THE FUNDAMENTAL SIGNIFICANCE OF APPROXIMATE METHODS IN THEORETICAL PHYSICS*

V. A. Fock, Leningrad

1. The task of theoretical physics is the mathematical formulation of the laws of nature. This task is closely connected with, but by no means identical to, the task of mathematical physics—the solution of the equations posed by theoretical physics.

The equations of theoretical physics are never, and cannot be, absolutely exact: in deriving them one always neglects certain secondary factors.

The correct accounting of all factors that are truly essential in a given physical problem may be called physical rigor. Physical rigor is just as necessary in the solution of physical problems as ordinary mathematical rigor is in the solution of problems of analysis.

One of the most important and difficult questions that arise in each separate field of theoretical physics is the question of the extent to which, in posing and solving the problems of that field, the requirements of physical and mathematical rigor are fulfilled. An investigation of this kind is especially important for the domain of elementary physical laws—the domain adjacent to the theory of the structure of matter.

The aim of the present article, however, is not to attempt to give such an analysis of the physical rigor of the existing formulation of elementary physical laws. I should like to dwell here on another question, though one closely connected with this: namely, on the very process of the formation of physical concepts.

Let us suppose that we have one more general physical theory and another, more special one; this special theory is contained in the general one as a special case. The transition from the special theory to the general one is undoubtedly connected with the introduction of new physical concepts. It is this side of the process of concept formation—the acquisition of new concepts as a result of the generalization of a theory—that most often attracts our attention. But the development of physics in recent decades has shown that the generalization of a theory is also connected with the reverse—

* Presented on March 21, 1936, at a meeting of the Mathematical Group of the Academy of Sciences of the USSR.

process, namely, the rejection of old physical concepts. Let us consider, for example, Newtonian classical mechanics and the mechanics of the special theory of relativity (this special theory is, in the present case, i.e. in comparison with Newtonian mechanics, a more general one). In Newtonian mechanics we dealt with absolute time; the concept of simultaneity of two events required no qualifications (specification of a coordinate system, etc.) and was in this sense an absolute concept. In the theory of relativity the concept of absolute simultaneity is lost; it may even happen that, without special qualifications, it will be impossible to say which of two events occurred earlier and which later. Simultaneity becomes an approximate concept, applicable only in those cases when one may neglect the interval of time spent by light in traversing the distance between the points at which the events under consideration occur (for events on the Earth and on the Sun this interval of time is about 500 sec.). Thus the generalization of physical theories is connected not only with the acquisition of new concepts, but also with the rejection of old ones. Here it is necessary to note the following psychological factor: the rejection of old, customary physical concepts is incomparably more difficult than the assimilation of new concepts not connected with such a rejection.

In order to form a clear idea of the necessity of this kind of rejection, we shall try to trace the process of the formation of concepts in a direction which, in a certain sense, is the reverse of the historical development of the theory. If we proceed in this direction, then we must start from the most general physical theory known at the given moment; we shall convince ourselves that with each simplification of it, with each transition to a more particular theory, ever newer physical concepts will arise. But then it will be clear that the reverse process—the transition from a more particular theory to a more general one—must be connected with the rejection of certain physical concepts.

We choose this method of consideration—successive simplification of the theory—because it is easier to trace from the mathematical point of view. The transition to a simplified theory means, in essence, the application of one or another approximate method, based on the possibility, in the given problem, of neglecting certain secondary factors, certain small quantities. The lawfulness of applying the given approximate method—its physical rigor—can be evaluated by means of ordinary mathematical inequalities characterizing the smallness of those quantities which are discarded in the given method. But this “error estimate” in the usual sense at the same time gives an estimate of the applicability of those physical concepts which are connected with the given approximate method. Thus the more difficult logical, or, if one wishes, even philosophical, question of the applicability of definite physical concepts here receives a concrete mathematical expression. Thereby

the fundamental importance of approximate methods in theoretical physics becomes clear.

