psi(x)$ is an eigenfunction of the energy operator $\hat{H}$ and corresponds to the eigenvalue $E$, and if $\hat{H}$ remains invariant under the action of the symmetry operation $A$, then $A\psi(x)$ will likewise be an eigenfunction of $\hat{H}$ corresponding to the same eigenvalue $E$.

Under a transformation of the coordinate system corresponding to an element of the group $\mathfrak{G}$, the wave function $\psi$ passes into some other function. Performing successively all $g$ transformations of the group ($g$ is the order of the group), we obtain from $\psi$ $g$ different functions. However, some of these functions may turn out to be linearly dependent. Therefore we obtain a system of $f$ ($f \leqslant g$) linearly independent functions $\psi_1,\psi_2,\ldots,\psi_f$, which under the symmetry transformations belonging to the group under consideration pass into the system $\psi'_1,\psi'_2,\ldots,\psi'_f$. Under such a transformation $A$ each new $\psi'_i$ is a linear combination of the functions $\psi_1,\psi_2,\ldots,\psi_f$, i.e.,

\[ \psi'_i=\sum_{k=1}^{f} a_{ik}\psi_k, \tag{9,11} \]

where $a_{ik}$ are constants depending on the transformation $A$. The system (9,11) can be written explicitly in the form

\[ \begin{pmatrix} \psi'_1\\ \cdot\\ \cdot\\ \cdot\\ \psi'_f \end{pmatrix} = \begin{pmatrix} a_{11} & \cdots & a_{1f}\\ \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot\\ a_{f1} & & a_{ff} \end{pmatrix} \begin{pmatrix} \psi_1\\ \cdot\\ \cdot\\ \cdot\\ \psi_f \end{pmatrix}, \tag{9,12} \]

where $(a_{ik})$ is the transformation matrix corresponding to a certain element of the symmetry group. Such a notation means that the elements of the group may be regarded as operators acting on the functions $\psi$, the set of which is regarded as a basis of a representation of the group. The set of functions is assumed to be orthonormalized. It follows from this that the concept of the transformation matrix of a group coincides with the concept of the matrix of an operator in the form in which it is usually employed.

are used in quantum mechanics, namely:

\[ a_{ik}=\int \psi_k^* A\psi_i\,d\tau . \tag{9,13} \]

It follows directly from Wigner’s theorem that, under symmetry transformations, the wave functions of stationary states of a system belonging to one and the same energy level are transformed into one another, i.e., they realize some representation of the group. What is essential is that this representation is irreducible.

Thus one may conclude that to each energy level of a system there corresponds a certain irreducible representation of its symmetry group. The dimensionality of this representation determines the degree of degeneracy of the given level, i.e., the number of different states with this energy. It should be emphasized that degeneracy is connected with the presence of certain symmetry groups which leave invariant the wave equation of quantum mechanics, and the removal of degeneracy, even partial, is connected with a lowering of symmetry*).

By specifying an irreducible representation, all the symmetry properties of the given state are determined, i.e., its behavior with respect to various symmetry transformations. Thus, by establishing and classifying the various possible representations of the group under consideration, we thereby obtain a classification of the energy eigenvalues and eigenfunctions of the system (atom, molecule, crystal). This is the basis of the group-theoretical systematics of terms.

10. CONSERVATION LAWS AND A COMPLETE SET OF PHYSICAL QUANTITIES¹⁰

It is well known that, both in classical and in quantum mechanics, for a physical system moving under certain definite conditions, the conservation laws of energy, momentum, and angular momentum are valid. It should be emphasized that the very existence and applicability of these laws are due to the properties of space-time in which the system moves, more precisely to the fact that the space-time continuum admits certain continuous groups of transformations. For systems moving in complicated force fields these laws cease to be valid, but it is natural to expect that the symmetry of the field will make it possible to determine new characteristics of the system that are conserved during the motion of the system in the given field.

*) We do not consider accidental degeneracy, which cannot be attributed either to the symmetry of the system or to the materiality of the Hamiltonian.

An integral of motion in quantum mechanics is any operator \(\hat A\) that has the property

\[ \hat A \hat H = \hat H \hat A, \tag{10,1} \]

i.e., an operator whose eigenvalue can be measured simultaneously with the value of the energy. Consequently, an integral of motion is any operator of the symmetry group that leaves the Hamiltonian operator invariant. It is clear that if we choose the quantities

\[ a_1,\ a_2,\ldots,\ a_n \tag{10,2} \]

of the eigenvalues of the operators

\[ \hat A_1,\ \hat A_2,\ldots,\ \hat A_n, \tag{10,3} \]

corresponding to a system of generating elements of the symmetry group,

\[ G=\{A_1,\ldots,\ A_m\}, \tag{10,4} \]

then all independent conserved physical quantities will be obtained. In other words, the conservation laws will be obtained for a physical system moving in a field of the given symmetry.

Whatever quantum-mechanical system we may be dealing with, if only an integral of motion \(\hat A\) has been found, then it is known that if in some state of motion the operator initially had eigenvalue \(a\), then it will always have the same eigenvalue thereafter, so that one can put different eigenvalues of \(\hat A\) in correspondence with different states and in this way obtain the classification of states that we need. Such a classification is not so direct when there are several integrals of motion (10,3) that do not commute with one another, since in this case there are no states in which all (10,2) would be simultaneously measurable. The existence of mutually noncommuting integrals of motion is a sign that the state is degenerate. Indeed, in the absence of degeneracy the Hamiltonian function by itself forms a complete set and therefore each integral of motion \(\hat A_i\), since it commutes with \(\hat H\), is a function of \(\hat H\) and, consequently, commutes with all the other \(\hat A_j\).

Passing to the degenerate case, in which the integrals of motion do not commute with one another, we must find such a function of these integrals of motion that would have one and the same value in all states corresponding to a definite energy level \(E\), so that with their aid it would be possible to classif-

fix the energy levels of the system. It is not difficult to see that this function must depend on $\hat H$ and therefore must also commute with every dynamical variable that commutes with $\hat H$, i.e., with every integral of motion. In other words, the problem reduces to finding such a function of the integrals of motion that would commute with all $\hat A_j$. If several such functions can be found, then they must all commute with one another, so that numerical values can be assigned to all of them simultaneously and thus a complete classification of the levels of the system can be obtained.

It is not difficult to see that such functions are the classes of conjugate elements of the group leaving the Hamiltonian operator invariant. Indeed, the classes $K_i$ of any group $\mathfrak G$ commute with one another, and each of them commutes with any element of the group (which is an integral of motion), i.e., the relations hold:
\[ K_iK_j=K_jK_i, \tag{10,5} \]
\[ A_jK_i=K_iA_j. \tag{10,6} \]

Since the classes of conjugate elements are mutually commutative, and also commute with the Hamiltonian of the system, the physical quantities corresponding to the operators $\hat K_i$ form a set of simultaneously measurable quantities. However, since relations of the type (3,5) exist between the classes of conjugate elements:
\[ K_iK_j=\sum_{k=1}^{r} c_{ijk}K_k, \]
the operators $\hat K_i$ are not all independent. From the complete set of classes one can select a system of independent classes that contain the generating elements of the group (10,4). It should be noted that, since the system of generating elements is not chosen uniquely, it may happen that different choices of the system of generators (10,4) correspond to different numbers of classes. It is convenient to choose the system of generating elements (10,4) so that the number of classes corresponding to it is minimal. Let these classes be
\[ K_1,\ K_2,\ldots,K_p \tag{10,7} \]
with eigenvalues
\[ \varkappa_1,\ \varkappa_2,\ldots,\varkappa_p. \tag{10,8} \]

These eigenvalues are certain functions of the eigenvalues (10,2) of the operators (10,3). Their numerical

the quantities are found from the formula

\[ x_j^{(i)}=\frac{h_j X_j^{(i)}}{n_i}, \tag{10,9} \]

where \(h_j\) is the number of elements in the class, \(n_i\) is the dimensionality of the representation, and \(X_j^{(i)}\) is the character of the \(j\)-th class in the \(i\)-th representation. Let us also note that the dimensionality of the representation is nothing other than the character of the identity element \(X^{(i)}(E)\).