2. As we have already said, the equations of theoretical physics are never absolutely exact. Even the theory which, at a given stage in the development of physics, is the most general cannot claim universality, since it contains within itself a number of physical neglects. Therefore, when setting about the formulation of such a theory, it is necessary first of all to determine what these neglects are and what limits the applicability of the basic physical concepts with which the given theory operates. One of the most general existing physical theories is quantum electrodynamics. This theory does not include in its consideration the nature of the atomic structure of matter, but takes it as an experimental fact. The structure of material particles is not considered in it, and these particles are characterized in aggregate by certain constants, in particular by their charge and mass. The mass of particles is an approximate concept. According to the theory of relativity, the mass \(m\) is connected with the energy \(mc^2\); but if two particles interact so strongly that their interaction energy becomes comparable with the energy \(mc^2\), then the concept of the mass of a particle loses its unambiguous meaning. Such a considerable interaction energy is observed in nuclear processes and is manifested in a violation of the strict additivity of masses; thus, for example, the mass of an \(\alpha\)-particle, consisting of two protons and two neutrons, turns out to be less than the sum of the masses of its constituent parts by an amount equal to their interaction energy divided by \(c^2\). For electrons, because of their small mass and large interaction energy, the concept of mass loses its meaning earlier than for heavier particles (protons and neutrons); therefore one cannot speak of electrons in the nucleus as of separate particles. All these intranuclear processes lie beyond the limits of quantum electrodynamics.

As for the charge of particles, it has the property of taking values that are multiples of a certain elementary charge, so that with respect to it additivity is apparently observed. But the number of material particles itself is not constant. Experiments of recent years have shown that, under the action of radiation of sufficiently high frequencies, pairs of particles with equal mass but charges of opposite sign (electrons and positrons) can be created.

If we do not consider intranuclear processes, are not interested in the structure of material particles, and neglect the possibility of pair creation*, then we may use quantum electrodynamics in the form in which it was developed, for example, in the works of Heisenberg and Pauli or of Dirac, mine and Podolsky.

The idea of quantum electrodynamics consists in the fact that an assembly of a given number \(n\) of material particles and an indefinite

* The possibility of pair creation can also be taken into account within the framework of the existing theory.

the number of light quanta is regarded as one system. This system, consisting of matter and the electromagnetic field, possesses an infinite number of degrees of freedom. If we denote the variables referring to an individual material particle (coordinates and the so-called spin) by the letter \(x\), and the variables referring to a light quantum (the wave vector and the polarization of the corresponding plane wave) by the letter \(k\), then the state of such a system can be described by means of the sequence of functions

\[ \psi_0,\ \psi_1,\ldots \psi_N,\ldots \tag{1} \]

where

\[ \psi_N=\psi(x_1,x_2,\ldots x_n;\ k_1,k_2,\ldots k_N), \tag{2} \]

where the function \(\psi_N\) must be symmetric with respect to the variables \(k_1,k_2,\ldots k_N\). Instead of this sequence of functions one may consider, as I have shown,\(^1\) a single quantity, namely the functional \(\Omega\) of a certain auxiliary function, \(\bar b(k)\). This functional has the form

\[ \Omega=\psi_0+\sum_{N=1}^{\infty}\frac{1}{\sqrt{N!}}\int \psi_N \bar b(k_1)\ldots \bar b(k_N)\,dk_1\ldots dk_N . \tag{3} \]

For the physical interpretation of this functional it suffices to give an expression for the “scalar product” of two functionals \(\Omega\) and \(\Omega'\). This scalar product \((\Omega,\Omega')\) has the same physical meaning as the scalar product

\[ (\psi,\psi')=\int \bar\psi\psi'\,d\tau \]

of two ordinary wave functions in quantum mechanics.

If the functional \(\Omega\) has the form (3), and the functional \(\Omega'\) is constructed in an analogous way with the aid of the functions \(\psi'_N\), then their scalar product \((\Omega,\Omega')\) will have the form

\[ (\Omega,\Omega')=\int dx_1\ldots dx_n \left\{\bar\psi_0\psi'_0+ \sum_{N=1}^{\infty}\int \bar\psi_N\psi'_N\,dk_1\ldots dk_N \right\}. \tag{4} \]

If here one sets \(\Omega'=\Omega\), so that \(\psi'_N=\psi_N\), then the individual terms of this expression can be interpreted as probabilities. Thus, for example, the integral

\[ \int dx_1\ldots dx_n\int \bar\psi_N\psi_N\,dk_1\ldots dk_N \tag{5} \]

is the probability that there are exactly \(N\) light quanta.

The time dependence of the functional \(\Omega\) is determined by an equation of the form

\[ H\Omega-ih\frac{\partial \Omega}{\partial t} = \sqrt{\alpha}\int \left\{ G^{+}(k)\frac{\delta'\Omega}{\delta \bar b(k)} + G(k)\bar b(k)\Omega \right\}\,dk, \tag{6} \]

where \(\dfrac{\delta'\Omega}{\delta \bar b(k)}\) is the functional derivative of \(\Omega\) with respect to \(b(k)\), defined as the coefficient of \(\delta \bar b(k)\) in the expression for the variation

\[ \delta \Omega=\int \frac{\delta'\Omega}{\delta \bar b(k)}\,\delta \bar b(k)\,dk . \tag{7} \]

The quantities \(G(k)\) and \(G^{+}(k)\) are certain known operators. The operator \(H\) standing on the left-hand side of equation (6) is the usual operator for the total energy of a system of \(n\) material particles,

\[ H=\sum_{s=1}^{n} T_s+\sum_{u>v=1}^{n}\frac{e_u e_v}{|r_u-r_v|}. \tag{8} \]

The right-hand side of equation (6), however, represents the operator of the energy of interaction of the particles with the radiation (taken with the opposite sign). On the right-hand side there stands, as a factor, the quantity \(\sqrt{\alpha}\), where \(\alpha\) is the so-called fine-structure constant,

\[ \alpha=\frac{e^2}{hc}=\frac{1}{137.3}. \tag{9} \]

The derivation of equation (6) is mathematically nonrigorous. For example, in forming the operator \(H\) [formula (8)] the calculations lead to a double sum in which there are terms with \(u=v\); these terms simply have to be subtracted, with the motivation that they represent an infinitely large constant. Nonrigorous procedures of this kind, which are a consequence of the approximations underlying the theory, are undoubtedly a major shortcoming of it. Nevertheless, in the present approximate theory these nonrigorous steps are physically necessary. They are essentially a crude device by means of which the deficiencies of the original formulation are corrected—deficiencies which, for example, fail properly to take into account the role of light quanta of very large energy.*

Not only the derivation of the wave equation (6), but also its solution, can in no way be called mathematically rigorous. Formally, the approximate methods for solving this equation are based on the smallness of the parameter \(\sqrt{\alpha}\) entering its right-hand side. Physically this smallness corresponds to the secondary role of the interaction,

* The assumptions of the theory undoubtedly cease to be valid at frequencies for which the energy of the quantum \(h\nu\) becomes of the order of \(\dfrac{mc^2}{\alpha}\), where \(m\) is the mass of the electron; meanwhile, in deriving the formulas of quantum electrodynamics one has to integrate with respect to \(\nu\) up to infinity.

particles with radiation in comparison with their interaction with one another—a fact established by experiment. However, the attempt to seek solutions of the wave equation in the form of a series arranged in powers of \(\sqrt{\alpha}\) leads to reasonable results only if one takes a small number of terms in this series, discarding terms of order higher than \(\alpha\) or higher than \(\sigma^2\). No physical meaning can be ascribed to the further terms, since they contain expressions of the following kind: a small coefficient multiplied by a divergent integral. Thus, in order to observe here a minimum of physical rigor, one has to proceed with mathematical rigor.

Nevertheless, in spite of all its shortcomings, the theory based on the wave equation (6) correctly describes a very extensive class of physical phenomena. It represents with a high degree of accuracy the processes of radiation by atoms and molecules and makes it possible to determine not only the frequencies but also the natural width of spectral lines. It leads to the correct formula for the scattering of light by free electrons (the Klein–Nishina formula). Finally, it gives corrections to Coulomb’s law of interaction between electrons; these corrections arise from the emission and absorption by electrons of light quanta (in the classical theory the corresponding correction was interpreted as taking account of the retardation of potentials).

1. From quantum electrodynamics, whose basic ideas we have tried to convey, we shall now pass to ordinary quantum mechanics.

The subject of quantum electrodynamics was, first of all, the formulation of the laws of interaction of material particles with radiation. When this interaction is taken into account, an atom in an excited state (i.e., possessing not the smallest possible energy, but a higher one) has a probability of emitting a light quantum and passing to a lower energy level. The duration of the atom’s stay in the excited state will not be unlimited, so that this state will not be strictly stationary (the natural width of spectral lines is also connected with this). If, however, we neglect the interaction of the atom with radiation, then we arrive at a new physical concept of the stationary state of an atom.