The identity element by itself constitutes a class, to which there must correspond a certain conserved quantity, since one always has

\[ \hat E \hat H=\hat H \hat E . \]

This relation therefore expresses the “law of conservation of the degeneracy of an expression”: if some energy level of a closed system possesses a definite degeneracy of expression, then this degeneracy is preserved in time. Recalling that different representations correspond to different energy levels and, generally speaking, not one-to-one, since the energy depends essentially on the form of the potential energy and not only on the symmetry properties of the potential, one may write:

\[ E(\nu,\ x_1,\ldots,x_p). \tag{10,10} \]

By assigning the quantities

\[ \nu,\ x_1,\ldots,x_p \tag{10,11} \]

the state of the system is still not completely determined, since, according to the theorem of § 9, together with any function \(\psi(x)\) the function \(A\psi(x)\) will also belong to the set (10,11), and consequently degeneracy will occur. The degeneracy of the expression is determined, according to general group-theoretical considerations, by the dimensionality of the representation according to which the functions of the given energy level transform. If part of the generating elements of the group commute with one another, then the eigenvalues of the corresponding operators may be found simultaneously with the set (10,11). Let their eigenvalues be

\[ \alpha_1,\ \alpha_2,\ldots,\alpha_q. \tag{10,12} \]

Then the state of the system is completely described by the set of quantities

\[ \nu,\ x_1,\ldots,x_p,\ \alpha_1,\ldots,\alpha_q, \tag{10,13} \]

which may be called the complete set of physical quantities necessary for describing the quantum-mechanical system under consideration. Thus, the state of the system will be described by a wave function of the form

\[ \psi=\psi_{\nu,\ x_i,\ a_j}(x). \tag{10,14} \]

Already here one can make the following remarks.

1. Every symmetry transformation may be regarded as a transformation taking the function \(\psi_{\nu, x_i, a_j}(x)\) into some function \(\psi'_{\nu, x_i, a_j}(x)\), or as a transformation of the set (10.12) into some other set. It follows from this that the complete removal of degeneracy will be effected under a symmetry that leaves the set (10.12) invariant. The presence of accidental degeneracy is dictated by the fact that two values of the set (10.12) are equivalent to one another for reasons due neither to the symmetry of the system nor to the reality of the Hamiltonian.

2. The method indicated by us for finding conserved quantities and a complete set makes it possible at once to indicate which of the quantities found will not lose their meaning when the symmetry is changed and, consequently, will be suitable for describing the new system.

3. In addition, such a consideration makes it possible to carry out a relative division of all the characteristics of the system into external and internal ones. The external characteristics describe the motion of the entire system as a whole, whereas the internal ones are due to the interaction of the separate parts of the system.

11. THE HYDROGEN ATOM PROBLEM

The Schrödinger equation describing the motion of an electron in a central field can be written in the form

\[ \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right) + \frac{1}{r^2}\left[ \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\,\frac{\partial\psi}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2\psi}{\partial\varphi^2} \right] + \frac{2m}{\hbar^2}\,[E - V(r)]\psi=0 . \tag{11.1} \]

We shall be interested in the solutions of this equation corresponding to the discrete branch of the spectrum of eigenvalues. In this case the solutions of (11.1) have the form

\[ \psi_{n,l,m_l}(r,\theta,\varphi)=R_{n,l}(r)\,Y_l^{m_l}(\theta,\varphi), \tag{11.2} \]

and the eigenvalues will be

\[ E=E(n,l), \tag{11.3} \]

where \(n\) is the principal quantum number, determined by the form of the potential energy; \(l\) is the quantum number of the square of the angular momentum; \(m_l\) is the quantum number of the projection of the angular momentum on the \(z\)-axis.

The symmetry group of the Schrödinger equation with a centrally symmetric potential is the group of complete spherical symmetry \(\mathfrak{K}_h\). This group contains rotations about any axis passing-

passing through the center, by any angle, and reflections in any plane passing through the same point. It contains, as a subgroup, the group \(\mathfrak{K}\) of all spatial rotations. The group \(\mathfrak{K}_h\) can be obtained from the group \(\mathfrak{K}\) by adding the center of symmetry

\[ \mathfrak{K}_h=\mathfrak{K}\times \mathfrak{C}_i . \tag{11,4} \]

The spherical functions \(Y_l^{m_l}(\theta,\varphi)\) entering the solution (11,2) are transformed according to the irreducible representations \(D_l\) of the rotation group. In other words, the representations \(D_l\) with integral \(l\) transform the eigenfunctions of electrons without taking spin into account. The representations \(D_j\) with half-integral index \(j\) give the transformation law of the angular part of the electron eigenfunctions when spin is taken into account. It should also be noted that the purely spin functions, which we shall denote by \(\xi(\sigma)\), \(\eta(\sigma)\), transform according to the representation \(D_{1/2}\).

However, since, starting from the Schrödinger equation, one cannot obtain the eigenfunctions of a “spin” electron, it is usually assumed that the eigenfunctions describing the states of an electron with spin taken into account are products of coordinate functions by purely spin functions, i.e.,

\[ \left. \begin{aligned} \psi&=\psi_{n,l,m_l}(r,\theta,\varphi)\xi(\sigma),\\ \psi&=\psi_{n,l,m_l}(r,\theta,\varphi)\eta(\sigma). \end{aligned} \right\} \tag{11,5} \]

These functions transform according to a representation which is the direct product of the representations \(D_l\) and \(D_{1/2}\). However, this representation is reducible:

\[ D_l \times D_{1/2}=D_{l+1/2}+D_{l-1/2}, \tag{11,6} \]

i.e., when spin is taken into account, a term with a given orbit splits into two levels, the energy of each of which is determined by the quantum number of the square of the electron’s total angular momentum

\[ j=l\pm \frac{1}{2}. \tag{11,7} \]

From (11,7) it is seen that one and the same value of \(j\) can be obtained in two ways:

\[ j=l+\frac{1}{2}, \qquad j=(l+1)-\frac{1}{2}. \tag{11,8} \]

The energy levels corresponding to these values of \(j\) differ in the behavior of their corresponding eigenfunctions under inversion, namely, functions with even \(l\) do not change their sign under inversion, whereas functions with odd \(l\) change their sign to the opposite one.

Thus, we may write that, when spin is taken into account, the electron states will be described by eigenfunctions of the form

\[ \psi=\psi_{n,j,l,m_j}(r,\theta,\varphi,\sigma) \tag{11,9} \]

with eigenvalues

\[ E=E(n,j,l). \tag{11,10} \]

We shall now carry out the investigation of this same problem according to the scheme of § 10. As was already indicated, the Hamiltonian of the system remains invariant under the action of the operations of the group \(\Re_h\). As generators of the group one may choose the infinitesimal transformations [cf. (7,10)]

\[ \left. \begin{aligned} \hat I_x&=-\left(y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}\right),\\ \hat I_y&=-\left(z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}\right),\\ \hat I_z&=-\left(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right), \end{aligned} \right\} \tag{11,11} \]

and the inversion \(I\). The operators of the infinitesimal transformations (11,11) coincide, to within a constant factor, with the operators of the projections of the angular momentum \(\hat J\) on the axes \(x\), \(y\), and \(z\), respectively. Thus, from the condition of invariance of the Hamiltonian with respect to rotations there follows the law of conservation of the angular momentum \(\hat J\).

Let us note now that the operators (11,11) do not commute with one another, and therefore their eigenvalues cannot be determined simultaneously. This means that branching will take place.

The group \(\Re_h\) decomposes into the following classes of conjugate elements:

\[ \begin{aligned} &E \text{ — the identity element;}\\ &K(u) \text{ — all rotations through a given angle } u;\\ &I \text{ — inversion;}\\ &IK(u) \text{ — rotations with inversion.} \end{aligned} \]

The characters of its representations are given in Table 2. Denoting a rotation through an angle \(\varphi\) about the axis \(z\) by \(C(\varphi)\), we may write:

\[ \hat C(\varphi)\psi=e^{im\varphi}\psi. \tag{11,12} \]

Thus, the operators representing a complete set of physical quantities will be

\[ \hat K(u),\quad \hat I,\quad \hat C(\varphi). \tag{11,13} \]

As is well known, any finite motion \(A(t)\) can be expressed in terms of the corresponding infinitesimal transformation \(X\) in the form

\[ A(t)=e^{Xt}. \tag{11,14} \]

Accordingly, a rotation through an angle \(u\) about some arbitrary axis can be written as follows:

\[ \exp\left[\left(I_x\sin\theta'\cos\varphi' +I_y\sin\theta'\sin\varphi' +I_z\cos\varphi'\right)u\right]. \tag{11,15} \]