We obtain the possibility of speaking of the atom as a mechanical system in the proper sense. Thus there arises the concept of a mechanical system, connected with neglecting the interaction of matter with radiation.

But we can take one more step: neglect the corrections due to the theory of relativity. This neglect is permissible under the condition that the velocities of the constituent parts of the material system are small in comparison with the speed of light:

\[ \frac{v}{c} \ll 1. \tag{10} \]

For electrons in an atom the ratio \(\frac{v}{c}\) will be of the order

\[ \frac{v}{c}\sim \alpha=\frac{1}{137.3}, \tag{11} \]

so that for electrons this neglect is, strictly speaking, connected with neglecting radiation. (Let us recall that in formula (6) the energy of interaction of material particles with radiation contains the factor \(\sqrt{\alpha}\).)

Neglecting the corrections from the theory of relativity is connected with the acquisition of new physical concepts concerning the separateness of space and time and absolute simultaneity.

The omissions made lead us to a simplified formulation of the problem: from quantum electrodynamics we pass to quantum mechanics (in the proper sense) and to Schrödinger’s theory. Formally, neglecting radiation corresponds to striking out the right-hand side of equation (6) and replacing the functional \(\Omega\) by its “zero” term of the expansion \(\psi_0\). Then, neglecting the corrections from the theory of relativity, we arrive at the ordinary Schrödinger equation

\[ H\psi=ih\frac{\partial\psi}{\partial t}, \tag{12} \]

in which the energy operator has the form (8). If to this operator one adds the potential energy of particles in an external field [omitted in equation (8)], then the operator \(H\) will take the form

\[ H=-\sum_{s=1}^{n}\frac{h^2}{2m_s}\Delta_s+\sum_{s=1}^{n}U_s(x_s)+\sum_{\mu>\nu=1}^{n}\frac{e_\mu e_\nu}{|r_\mu-r_\nu|}, \tag{13} \]

where \(\Delta_s\) is the Laplace operator acting on the coordinates of particle number \(s\).

A theory based on Schrödinger’s equation is much more satisfactory mathematically than quantum electrodynamics. Here the problem can be formulated quite rigorously (in the mathematical sense). The non-rigorous points allowed in solving it are due exclusively to the complexity of the problem, and not to shortcomings of the initial equations, as is the case in quantum electrodynamics.

In the physical respect this theory—quantum mechanics in the proper sense—constitutes a complete logical scheme, operating with concepts that are defined with sufficient rigor. In all those cases where the assumptions underlying this theory prove to be fulfilled (the chief of which were enumerated above by us), calculations carried out on the basis of quantum mechanics, provided only that they are carried out with sufficient accuracy, lead to complete agreement with experiment. The assumptions of the theory

are carried out in the majority of problems concerning atoms and molecules; thus quantum mechanics embraces the theory of the structure and interaction of atoms and molecules, including all of chemistry.

According to quantum mechanics, the state of a mechanical system is described by a wave function \(\psi\), satisfying the Schrödinger equation (12). Knowledge of the wave function gives us all information about the system that can be obtained as a result of a definite maximally precise experiment upon it. By a maximally precise experiment we mean one as a result of which the values are obtained of all those mechanical quantities that in general can be measured simultaneously (measurements of different quantities may interfere with one another). Maximally precise experiments may be different, depending on the choice of the quantities measured. It may also be said that the physical meaning of the wave function consists in the fact that it represents a record of information about the system obtained as a result of a definite maximally precise experiment. The meaning of the wave equation (12), in turn, consists in the fact that it makes it possible to pass from experimental data or information referring to the initial instant of time (the initial value of the wave function), to information referring to a later instant of time (the value of \(\psi\) at time \(t\)). Information about the system recorded in the form of the wave function also makes it possible to compute the probabilities of various results of a subsequent measurement of different quantities, as well as the mathematical expectations of these quantities.

We shall not dwell here on the question of recording data obtained from experiment for the case in which this experiment is not maximally precise*, but shall draw attention to the following circumstance, which is important for our analysis of the process of formation of physical concepts.

In quantum mechanics the concept of the state of a system merges with the concept of the maximally precise information or data that can be obtained about the state. Connected with this is the fact that the laws of quantum mechanics lead to the conclusion of the impossibility of an objective description of the detailed course of physical processes. Indeed, description by means of the wave function is not objective in the usual sense**. This is seen especially clearly in the example of the so-called spreading of a wave packet. According to quantum mechanics, a free particle whose initial position and momentum are known with the accuracy permitted by the Heisenberg inequality

\[ \Delta p \Delta x \gg h, \tag{14} \]

* The mathematical side of this question is treated in detail in J. von Neumann’s book\(^2\).

** On this question see the papers by A. Einstein and N. Bohr, which were published together with the author’s introductory article under the general title “Can the quantum-mechanical description of physical reality be considered complete?”\(^3\).

is described by the function \(\psi\), which represents a “wave packet,” i.e., is appreciably different from zero only in some not sharply bounded region of space. This packet consists of a superposition of plane waves, whose directions and frequencies are not quite the same. The Schrödinger equation shows that with the passage of time this packet spreads out, i.e., the wave function becomes different from zero over an ever broader region. Meanwhile, since the particle is free, clearly no objective changes take place with it in this process. What occurs is only the loss of our information about the localization of the particle in space. (This is connected with the fact that, in the language of classical mechanics, the free motion of a particle is unstable in the Lyapunov sense.) Hence it is clear that the description of a state by means of a wave function is not objective. At the same time, this description fully conveys everything that we can obtain as a result of maximally precise measurements, and is therefore an expression of objectively existing laws of nature. The necessity of precisely such a nonobjective description follows from the impossibility of reconciling in any other way the wave and corpuscular nature of matter, both of which have been firmly established by experience.

Another characteristic feature of quantum mechanics is that knowledge of the state (wave function) of an entire system, for example of an entire atom, does not yet imply knowledge of the states (wave functions) of the individual parts of the system, for example of the individual electrons in the atom. Connected with this is the fact that an experiment which is maximally precise with respect to the entire system is not such with respect to the individual parts of the system.

Thus in quantum mechanics there already exist the concepts of the mass and charge of particles, of a mechanical system in the proper sense (i.e., one not dependent on radiation), and of stationary states of a system (for example, an atom or a molecule), but there are as yet no concepts of the state of the individual particles forming the system (electrons in an atom), nor of an objective description of the detailed course of physical processes.

1. We have clarified with what physical concepts quantum mechanics operates. Let us now turn to the consideration of approximate methods for solving problems of quantum mechanics and to clarifying those new physical concepts that arise in the process.

The problem of determining the stationary states of systems reduces to solving an equation of the form

\[ H\psi = E\psi, \tag{15} \]

i.e., to finding the eigenfunctions of the energy operator \(H\). If the system consists of identical particles, for example of electrons located in the field of an atomic nucleus, then the wave function \(\psi\) must satisfy, in addition to equation (15), the requirement of antisymmetry: it must change sign when the arguments belonging to two electrons are interchanged. For example:

\[ \psi(x_1, x_2,\ldots,x_n) = -\psi(x_2, x_1,\ldots,x_n), \tag{16} \]

This condition, which is an expression of the Pauli principle, is compatible with equation (15), since in the case of identical particles the energy operator \(H\) is symmetric with respect to them. In the case of electrons each argument \(x\) of the function \(\psi\) represents the set of three spatial coordinates \(x, y, z\) and one more variable \(\sigma\), taking only two values. This variable corresponds, in a known sense, to the orientation of the electron (the so-called spin). Thus in the case of \(n\) electrons it is required to find one function of the variables

\[ x_s, y_s, z_s, \sigma_s \quad (s = 1, 2, \ldots n), \tag{17} \]

or, what is the same thing, \(2^n\) functions of \(3n\) variables

\[ x_1, y_1, z_1; \ x_2, y_2, z_2; \ \ldots \ x_n, y_n, z_n. \tag{18} \]

In order to obtain an idea of the degree of complexity of this problem, it is enough to recall that, for example, for the sodium atom \(n = 11\), so that it is necessary to find \(2^{11} = 2048\) functions, each of 33 variables, while for the copper atom \(n = 29\), and there it is necessary to determine more than half a billion functions of 87 variables.