In order to obtain the class operator, one must integrate

Table 2

\(\mathfrak{R}_h\)	\(E\)	\(K(u)\)	\(I\)	\(I\cdot K(u)\)
\(\Gamma_j\)	\((2j+1)\)	\(\displaystyle \frac{\sin\left(j+\frac12\right)u}{\sin\frac12 u}\)	\((2j+1)\)	\(\displaystyle \frac{\sin\left(j+\frac12\right)u}{\sin\frac12 u}\)
\(\Gamma'_j\)	\((2j+1)\)	\(\displaystyle \frac{\sin\left(j+\frac12\right)u}{\sin\frac12 u}\)	\(-(2j+1)\)	\(\displaystyle -\frac{\sin\left(j+\frac12\right)u}{\sin\frac12 u}\)

(11,15) over all possible directions of rotation. Consequently,

\[ \hat K(u)=\frac{1}{4\pi}\int_0^{2\pi}\int_0^\pi \exp\left[\left(I_x\sin\theta'\cos\varphi' +I_y\sin\theta'\sin\varphi' +I_z\cos\theta'\right)u\right]\sin\theta'\,d\varphi'\,d\theta'. \tag{11,16} \]

The eigenvalues of this operator are found from formula (10,9), taking Table 2 into account, and are equal to

\[ \frac{\sin\left(j+\frac12\right)u}{(2j+1)\sin\frac12 u} =\varkappa(j,u), \tag{11,17} \]

i.e., the equality

\[ \hat K(u)\psi=\varkappa(j,u)\psi \tag{11,18} \]

holds. If we now expand the left- and right-hand sides of (11,18) in a series in \(u\)

and, considering \(u\) small, confine ourselves to the first terms of the expansion, then we shall have*):

\[ \left[1+\frac{u^2}{3}\left(\hat l_x^2+\hat l_y^2+\hat l_z^2\right)\right]\psi = \left[1-\frac{u^2}{3}j(j+1)\right]\psi . \tag{11,19} \]

It follows from this that

\[ \left(\hat l_x^2+\hat l_y^2+\hat l_z^2\right)\psi = -j(j+1)\psi \tag{11,20} \]

or

\[ \hat l^2\psi=j(j+1)\psi . \tag{11,21} \]

Thus, one may say that the operator (11,16) with the eigenvalues (11,17) represents a quantity closely connected with the square of the angular momentum.

The operator \(\hat I\) has eigenvalues equal to \(\pm 1\), and, as is well known, characterizes the parity of the state. Consequently, we may finally write:

\[ E=E(n,j,\pm 1). \tag{11,22} \]

(We note once more that the quantum number \(n\) cannot be determined from the symmetry of the problem.) Since terms with a given \(j\) and a given parity are \((2j+1)\)-fold degenerate, the different states belonging to a given level can be distinguished by the eigenvalues of the operator \(\hat C(\varphi)\). It is easy to show that, in the limiting case of small \(\varphi\), this operator becomes the operator of the projection of the angular momentum on the \(z\)-axis, with eigenvalues \(m_j\). Consequently, for the wave function of an electron in the field of central forces we obtain:

\[ \psi=\psi_{n,j,\pm 1,m_j}(r,\theta,\varphi,\sigma). \tag{11,23} \]

12. STATES OF AN ELECTRON IN A CRYSTALLINE FIELD \(^{4,11}\)

When an electron is in a crystalline field, the Schrödinger equation for it remains invariant only under the action of the symmetry operations of this crystalline field. In other words, among the motions of the electron that leave the Schrödinger equation invariant, there will no longer be infinitely small transformations,

*) On the left-hand side, the integrand expression (11,16) is expanded in a series in \(u\); then, after carrying out term-by-term integration over \(\theta'\) and \(\varphi'\), one obtains the result (11,19).

that in turn leads to the violation of the usual conservation laws. Now the role of conserved quantities will be played by certain quasi-quantities connected, as was shown earlier, with the symmetry of the field under investigation.

The symmetry group of the Schrödinger equation describing the stationary states of an electron in a cubic crystal is the group \(D_h\). As its generating elements one may choose:

\[ A \text{— rotation by } \frac{\pi}{2} \text{ about the } x\text{-axis;} \]

\[ B \text{— rotation by } \frac{\pi}{2} \text{ about the } y\text{-axis;} \]

\[ I \text{— inversion with respect to the origin.} \]

Since \(A\) and \(B\) do not commute with each other, the eigenvalues of the corresponding operators cannot be determined simultaneously; hence it follows that degeneracy will occur. In order to find the complete set of conserved quantities, one must first divide the elements of the group \(D_h\) into classes of conjugate elements and select independent ones among them.

It is well known that in any group the identity element forms a class by itself. The character of this class is equal to the dimension of the representation, i.e., to the multiplicity of degeneracy of the corresponding level. Moreover, since inversion commutes with all elements of any point crystallographic group, it also constitutes a separate class. The eigenvalues of the inversion operator are equal to \(\pm 1\) and, as has already been said, characterize the parity of the state. Both of these characteristics—the dimension and the parity—can be determined at once, and we shall henceforth regard them as already known. Then it will be sufficient to consider only the class consisting of rotations by \(\pi/2\) about the coordinate axes. If the operator of the class is taken to be equal to the sum of the operators corresponding to all group elements contained in it, divided by the number of elements in the class, then the equation for determining the eigenvalues and eigenfunctions of the class operator has the form

\[ \hat K \psi_i = \frac{X^{(i)}}{n_i}\,\psi_i, \tag{12,1} \]

where \(X^{(i)}\) is the character of the class; \(n_i\) is the dimension of the representation.

Thus, one may say that the energy term in a cubic crystal will be characterized by a definite multiplicity of degeneracy, a definite parity, and a definite eigenvalue \(x\) of the operator \(\hat K\), which are found from formula (12,1). Degenerate states can be distinguished from one another by the eigenvalues \(\alpha\) of the operator \(\hat C\), corresponding to the opera-

... of a rotation about the \(z\)-axis through an angle \(\dfrac{\pi}{2}\). Thus, in a cubic crystal the following types of energy states are possible (Table 3).

Table 3

Energy term	Degeneracy multiplicity	Parity	\(\chi\)	\(a\)
\(E_1\)	1	Even	\(1\)	\(1\)
\(E'_1\)	1	Odd	\(1\)	\(1\)
\(E_2\)	1	Even	\(-1\)	\(-1\)
\(E'_2\)	1	Odd	\(-1\)	\(-1\)
\(E_3\)	2	Even	\(0\)	\(1,-1\)
\(E'_3\)	2	Odd	\(0\)	\(1,-1\)
\(E_4\)	3	Even	\(-\dfrac{1}{3}\)	\(-1,i,-i\)
\(E'_4\)	3	Odd	\(-\dfrac{1}{3}\)	\(-1,i,-i\)
\(E_5\)	3	Even	\(\dfrac{1}{3}\)	\(1,i,-i\)
\(E'_5\)	3	Odd	\(\dfrac{1}{3}\)	\(1,i,-i\)

In order to clarify what physical meaning the operators \(\hat K\) and \(\hat C\) have, it is convenient to represent them in terms of infinitesimal transformations; then the operators of rotations about the coordinate axes through a finite angle can be written in the form

\[ \left. \begin{aligned} \hat A(u)&=e^{\hat I_x u},\\ \hat B(u)&=e^{\hat I_y u},\\ \hat C(u)&=e^{\hat I_z u}, \end{aligned} \right\} \tag{12,2} \]

where \(\hat I_x,\ \hat I_y,\ \hat I_z\) are the operators of infinitesimal rotations about the \(x\)-, \(y\)-, and \(z\)-axes, respectively.