It is quite clear that a rigorous solution of such a problem is impossible and that it is necessary to resort to approximate methods.

One of the principal methods used in quantum mechanics for solving problems of this kind is the well-known Ritz method, or rather its modification. The possibility of applying it is based on the fact that the problem under consideration can be formulated in the form of the following variational problem: it is required to find the minimum of the integral

\[ W = \int \psi H \psi \, d\tau, \tag{19} \]

representing the energy of the atom, under the condition

\[ \int \psi \psi \, d\tau = 1 \tag{20} \]

i.e. under the condition of normalization of the wave function.

The required function \(\psi\) can be approximately represented in the form of a linear combination of products of functions \(\varphi_s(x_n)\), each depending on the variables of one electron. In order to satisfy the Pauli principle (16), it is necessary to take the antisymmetric combination

\[ \psi = \frac{1}{\sqrt{n!}} \sum_{\alpha_1 \ldots \alpha_n} \varepsilon_{\alpha_1 \ldots \alpha_n} \varphi_{\alpha_1}(x_1) \cdots \varphi_{\alpha_n}(x_n), \tag{21} \]

where \(\varepsilon\) is a quantity antisymmetric with respect to its indices, with \(\varepsilon_{123\ldots n} = 1\). The functions \(\varphi_s(x)\) may be assumed orthogonal and normalized:

\[ \int \varphi_s(x)\varphi_r(x)\,dx = \delta_{sr}. \tag{22} \]

If in (19) one substitutes the expression (21) for \(\psi\) (which may also be written in the form of a determinant), then the energy of the atom \(W\) is obtained,

expressed through the functions $\varphi_s(x)$. Varying the energy $W$ with respect to the functions $\varphi_s(x)$ under the supplementary conditions (22), we obtain for these functions equations of the form

\[ -\frac{h^2}{2m}\Delta \varphi_s(x) + U_s(x)\varphi_s(x) - A_s\varphi_s(x) = E_s\varphi_s(x). \tag{23} \]

Here $U_s(x)$ is a certain function of the coordinates, depending on all the unknown functions except $\varphi_s(x)$; the quantity $U_s(x)$ plays the role of potential energy. The symbol $A_s$ denotes a certain integral operator, likewise depending on all the unknown functions except $\varphi_s(x)$. Thus, if all functions except $\varphi_s(x)$ are regarded as known, then for $\varphi_s(x)$ one obtains a linear integro-differential equation.

Equations (23) were first derived by me in 1929–1930. Let us note that in the cases calculated (atoms of lithium, sodium, etc.) the energy levels obtained are very close to the experimental ones; the error does not exceed 1–2%. Thus the approximate method under consideration gives a fairly accurate formulation of our very complex physical problem.

What, then, is the fundamental significance of this method? In it the wave function of the whole atom is approximately expressed through the wave functions of the individual electrons. But this means that we have acquired a new physical concept, namely the concept that the electrons in the atom are also in definite states. (As we noted above, in the exact Schrödinger theory this concept was absent.)

This concept is especially important because Bohr’s scheme of the electronic shells of the atom, explaining the structure of Mendeleev’s periodic system of elements, is built on its basis.

But this is not all. In our equations (23), besides the term $U_s(x)$, which can be interpreted as the potential energy arising from the atomic nucleus and from the remaining electrons, there enters one more term, namely $A_s$. This term represents a new kind of energy, unknown both to classical mechanics and to the exact Schrödinger theory, namely the so-called energy of quantum exchange. (This name arose from the fact that the term $A_s$ is obtained as a consequence of taking into account the Pauli principle, which is the expression of the identity of electrons, i.e. of the fact that nothing changes if two electrons exchange places.)

Thus we have acquired yet another new physical concept—the concept of the energy of quantum exchange. This concept plays a very large role both in the theory of atoms (discarding the term $A_s$ increases the error from 1–2% to 20–30%), and especially in the theory of molecules (this concept was first introduced by Heisenberg and Heitler in application to molecules). It turns out that the very existence of homopolar molecules (composed of identical atoms) cannot be explained without taking into account the energy of quantum exchange. Thus this new concept plays a decisive role in quantum chemistry,

1. A further and, perhaps, the most important method for the approximate solution of problems of quantum mechanics is perturbation theory. Its idea consists, as is well known, in the fact that the Hilbert space characterizing the manifold of all possible states of the system and possessing an infinite number of dimensions is replaced by a certain subspace with a finite number of dimensions. This subspace is chosen so that it corresponds precisely to those states of the system which play the most important role in the given problem. If the number of dimensions of this subspace is in turn large, then in order to investigate it one has to resort to special methods, most often to the methods of group theory.