Since, by definition, the operator of the class under consideration can be represented in the form

\[ \hat K(u)=\frac{1}{6}\left[\hat A(u)+\hat A^{-1}(u)+\hat B(u)+\hat B^{-1}(u)+\hat C(u)+\hat C^{-1}(u)\right], \tag{12,3} \]

then, taking (12,2) into account and recalling that the infinitesimal operators

rotations are connected with the operators of projections of the angular momentum on the coordinate axes by the relations (6.11)

\[ \hat L_x=i\hat l_x,\quad \hat L_y=i\hat l_y,\quad \hat L_z=i\hat l_z, \]

we shall finally have:

\[ \hat K(u)=\frac{1}{3}\left[\cos \hat L_x u+\cos \hat L_y u+\cos \hat L_z u\right], \tag{12.4} \]

and also

\[ \hat C(u)=e^{-i\hat L_z u}. \tag{12.5} \]

Consequently, one may conclude that the operator \(\hat K(u)\) with eigenvalues \(\chi\) represents a certain conserved physical quantity closely connected with the total angular momentum of the system, while the operator \(\hat C(u)\) with eigenvalues \(\alpha\) represents a physical quantity analogous to the projection of angular momentum on the \(z\)-axis. In order to determine the eigenfunctions of these operators, it is necessary to solve the equations:

\[ \hat K(u)\psi(\theta,\varphi)=\chi\psi(\theta,\varphi), \tag{12.6} \]

\[ \hat C(u)\psi(\theta,\varphi)=\alpha\psi(\theta,\varphi). \tag{12.7} \]

However, solving the problem thus posed is associated with great mathematical difficulties, and therefore we shall approach the investigation of the question in a somewhat different way, namely we shall try to determine which combinations of spherical functions \(Y_l^m(\theta,\varphi)\) \((m=-l,\ldots,+l)\), i.e. eigenfunctions of the square of the angular momentum of an electron in an atom, satisfy these conditions. The spherical functions used will be assumed normalized. Then three ways of solving the question are possible, two of which are briefly set forth below, and the third in more detail in § 13.

1. Let us expand the operator (12.4) in a series in powers of \(u\):

\[ \hat K(u)=1+\frac{1}{3}\sum_{k=1}^{\infty}(-1)^k\frac{u^{2k}}{(2k)!}\hat L_k, \tag{12.8} \]

where

\[ \hat L_k=\hat L_x^{2k}+\hat L_y^{2k}+\hat L_z^{2k}. \tag{12.9} \]

Knowing how the operators \(\hat L_x,\ \hat L_y,\ \hat L_z\) act on the functions \(Y_l^m\ (m=-l,\ldots,+l)\), one can determine the result of the action on these functions

and of the operators \(\hat L_x^{2k}, \hat L_y^{2k}, \hat L_z^{2k}\). Then one can find the eigenfunctions and eigenvalues of the operators \(\hat L_R\) and, consequently, also the eigenfunctions and eigenvalues of the operator \(\hat K(u)\).

1. It is known that spherical functions transform according to the irreducible representations \(D_l\) of the rotation group. Consequently, the operators (12,2), and together with them also the operator \(\hat K(u)\) in the form (12,3), can be represented by the matrices corresponding to them in this representation. By diagonalizing the matrix thus obtained for \(K(u)\) and finding its eigenvectors, we also determine the eigenfunctions and eigenvalues of the indicated operator.

Omitting the elementary but cumbersome calculations, we give only the table of final results.

The functions given in Table 4 are eigenfunctions of the operators (12,4) and (12,5). But it is clear from the table that several eigenfunctions can correspond to the same eigenvalues of these operators. Consequently, in order to obtain eigenfunctions satisfying the Schrödinger equation of the given perturbed problem and possessing the required symmetry, it is necessary to form linear combinations of functions belonging to one and the same eigenvalues of equations (12,6) and (12,7). It is also clear from the table that the energy terms differ by the quantity

\[ \frac{1}{n}\sum_\mu \cos \mu u, \tag{12,10} \]

where \(n\) is the multiplicity of the representation, and the states by the quantity

\[ e^{i\mu u}, \tag{12,11} \]

where

\[ \mu = 0,\ \pm 1,\ 2 . \tag{12,12} \]

It is natural to call \(\mu\) the projection of the quasimomentum on the \(z\)-axis.

On the basis of all the foregoing one can draw the following conclusions. For an electron in a field of cubic symmetry the conserved quantities are the electron energy, the quasimomentum of the amount of motion, described by the class operator \(\hat K(u)\), and the quasiprojection of the momentum of the amount of motion on the \(z\)-axis, described by the operator \(e^{-i\hat L_z u}\). The energy levels are numbered by the quantum numbers \(\nu\) and \(\chi\), where \(\nu\) depends on the form of the potential energy and cannot be determined from symmetry considerations, while \(\chi\) is the eigenvalue of the operator \(\hat K\). The expressed states are numbered by the quantum number \(\mu\), defined up to comparison modulo 4.

Table 4

Atomic term	Eigenvalue of the operator $\tilde K(u)$	$\tilde K\left(\dfrac{\pi}{2}\right)$	$C(u)$ at $u=\dfrac{\pi}{2}$	Eigenfunctions	Term
$s$	$1$	$1$	$1=1$	$Y_0^0$	$E_1$
$p$	$\dfrac{1}{3}\,[1+2\cos u]$	$\dfrac{1}{3}$	$e^{iu}=i$	$Y_1^{-1}$	$E'_5$
$p$	$\dfrac{1}{3}\,[1+2\cos u]$	$\dfrac{1}{3}$	$1=1$	$Y_1^0$	$E'_5$
$p$	$\dfrac{1}{3}\,[1+2\cos u]$	$\dfrac{1}{3}$	$e^{-iu}=-i$	$Y_1^1$	$E'_5$
$d$	$\dfrac{1}{2}\,[1+\cos 2u]$	$0$	$e^{2iu}=-1$	$\dfrac{\sqrt{2}}{2}\,(Y_2^2+Y_2^{-2})$	$E_3$
$d$	$\dfrac{1}{2}\,[1+\cos 2u]$	$0$	$1=1$	$Y_2^0$	$E_3$
$d$	$\dfrac{1}{3}\,[2\cos u+\cos 2u]$	$-\dfrac{1}{3}$	$e^{2iu}=-1$	$\dfrac{\sqrt{2}}{2}\,(Y_2^2-Y_2^{-2})$	$E_4$
$d$	$\dfrac{1}{3}\,[2\cos u+\cos 2u]$	$-\dfrac{1}{3}$	$e^{iu}=i$	$Y_2^{-1}$	$E_4$
$d$	$\dfrac{1}{3}\,[2\cos u+\cos 2u]$	$-\dfrac{1}{3}$	$e^{-iu}=-i$	$Y_2^1$	$E_4$

Continuation of Table 4

Atomic term	Eigenvalue of the operator $\hat K(u)$	$\hat K\left(\dfrac{\pi}{2}\right)$	$C(u)$ at $u=\dfrac{\pi}{2}$	Eigenfunctions	Term
$f$	$\cos 2u$	$-1$	$e^{2iu}=-1$	$\dfrac{\sqrt{2}}{2}\left(Y_3^{2}-Y_3^{-2}\right)$	$E_2'$
$f$	$\dfrac{1}{3}\left[\dfrac{5}{4}\cos u+\cos 2u+\dfrac{3}{4}\cos 3u\right]$	$-\dfrac{1}{3}$	$e^{2iu}=-1$	$\dfrac{\sqrt{2}}{2}\left(Y_3^{2}+Y_3^{-2}\right)$	$E_4'$
$f$	$\dfrac{1}{3}\left[\dfrac{5}{4}\cos u+\cos 2u+\dfrac{3}{4}\cos 3u\right]$	$-\dfrac{1}{3}$	$e^{iu}=i$	$\dfrac{\sqrt{6}}{4}Y_3^{3}-\dfrac{\sqrt{10}}{4}Y_3^{-1}$	$E_4'$
$f$	$\dfrac{1}{3}\left[\dfrac{5}{4}\cos u+\cos 2u+\dfrac{3}{4}\cos 3u\right]$	$-\dfrac{1}{3}$	$e^{-iu}=-i$	$\dfrac{\sqrt{6}}{4}Y_3^{-3}-\dfrac{\sqrt{10}}{4}Y_3^{1}$	$E_4'$
$f$	$\dfrac{1}{3}\left[1+\dfrac{3}{4}\cos u+\dfrac{5}{4}\cos 3u\right]$	$\dfrac{1}{3}$	$e^{iu}=i$	$\dfrac{\sqrt{10}}{4}Y_3^{3}+\dfrac{\sqrt{6}}{4}Y_3^{-1}$	$E_5'$
$f$	$\dfrac{1}{3}\left[1+\dfrac{3}{4}\cos u+\dfrac{5}{4}\cos 3u\right]$	$\dfrac{1}{3}$	$1=1$	$Y_3^{0}$	$E_5'$
$f$	$\dfrac{1}{3}\left[1+\dfrac{3}{4}\cos u+\dfrac{5}{4}\cos 3u\right]$	$\dfrac{1}{3}$	$e^{-iu}=-i$	$\dfrac{\sqrt{10}}{4}Y_3^{-3}+\dfrac{\sqrt{6}}{4}Y_3^{1}$	$E_5'$

Continuation of Table 4

Atomic term	Eigenvalue of the operator $\hat K(u)$	$\hat K\left(\dfrac{\pi}{2}\right)$	$C(u)$ for $u=\dfrac{\pi}{2}$	Eigenfunctions	Term
$g$	$\left[\dfrac{7}{12}+\dfrac{5}{12}\cos 4u\right]$	$1$	$1=1$	$\sqrt{\dfrac{7}{12}}\,Y_4^0+\sqrt{\dfrac{5}{24}}\,(Y_4^4+Y_4^{-4})$	$E_1$
$g$	$\dfrac{1}{2}\left[\dfrac{5}{12}+\cos 2u+\dfrac{7}{12}\cos 4u\right]$	$0$	$1=1$	$\sqrt{\dfrac{5}{12}}\,Y_4^0-\sqrt{\dfrac{7}{24}}\,(Y_4^4-Y_4^{-4})$	$E_3$
$g$	$\dfrac{1}{2}\left[\dfrac{5}{12}+\cos 2u+\dfrac{7}{12}\cos 4u\right]$		$e^{2iu}=-1$	$\dfrac{\sqrt{2}}{2}\,(Y_4^2+Y_4^{-2})$
$g$	$-\dfrac{1}{3}\left[\dfrac{1}{4}\cos u+\cos 2u+\dfrac{7}{4}\cos 3u\right]$	$-\dfrac{1}{3}$	$e^{2iu}=-1$	$\dfrac{\sqrt{2}}{2}\,(Y_4^2-Y_4^{-2})$	$E_4$
$g$	$-\dfrac{1}{3}\left[\dfrac{1}{4}\cos u+\cos 2u+\dfrac{7}{4}\cos 3u\right]$		$e^{iu}=i$	$\sqrt{\dfrac{1}{8}}\,Y_4^{-1}-\sqrt{\dfrac{7}{8}}\,Y_4^3$
$g$	$-\dfrac{1}{3}\left[\dfrac{1}{4}\cos u+\cos 2u+\dfrac{7}{4}\cos 3u\right]$		$e^{-iu}=-i$	$\sqrt{\dfrac{1}{8}}\,Y_4^1-\sqrt{\dfrac{7}{8}}\,Y_4^{-3}$
$g$	$\dfrac{1}{3}\left[\dfrac{7}{4}\cos u+\dfrac{1}{4}\cos 3u+\cos 4u\right]$	$\dfrac{1}{3}$	$e^{iu}=i$	$\sqrt{\dfrac{7}{8}}\,Y_4^{-1}+\sqrt{\dfrac{1}{8}}\,Y_4^3$	$E_5$
$g$	$\dfrac{1}{3}\left[\dfrac{7}{4}\cos u+\dfrac{1}{4}\cos 3u+\cos 4u\right]$		$1=1$	$\dfrac{\sqrt{2}}{2}\,(Y_4^4-Y_4^{-4})$
$g$	$\dfrac{1}{3}\left[\dfrac{7}{4}\cos u+\dfrac{1}{4}\cos 3u+\cos 4u\right]$		$e^{-iu}=-i$	$\sqrt{\dfrac{7}{8}}\,Y_4^1+\sqrt{\dfrac{1}{8}}\,Y_4^{-3}$

13. SPLITTING OF ATOMIC TERMS IN A CRYSTAL WITHOUT TAKING SPIN INTO ACCOUNT4, 12

The perturbation of a free atom when it is introduced into a crystal may occur for two reasons: on the one hand, the atom enters into electronic exchange with other atoms of the crystal; on the other hand, an electric field of definite symmetry, due to the remaining atoms, acts on the atom in the crystal.

The electric field of the crystal causes a splitting of the terms of the unperturbed atom, similar to the splitting in the Stark effect and characteristic of the symmetry of the crystal field. It was noted in § 9 that if the degeneracy of the energy levels of a system is connected with the presence of identical symmetry elements, then, obviously, the removal of degeneracy (splitting of terms) is caused by a lowering of the symmetry. Indeed, if a system, for example an atom, is subjected to the action of some external force field with a symmetry lower than the symmetry of the field-free system, then the latter, as it were, adapts itself to the symmetry of the perturbing field and lowers its symmetry to the symmetry of the perturbation. The number of components into which the term of a free atom is split increases as the symmetry is lowered.

Depending on the magnitude of the splitting, three cases are distinguished.

1. The splitting caused by the crystal field is considerably greater than the distances between different multiplets ($s - p$, $p - d$, ... etc.), i.e. the crystal field is stronger than the interaction of the electrons in the atom. In this case one starts from a model of a free atom with quantum numbers $n_i$ and $l_i$ for an individual electron, and the splitting of terms due to exchange interaction is neglected. In the first approximation, the perturbation of individual electrons by the crystal field is taken into account, i.e. all possible orientations of the angular momentum $l_i$ of an individual electron relative to the crystallographic axes are considered, as well as the resulting splitting of the atomic term. In the second approximation, the exchange of electrons within the atom must be taken into account. This leads to a further splitting of the term, whose order is comparable with the distance between different multiplets. Finally, the spin-orbit interaction gives the usual multiplet splitting.

2. The splitting caused by the crystal field is small in comparison with the distance between different multiplets, but large in comparison with the splitting within an individual multiplet. In this case one starts from a model of a free atom with the splitting of the term due to exchange interaction taken into account, but without taking the spin-orbit interaction into account. Such an atom should be placed in the crystal, and the orientation of its total angular momentum $l$ relative to the axes of the crystal and the resulting splitting of the atomic term should be investigated. In the second approximation the spin-orbit interaction must be taken into account.

3. Splitting caused by the crystal field, considerably smaller than the distances between terms within a multiplet. The completely “ready-made” atom (with spin–orbit interaction taken into account) is placed in the crystal field, and the splitting of its terms by this field is investigated. The spin–orbit coupling is not destroyed by the crystal field, and the orientation of the total angular momentum \(j\) in the crystal is considered.

Here we consider the characteristic perturbation of the atom caused by the symmetry of the field, and electron exchange is not taken into account. It should be noted at once that the indicated treatment is not strictly valid in a real crystal, but rather for an atom placed in a field of a definite point symmetry, since crystalline translational invariance causes the splitting of levels into bands, i.e. leads to a dependence of the energy on the quasi-momentum.

The method of calculation is based on the fact that the Schrödinger equation for a physical system is invariant with respect to the symmetry transformations of this system. Under symmetry transformations, the wave functions of the stationary states of the system that belong to one and the same energy level transform into one another, i.e. they realize a representation of the symmetry group. The dimensionality of this representation determines the degeneracy multiplicity of the given level, i.e. the number of different states with this energy. However, it should be stipulated that if some set of functions and the set of functions complex-conjugate to it realize different irreducible representations of the group, then these two complex-conjugate representations must, from the physical point of view, be considered together, as one representation of doubled dimensionality.

Let the physical system be subjected to the action of some perturbation. Let us consider when this perturbation can lead to a splitting of degenerate levels. The external field (perturbation) has by itself some intrinsic symmetry. If the symmetry of the field is the same as or higher than the symmetry of the unperturbed system, then the symmetry of the perturbed Hamiltonian

\[ \hat H = \hat H_0 + \hat W \]

coincides with the symmetry of the unperturbed operator \(\hat H_0\), and no splitting of levels will occur. If, however, the symmetry of the perturbation is lower than the symmetry of the unperturbed system, then the symmetry of the Hamiltonian \(\hat H\) will coincide with the symmetry of the perturbation \(\hat W\). The wave functions which realized an irreducible representation of the symmetry group of the operator \(\hat H_0\) will also realize a representation of the symmetry group of the perturbed operator \(\hat H\), but this representation may prove to be reducible, which will ...

means the splitting of a degenerate level, i.e., the complete set of functions will be divided into such subsets whose functions transform only into one another under the action of symmetry operations.

Let us see how terms split in passing from a free atom to a cubic crystal (in this paragraph we shall neglect the presence of spin). For this purpose we shall find the characters of the classes of conjugate elements of the group $\mathfrak{D}$ in the representation carried out by means of the matrices of the representation of the rotation group. Let us also note that, when spin is neglected, we do not need to take into account the parity of states with respect to inversion, since it is already automatically taken into account by the quantum number $l$. Recalling which operations each class contains, from formula (6.21) we find:

\[ \begin{aligned} X_1 &= X_l(0)=2l+1,\\ X_2 &= X_4 = X_l(\pi)=(-1)^l,\quad X_5=X_l\!\left(\frac{2\pi}{3}\right)= \begin{cases} 1, & l \equiv 0 \pmod 3,\\ 0, & l \equiv 1 \pmod 3,\\ -1, & l \equiv 2 \pmod 3, \end{cases}\\ X_3 &= X_l\!\left(\frac{\pi}{2}\right)=(-1)^{\left[\frac{l}{2}\right]} . \end{aligned} \]

From the formulas obtained it is easy to compile a table of the characters of the representations obtained for various values of $l$. As has already been said, these representations will, generally speaking, be reducible, and consequently they can be decomposed into irreducible parts, i.e., into irreducible representations of the group $\mathfrak{D}$. The results are collected in Table 5.