A simplifying circumstance in investigations of this kind is, first, the property of antisymmetry of the wave function, and, second, the fact that although the wave function itself depends on spin variables, the energy operator in the Schrödinger approximation does not depend on them. This allowed Dirac to formulate the problem of perturbation theory by introducing operators (finite matrices) acting on the spin variables. These operators may be interpreted as spin angular momenta of individual electrons, and the corresponding terms in the expression for the energy—as the energy of interaction of these spin momenta. This new physical concept proves to be extremely useful in interpreting the spectra of complex atoms.

One of the most remarkable applications of perturbation theory, based on the same ideas, is the theory of spin invariants proposed by Weyl. The mathematical scheme of this theory proves to be completely equivalent to the scheme of chemical compounds obtained on the basis of the theory of valence*. Thus this application of the approximate methods of perturbation theory leads to a new physical (or, if one likes, chemical) concept of chemical valence.

We shall now pass to the domain bordering between quantum mechanics and classical mechanics, and, for simplicity, consider the Schrödinger equation for one particle

\[ -\frac{h^2}{2m}\Delta\psi+U(x,y,z)\psi=ih\frac{\partial\psi}{\partial t}. \tag{24} \]

If the corresponding classical motion of the material particle is such that for it the inequality

\[ \frac{mv^3}{w}\gg h, \tag{25} \]

holds, where \(v\) is the velocity of the particle and \(w\) is its acceleration, then the solution of the Schrödinger equation (24) can, with a high degree of approxi-

* See, for example, the article by M. Born.

is expressed through the solution of the classical Hamilton–Jacobi equation

\[ \frac{1}{2m}(\operatorname{grad} S)^2+U(x,y,z)-\frac{\partial S}{\partial t}=0. \tag{26} \]

If \(S\) is a complete integral of equation (26), containing three arbitrary constants \(c_1, c_2\), and \(c_3\), then the wave function \(\psi\) is approximately expressed in terms of \(S\) in the following way:

\[ \psi=\sqrt{\left\lVert \frac{\partial^2 S}{\partial x_i \cdot \partial c_k}\right\rVert}\cdot e^{\frac{i}{h}S}. \tag{27} \]

For stationary states one may put

\[ S=-Et+V(x,y,z) \tag{28} \]

and replace the exponential function by an expression of the form

\[ e^{-\frac{i}{h}Et}\cos\left(\frac{V}{h}+\alpha\right), \tag{29} \]

where \(\alpha\) is a constant phase.

This approximate method, due to Wentzel and Brillouin, leads us to the so-called old quantum mechanics and to all the physical concepts connected with it. From the boundary conditions for the wave function one obtains the quantization rules characteristic of this theory.

Finally, in some phenomena the restrictions imposed by the Heisenberg inequalities (14) may become insignificant, as a result of which the wave character of matter recedes into the background. Then classical mechanics, with the notions familiar to us, comes into force. New (and historically old) physical concepts appear concerning the objective description of the course of processes, the trajectory of a particle, and the fact that every mechanical quantity always has a definite value.

Our survey, in which the course of the historical development of mechanics and physics has to some extent been reversed, may perhaps help in understanding the historical course of the development of physical concepts as well. We wanted to show that every physical theory, every physical concept, is in essence only approximate. Every major advance in physical science is connected not only with the creation of new concepts, but also with a critical reexamination of the old ones. And if it is thereby proved that some of the old concepts are inapplicable to newly discovered phenomena, then one must part with them without regret.

References

1. V. Fock, Zur Quantenelektrodynamik, Sow. Phys., 6, 425, 1934.
2. J. v. Neumann, Mathematische Grundlagen der Quantenmechanik, Berlin, Springer, 1932.
3. V. Fock, A. Einstein, B. Podolsky, and N. Rosen, On Problems of Physical Sciences, 16, 436, 1936.
4. M. Born, Chemical Bond and Quantum Mechanics, ONTIVU, Kharkov, 1932.