Thus, the indicated method allows us to determine into which components the term of an unperturbed system splits when it is placed in a perturbing field, what the multiplicity of degeneracy of the new terms will be. Naturally the question arises as to which eigenfunctions of the unperturbed problem correspond to each of the new terms, and for this one must know the eigenfunctions corresponding to each irreducible representation.

Suppose we have a set $\psi_1,\psi_2,\ldots,\psi_n$ of eigenfunctions of the unperturbed problem, which furnish a certain representation $\Gamma$ for the symmetry group of the perturbing field, and let $\Gamma_1,\Gamma_2,\ldots$ be the irreducible representations of this group. Acting on a set of eigenfunctions by the symmetry operations of the group, we first select from them those which transform according to the representation $\Gamma_1$, then those which transform according to the representation $\Gamma_2$, and so on. In this way we learn what type of functions correspond to the various irreducible representations, and hence also which functions will belong to the new terms. The result will be the same as in § 12. It should be emphasized, however, that §§ 12 and 13 solve, generally speaking, entirely different problems. In the first case the question of the description of electronic states in the field of a crystal of cubic symmetry is discussed; in the second—the question of

Table 5

\(l\)	\(K_1\)	\(K_2\)	\(K_3\)	\(K_4\)	\(K_5\)	Decomposition into irreducible representations	Number of terms
0	1	1	1	1	1	\(\Gamma_1\)	1
1	3	\(-1\)	1	\(-1\)	0	\(\Gamma_5\)	1
2	5	1	\(-1\)	1	\(-1\)	\(\Gamma_3+\Gamma_4\)	2
3	7	\(-1\)	\(-1\)	\(-1\)	1	\(\Gamma_2+\Gamma_4+\Gamma_5\)	3
4	9	1	1	1	0	\(\Gamma_1+\Gamma_3+\Gamma_4+\Gamma_5\)	4
5	11	\(-1\)	1	\(-1\)	\(-1\)	\(\Gamma_3+\Gamma_4+2\Gamma_5\)	4
6	13	1	\(-1\)	1	1	\(\Gamma_1+\Gamma_2+\Gamma_3+2\Gamma_4+\Gamma_5\)	6
7	15	\(-1\)	\(-1\)	\(-1\)	0	\(\Gamma_2+2\Gamma_3+2\Gamma_4+2\Gamma_5\)	6
8	17	1	1	1	\(-1\)	\(\Gamma_1+2\Gamma_3+2\Gamma_4+2\Gamma_5\)	7
9	19	\(-1\)	1	\(-1\)	1	\(\Gamma_1+\Gamma_2+\Gamma_3+2\Gamma_4+3\Gamma_5\)	8
10	21	1	\(-1\)	1	0	\(\Gamma_1+\Gamma_2+2\Gamma_3+3\Gamma_4+2\Gamma_5\)	9
11	23	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(\Gamma_2+2\Gamma_3+3\Gamma_4+3\Gamma_5\)	9
12	25	1	1	1	1	\(R+\Gamma_1\)	11
\(12p+q\)	Here \(R\) denotes the regular representation	Here \(R\) denotes the regular representation	Here \(R\) denotes the regular representation	Here \(R\) denotes the regular representation	Here \(R\) denotes the regular representation	\(Rp+\) the constituent parts for \(l=q\)

changes of atomic states under the action of such a perturbing field. The agreement of the results begins from the moment when it is a question of determining the eigenfunctions of the zero approximation, since in both cases the initial functions are spherical functions.

14. SPLITTING OF ATOMIC TERMS IN A CRYSTAL WITH SPIN TAKEN INTO ACCOUNT4, 12, 13

In studying the question of the splitting of atomic terms in a crystal with spin taken into account, i.e. for half-integral \(j\), a difficulty arises connected with the fact that the representations of the group \(\Re\) are double-valued. The difficulty consists in the fact that, in order to decompose these representations into irreducible representations of the crystallographic groups, it is necessary to know from the very beginning the double-valued irreducible representations of the crystallographic groups, whereas the usual procedure gives only single-valued representations. To find the double-valued representations it is convenient to use the following artificial device. Let us introduce, in a purely formal way, the concept of a new element of the group (we shall denote it by \(R\))—a rotation by \(2\pi\) about an arbitrary axis, as an element different from the identity element but coinciding with \(E\) when applied twice: \(R^2 = E\). In accordance with this, rotations \(A_n\) about axes of symmetry of order \(n\) will give the identity transformation only after \(2n\)-fold, and not \(n\)-fold, application:

\[ A_n^n = R, \quad A_n^{2n} = E. \]

Inversion \(I\), as an operation commuting with every rotation, must, upon double application, still give \(E\). Supplementing the elements of the crystal group by the elements obtained when they are multiplied by \(R\), we obtain the so-called double point group. The order of the double group is twice the order of the original group. The double-valued representations of the actual point group will obviously be single-valued representations of the corresponding double group, so that the usual methods can be used to find them.

The number of classes in the double group is greater than in the original one, but, generally speaking, not twice as great. The element \(R\) commutes with all other elements of the group and therefore constitutes a class by itself. Indeed, \(R\) commutes both with inversion and with any rotation. For the operation of reflection this follows from the fact that the latter can always be expressed as the product of a rotation and inversion.

If the axis of symmetry is twofold, then in the double group this means that the elements \(A_n^k\) and \(RA_n^{\,n-k} = A_n^{\,2n-k}\) are conjugate. In this connection, in the presence of axes of the second order, the distribution of elements

whether the axes are two-sided also depends on the classes. In ordinary groups this is immaterial, since \(A_2=A_2^{-1}\).

Among all irreducible representations of the double point group there enter, first, representations coinciding with the single-valued representations of the ordinary group, with the element \(R\), as with \(E\), corresponding to the identity matrix, and, second, two-valued representations of the ordinary group, with the element \(R\) corresponding to the negative identity matrix.

Let us now consider the double group of the octahedron. This group contains 48 elements distributed among eight classes, since to the classes \(K_1, K_3, K_5\) of the ordinary group there must correspond two classes each of the double group, i.e. we have:

\[ \begin{aligned} K_1 &\to K_1',\ K_2',\\ K_2 &\to K_2,\\ K_3 &\to K_3',\ K_3'',\\ K_4 &\to K_4,\\ K_5 &\to K_5',\ K_5''. \end{aligned} \]

The double group of the octahedron, therefore, has three two-valued representations, including one four-dimensional and two two-dimensional ones (since \(48=24+2^2+2^2+4^2\), where 24 corresponds to the sum of the squares of the dimensions of the single-valued representations). The characters of the classes of the double group of the octahedron that give two-valued representations of the ordinary octahedron group are given in Table 6.

Table 6

\(\mathfrak{D}'\)	\(K_1'\)	\(K_1''\)	\(K_2\)	\(K_3'\)	\(K_3''\)	\(K_4\)	\(K_5'\)	\(K_5''\)
\(\Gamma_6\)	2	\(-2\)	0	\(\sqrt{2}\)	\(-\sqrt{2}\)	0	1	\(-1\)
\(\Gamma_7\)	2	\(-2\)	0	\(-\sqrt{2}\)	\(\sqrt{2}\)	0	1	\(-1\)
\(\Gamma_8\)	4	\(-4\)	0	0	0	0	\(-1\)	1

Now let us proceed to discuss the question of the splitting of terms in a cubic crystal when the electron spin is taken into account. In § 13 it was already indicated that in this case two possibilities must be distinguished. First, the crystalline field may destroy the spin-orbit coupling. Then, under the action of this field, both the orbit \(l\) and the spin \(s\) are separately oriented in the crystal. The orientation of the orbit leads to the fact that terms with a given \(l\) split into

some components characteristic of the symmetry of the field. The subsequent allowance for the “oriented” spin gives an additional splitting of these component parts. Secondly, the spin-orbit coupling may also be preserved in the crystalline field. In this case one must take into account from the very beginning the fact that the presence of the electron spin leads to the splitting of the term with a given orbit \(l\) into two terms characterized by the values of the quantum number of the total angular momentum \(j\left(j=l\pm\dfrac{1}{2}\right)\). Therefore in the crystal it will already be the total angular momentum of the electron that is oriented, which likewise leads to a splitting of the terms with a given \(j\).

Let us begin the consideration with the first case. For this we shall use the fact that the splitting of terms with a definite \(l\) has already been given in Table 5. In addition, we note that the purely spin functions, under transformations of the octahedral group, transform according to the representation \(\Gamma_6\). Then it is sufficient to consider the products \(\Gamma_i \times \Gamma_6\) \((i=1,2,3,4,5)\), i.e., to investigate the splitting of crystalline terms when spin is taken into account. The results of the calculations are collected in Table 7.

Table 7

Representation	Decomposition of \(\Gamma_i \times \Gamma_6\) into irreducible components	Number of terms	Representation	Decomposition of \(\Gamma_i \times \Gamma_6\) into irreducible components	Number of terms
\(\Gamma_1\)	\(\Gamma_1 \times \Gamma_6 = \Gamma_6\)	1	\(\Gamma_4\)	\(\Gamma_4 \times \Gamma_6 = \Gamma_7 + \Gamma_8\)	2
\(\Gamma_2\)	\(\Gamma_2 \times \Gamma_6 = \Gamma_7\)	1	\(\Gamma_5\)	\(\Gamma_5 \times \Gamma_6 = \Gamma_6 + \Gamma_8\)	2
\(\Gamma_3\)	\(\Gamma_3 \times \Gamma_6 = \Gamma_8\)	1

In order to consider the second case, following the usual procedure, we find the characters of the representations of the double octahedral group realized by the matrices of the rotation group:

\[ X'_1=-X''_2=(2j+1), \qquad X_2=X_4=0, \]

\[ X'_3=-X''_3= \begin{cases} \sqrt{2}, & j\equiv \dfrac{1}{2}\pmod 4,\\ 0, & j\equiv \dfrac{3}{2},\,\dfrac{7}{2}\pmod 4,\\ -\sqrt{2}, & j\equiv \dfrac{5}{2}\pmod 4, \end{cases} \]

\[ X'_5=-X''_5= \begin{cases} 1, & j\equiv \dfrac{1}{2}\pmod 3,\\ -1, & j\equiv \dfrac{3}{2}\pmod 3,\\ 0, & j\equiv \dfrac{5}{2}\pmod 3. \end{cases} \]

Then we decompose them into irreducible representations of the double group of the octahedron. The results are given in Table 8, from which it is seen into how many components a term with half-integral \(j\) splits.

Table 8

\(j\)	\(K_1'\)	\(K_3'\)	\(K_5'\)	Decomposition into irreducible representations	Number of terms
\(\dfrac{1}{2}\)	2	\(\sqrt{2}\)	1	\(\Gamma_6\)	1
\(\dfrac{3}{2}\)	4	0	\(-1\)	\(\Gamma_8\)	1
\(\dfrac{5}{2}\)	6	\(-\sqrt{2}\)	0	\(\Gamma_7+\Gamma_8\)	2
\(\dfrac{7}{2}\)	8	\(\sqrt{2}\)	1	\(\Gamma_6+\Gamma_7+\Gamma_8\)	3
\(\dfrac{9}{2}\)	10	0	\(-1\)	\(\Gamma_6+2\Gamma_8\)	3
\(\dfrac{11}{2}\)	12	\(-\sqrt{2}\)	0	\(\Gamma_6+\Gamma_7+2\Gamma_8\)	4
\(\dfrac{13}{2}\)	14	\(\sqrt{2}\)	1	\(\Gamma_6+2\Gamma_7+2\Gamma_8\)	5
\(\dfrac{15}{2}\)	16	0	\(-1\)	\(\Gamma_6+\Gamma_7+3\Gamma_8\)	5
\(\dfrac{17}{2}\)	18	\(-\sqrt{2}\)	0	\(2\Gamma_6+\Gamma_7+3\Gamma_8\)	6
\(\dfrac{19}{2}\)	20	\(\sqrt{2}\)	1	\(2\Gamma_6+2\Gamma_7+3\Gamma_8\)	7
\(\dfrac{21}{2}\)	22	0	\(-1\)	\(\Gamma_6+2\Gamma_7+4\Gamma_8\)	7
\(\dfrac{23}{2}\)	24	\(-\sqrt{2}\)	0	\(2\Gamma_6+2\Gamma_7+4\Gamma_8\)	8
\(\dfrac{25}{2}\)	26	\(\sqrt{2}\)	1	\((2\Gamma_6+2\Gamma_7+4\Gamma_8)+\Gamma_6\)	9
\(12p+q\)				\(p(2\Gamma_6+2\Gamma_7+4\Gamma_8)+\) components for \(j=q\)

Let us trace, for example, the evolution of a \(p\)-term under both methods of calculation. As is well known, the functions of a \(p\)-term transform according to the representation \(D_1\) of the rotation group. In a cubic crystalline field they will transform according to the representation \(\Gamma_5\). Forming the product \(\Gamma_5\times\Gamma_6=\Gamma_6+\Gamma_8\), we obtain that, when the spin is taken into account,

splitting occurs. Following the other method, we must form the product \(D_1 \times D_{1/2}=D_{1/2}+D_{3/2}\). On lowering the symmetry to cubic, the right-hand side again gives \(\Gamma_6+\Gamma_8\).

The coincidence of the results (such a coincidence can also be proved in general form) should not seem strange, since the total number of components into which the term is split must be the same regardless of whether the crystalline field is first taken into account and then the spin, or conversely. The strength of the field will affect only the magnitude of the splitting. It should be emphasized, however, that both methods of consideration are expedient, since at the second stage of the calculations they answer, in essence, completely different questions. The first method actually indicates how the crystalline term is split when spin is taken into account; the second—how the atomic term with spin included is split in a crystalline field.

15. SELECTION RULES \(^{12,14}\)

One of the most important problems of quantum physics is the determination of the probabilities of transition from one quantum state to another. This transition probability, according to the general rule of quantum mechanics, is proportional to the square of the matrix element of the interaction energy of the system with the perturbing field. The matrix element can be represented in the form

\[ \int \Psi^* \hat{\Lambda}\Phi\,d\tau, \tag{15,1} \]

where \(\Phi\) is the wave function of the initial state of the system; \(\hat{\Lambda}\) is the interaction operator causing the transition, and \(\Psi\) is the final state of the system after the transition. In order to find the transition probability, it is of course necessary to calculate the matrix element (15,1). However, in many cases, proceeding from the general symmetry properties of the system, it is possible to establish (without any calculation of the matrix element) that the interaction \(\hat{\Lambda}\) cannot cause a transition to a certain final state \(\Psi\). In this case the matrix element of the interaction energy vanishes. Usually the equality or inequality to zero of the matrix element is expressed through certain relations between the quantum numbers of the initial and final states of the system. Such rules, permitting only certain transitions, are called selection rules.

Group theory, relying on the symmetry properties of the system, makes it possible to find selection rules for the matrix elements of any interactions without carrying out an explicit calculation of the integral (15,1). The method is based on the following assertion. If \(\psi_i\) and \(\varphi_j\) are functions of the bases of irreducible representations \(A\) and \(B\) of the symmetry group \(\mathfrak{G}\), then the integral of their product, taken over the whole

with respect to the space of change of variables, is equal to zero if \(A\) and \(B\) are not equivalent.

Indeed, since the integral taken over the entire space is invariant with respect to any transformation of the coordinate system, including any transformation \(Q\) of the symmetry group \(\mathfrak{G}\), we have:

\[ \int \psi_i^* \varphi_j\,d\tau = Q \int \psi_i^* \varphi_j\,d\tau = \int \sum_{k=1}^{n} a_{ik}^* \psi_k^* \sum_{l=1}^{m} b_{jl}\varphi_l\,d\tau . \tag{15,2} \]

Summing this equality over all elements of the group, taking into account the orthogonality relations (4,6) and the fact that the integral on the left is simply multiplied by the order of the group \(g\), we obtain:

\[ g \int \psi_i^* \varphi_j\,d\tau = \sum_{k,l}\int \left( \sum_Q a_{ik}^* b_{jl} \right)\psi_k^*\varphi_l\,d\tau = \begin{cases} 0, & \text{for } A \not\sim B,\\[6pt] \dfrac{g}{n}\delta_{ij}\displaystyle\sum_{k=1}^{n}\int \psi_k^*\varphi_k\,d\tau, & \text{for } A = B. \end{cases} \tag{15,3} \]

Thus our assertion is proved. Let us also note that in the case when the representations \(A\) and \(B\) are identical, whether the integral is equal to zero or not is determined by which functions of the bases of the irreducible representations enter into the product under the integral sign.

If we now form the expression

\[ \int \Psi^*\Phi'\,d\tau, \tag{15,4} \]

where \(\Psi\) and \(\Phi\) are two arbitrary functions, and not necessarily functions of the bases of irreducible representations, and take into account that any wave function \(\Psi\) can be decomposed into component parts \(\psi_i\), each of which is transformed according to a representation \(\Gamma_i\), then from the preceding assertion the following immediately follows. In order that the integral (15,4) be different from zero, it is necessary that in the decompositions of the functions \(\Psi\) and \(\Phi'\) there be functions transforming according to identical representations.

Under the action of transformations of the group \(\mathfrak{G}\), the wave functions \(\Psi\), \(\Phi\), and the interaction operator \(\hat{\Lambda}\) will transform according to certain representations \(\Gamma_\psi\), \(\Gamma_\varphi\), and \(\Gamma_\lambda\) of this group, respectively. Generally speaking, these representations will be reducible. The wave function

\[ \Phi'=\hat{\Lambda}\Phi \tag{15,5} \]

is transformed according to the direct product of representations

\[ \Gamma_\lambda \times \Gamma_\varphi = \Gamma_{\varphi'}, \tag{15,6} \]

which is also reducible in the general case. Then the matrix element (15,4) vanishes if the reducible representations \(\Gamma_{\varphi'}\) and \(\Gamma_\psi\) do not contain common irreducible representations \(\Gamma_i\).

The application of this method is considerably simplified in those cases when the representations \(\Gamma_\psi\), \(\Gamma_\varphi\), and \(\Gamma_\lambda\), according to which \(\Psi\), \(\Phi\), and \(\hat{\Lambda}\) transform, are irreducible. Then it is necessary only to investigate whether the product \(\Gamma_{\varphi'}\) contains the representation \(\Gamma_\psi\). If \(\Gamma_{\varphi'}\) contains \(\Gamma_\psi\), then the matrix element is different from zero; otherwise it is equal to zero.

Thus, we can establish selection rules for transitions of a system from one energy level to another. However, we still do not know from which state of the given level into which state of the final level the system passes. For this it is necessary to take into account that a nonzero integral is obtained, generally speaking, only when the indices of the delta-symbol in formula (15,3) coincide, i.e., when \(i=j\). If we number the states by the eigenvalues of one and the same physical quantity for all terms, this will give us selection rules for the quantum numbers corresponding to this quantity.

Diagonal matrix elements (as distinct from the elements for transitions between different energy levels of one type) require special consideration. In this case there is only one, and not two, different systems of functions, and their pairwise products with one another realize the symmetric product of the representation \(\Gamma_i\) with itself, and not the direct product \(\Gamma_i \times \Gamma_i\). Therefore the existence of diagonal matrix elements of a vector quantity requires the presence of the unit representation in the decomposition of the product \([\Gamma_i^2]\times \Gamma_\lambda\), or, what is the same thing, the presence of \(\Gamma_\lambda\) in \([\Gamma_i^2]\).

Let us illustrate all that has been said above by the example of electronic transitions in a cubic crystal. We shall assume that the states are described with the aid of the quantum numbers used in § 12. Consider dipole transitions. In this case the interaction operator can be written in the form

\[ \hat{\Lambda}= e\mathbf r \tag{15,7} \]

and, under the action of the transformations of the group, is transformed according to the irreducible representation \(\Gamma_5\). Let us determine the selection rules for parity. Since the operator itself changes sign under inversion, in order that the integral

\[ \int \psi_i^* e\mathbf r \varphi_j\,dx \tag{15,8} \]

was different from zero, it is necessary that the functions be of different parity. Thus, we find that transitions are possible only from an even state to an odd one and conversely.

Now let us find from which energy states into which transitions are possible. For this purpose we form the products

\[ \begin{aligned} \Gamma_5 \times \Gamma_1 &= \Gamma_5, \\ \Gamma_5 \times \Gamma_2 &= \Gamma_4, \\ \Gamma_5 \times \Gamma_3 &= \Gamma_4 + \Gamma_5, \end{aligned} \qquad \begin{aligned} \Gamma_5 \times \Gamma_4 &= \Gamma_2 + \Gamma_3 + \Gamma_4 + \Gamma_5, \\ \Gamma_5 \times \Gamma_5 &= \Gamma_1 + \Gamma_3 + \Gamma_4 + \Gamma_5. \end{aligned} \]

The symmetric products of the irreducible representations of the group \(\mathfrak{S}\) will be

\[ \begin{aligned} [\Gamma_1^2] &= [\Gamma_2^2] = \Gamma_1,\\ [\Gamma_3^2] &= \Gamma_1 + \Gamma_3,\\ [\Gamma_4^2] &= [\Gamma_5^2] = \Gamma_1 + \Gamma_3 + \Gamma_4. \end{aligned} \]

\(\Gamma_5\) is not contained in any of them, and therefore the diagonal matrix elements are absent. Consequently, the following transitions are possible:

\[ \Gamma_4 \leftrightarrow \Gamma_2,\ \Gamma_3,\ \Gamma_5,\qquad \Gamma_5 \leftrightarrow \Gamma_1,\ \Gamma_3,\ \Gamma_4 \]

and, in addition,

\[ \Gamma_4 \leftrightarrow \Gamma_4,\qquad \Gamma_5 \leftrightarrow \Gamma_5 \]

for different levels of the same type. Therefore, for the quantum number \(\chi\), on the basis of Table 3 we obtain the selection rules

\[ \left. \begin{aligned} \chi' &= \chi \pm \frac{1}{3},\\ \chi' &= \chi \pm \frac{2}{3}, \end{aligned} \right\} \tag{15,9} \]

if we restrict ourselves to terms of different types, and the additional transitions

\[ \chi'=\chi \tag{15,10} \]

for triply degenerate levels \(\bar E_4\) and \(E_5\) in transitions between different levels of the same type.

Now let us determine the states for which transitions are possible, i.e., the selection rules for the quantum number \(\mu\). Suppose, for example, that we are required to determine the states between which transitions occur for the \(z\)-component of the electric moment, i.e.

one must determine when the matrix element is different from zero

\[ \int \psi_{\mu'}^{*} z \psi_\mu d\tau . \tag{15,11} \]

Since the function \(z\psi_\mu=\psi_\mu'\), it follows from equality (15,3) that \(\mu'\) must be equal to \(\mu\). Consequently, in the end we have \(\mu'=\mu\). Carrying out analogous arguments for the components \(x+iy\) and \(x-iy\), we find \(\mu'=\mu\pm 1\).

16. CONCLUSION

The general principles of investigating quantum-mechanical systems by means of group theory, set forth in the present review, show that questions lying at the foundation of quantum mechanics and of the quantum physics of the solid state can be considered with the greatest depth and completeness only when the apparatus of group theory is used.

The symmetry properties of physical systems should be used more widely in solving concrete problems, and especially in the case of complex quantum-mechanical systems, where exact quantitative calculations cannot be carried out and it is therefore important to obtain as many results as possible by general methods. Moreover, the conclusions obtained with the aid of group theory are the most rigorous, owing to the phenomenological character of the theory of symmetry itself.

The general principles set forth in §§ 9 and 10 can be applied to the investigation of any quantum-mechanical systems and will give results that are the more complete, the higher the symmetry of the problem. In the review these general principles were illustrated on three concrete problems. First, the problem of determining the electronic states of a system according to its symmetry properties was investigated. §§ 11 and 12 were devoted to the exposition of this question. Second, the problem of the splitting of atomic terms in a field of definite symmetry was solved (§§ 13, 14). Third, selection rules for quantum transitions in a crystal were found (§ 15).

However, it is already clear that group theory acquires still greater importance in the investigation of the band structure of the energy spectrum of solids. In particular, the question of the closing of bands when the full space symmetry is taken into account is especially important. The presence of spin-orbit coupling leads to a partial removal of this closing. A complete and sufficiently general study of this question is undoubtedly important for solid-state physics.

Finally, the point of view on the many-electron theory set forth in the works of Wolff and Haken\(^{15}\) makes it possible to apply group-theoretical methods also when the interaction between electrons is taken into account, i.e. in a purely many-electron problem.

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