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Spectral Formulas and Their Graphical Representation
G. S. Landsberg.
At the present time it may be considered well known that the fundamental difficulties standing in the way of constructing a model of the atom are, in essence, difficulties connected with the possibility of applying the classical laws of mechanics and electrodynamics—that is, laws derived for the case of comparatively slow motions and oscillations—to the case of those extremely rapid periodic processes that take place within the atom. And indeed, the flawless application of these classical laws led, as is known, to the law of the distribution of energy in the spectrum of an absolutely black body, a law unquestionably different from those given by experiment. H. A. Lorentz formulates the impotence of classical theory in the following terms: “In creating the theory of thermal radiation, one cannot fully trust the equations of classical mechanics and electrodynamics. These equations are not capable of explaining why a cooling stove does not emit, for example, yellow rays, whereas it emits radiation of greater wavelength.”
The way out of the difficulties indicated by M. Planck consists in the assumption that not every state of a system consistent with the laws of mechanics and electrodynamics is possible. The possible states must, in addition, satisfy the prescriptions of the quantum hypothesis. The restrictions imposed by quantum theory may be formulated in one way or another, depending on the character of the problem with which we are dealing. Historically, the first formulation, which enabled Planck to resolve the indicated contradiction between experiment and the theory of black radiation, consisted in the assertion that radiant energy possesses, to a certain degree, an atomistic structure. This means that the distribution of energy among various systems—for example, among atomic resonators and radiation—cannot proceed in arbitrary portions, as would take place with an unlimited divisibility of energy; in its distribution, each system always receives some whole number of individual elementary portions, or quanta, of energy, which ...
play the role of energy atoms. Characteristic of such a structure is the circumstance that the magnitude of the quantum is determined by the frequency of the kind of rays under consideration and is equal to \(h\nu\), where \(\nu\) is the frequency, and \(h\) is a constant universal for all kinds of rays, associated with the name of M. Planck.
Thus, among all possible states of the system, only those states will have real significance which satisfy the indicated restriction: in passing from one state to another, the energy reserve of the system changes not arbitrarily, but by an amount necessarily representing one or several whole quanta. These discrete states, of which the system proves capable, can be established by adding to the ordinary laws of mechanics and electrodynamics an additional quantum requirement, whose form for periodic and similar processes is rather simple. Let us consider, for example, a system with one degree of freedom, i.e. a system whose state is determined by a single coordinate \(q\). If, for example, our system represents a point moving in a circle, then such a coordinate could be the angle \(\varphi\) of the radius vector directed toward the point with some specified direction. For a point performing harmonic oscillations along a straight line (a linear oscillator), the coordinate could be the distance of the point from the position of equilibrium (elongation), etc. It is known that, among the various forms that can be given to the equation of motion of our system, the canonical form of Hamilton’s equations is distinguished by the greatest generality and simplicity. In this form, as the quantity by means of which the motion of the system can be characterized, one chooses not the rate of change of the coordinate \(q\) (i.e. \(\dot q = \dfrac{dq}{dt}\)), but a quantity \(p\) connected with this velocity, called “momentum” and defined by the condition:
\[ p=\frac{\partial L}{\partial \dot q} \]
where \(L\) is the so-called Lagrangian function of the motion. For most simple mechanical problems this Lagrangian function is the difference between the kinetic and potential energy of the system, i.e.
\[ L=T-U. \]
Consequently,
\[ p=\frac{\partial T}{\partial \dot q}, \]
since \(U\) depends only on \(q\) and does not depend on \(\dot q\). With the indicated choice of parameters (\(q\)—the coordinate determining the state of the system; \(p\)—co-
corresponding momentum, characterizing the change of state, i.e. the process under consideration) the equation of motion may be given the following “canonical” form:
\[ \frac{dq}{dt}=\frac{\partial H}{\partial p} \]
\[ \frac{dp}{dt}=-\frac{\partial H}{\partial q}, \]
where \(H\) is Hamilton’s function, equal, for the cases discussed above, to the sum of the kinetic and potential energies, i.e.
\[ H=T+U. \]
Solving this system of equations, we determine \(q\) and \(p\) as functions of time, i.e. we solve the problem of the motion of our point. However, the quantum condition says that not all values of \(H\) are possible. Accordingly, not every combination of \(q\) and \(p\) satisfying the equations of motion corresponds to a real state of the system. The quantum condition imposes a new restriction on \(p\) and \(q\), which must be appended to the equations of motion. This restriction, as Planck and Sommerfeld showed, may be expressed in the following form:
\[ \oint p\cdot dq=n\cdot h, \]
where the integration extends over the whole region of variation of the variable \(q\), \(n\) is an integer \((1, 2, 3\ldots)\), and \(h\) is Planck’s constant. Thus, for the type of periodic motions under consideration, the quantum condition takes a simple mathematical form.
In atomic theory of great importance is the simple case of periodic motion with one degree of freedom, consisting in the uniform rotation of an electron with charge \(-e\) about a stationary nucleus with charge \(+E\). For this case the coordinate \(q\) is the angle \(\varphi\); the kinetic energy is
\[ T=\frac{m_0 v^2}{2}. \]
The linear velocity
\[ v=r\cdot \frac{d\varphi}{dt}=r\cdot \dot{\varphi}, \]
since \(\dot{\varphi}\) is the angular velocity of the motion. Consequently,
\[ T=\frac{m_0 r^2\dot{\varphi}^{\,2}}{2};\qquad p=\frac{\partial T}{\partial \dot{\varphi}}=m_0 r^2\dot{\varphi} \]
or
\[ \dot{\varphi}=\frac{p}{m_0 r^2} \]
and, consequently,
\[ T=\frac{p^2}{2m_o r^2}. \]
The potential energy
\[ U=-\frac{eE}{r}. \]
The total energy, or Hamiltonian function,
\[ H=T+U=\frac{p^2}{2m_o r^2}-\frac{eE}{r}. \]
Equations of motion:
\[ \frac{d\varphi}{dt}=\frac{\partial H}{\partial p}=\frac{p}{m_o r^2} \]
\[ \frac{dp}{dt}=-\frac{\partial H}{\partial q}=\frac{\partial H}{\partial \varphi}=0. \]
The second of them gives:
\[ p=\mathrm{const}, \]
i.e. \(p\) does not depend on time.
The quantum requirement states:
\[ \oint p\,d\varphi=n\cdot h, \]
where the integral is taken over the entire range of variation of \(\varphi\).
Since \(p\) does not depend on time, and consequently also not on \(\varphi\), it can be taken outside the integral sign. The complete range of variation of the variable \(\varphi\) is from \(0\) to \(2\pi\); thus the integration must be carried out within the indicated limits. We obtain:
\[ p\int_0^{2\pi} d\varphi=nh \]
or
\[ p\cdot 2\pi=n\cdot h, \]
i.e.
\[ p=\frac{nh}{2\pi}. \]
Since
\[ p=m_o r^2\dot{\varphi}=m_o r\cdot\dot{\varphi}\cdot r=m_o vr, \]
the quantum constraint found for the case of the motion of a point in a circle gives:
\[ m_o vr=\frac{nh}{2\pi}. \]
relation encountered in the elementary theory of the atom. The quantum requirement therefore makes it possible, in this case as well, to select from all conceivable mechanical and electrodynamic motions of the electron about the nucleus the discrete, actually existing motions, for for \(r\), and consequently also for \(v\), only those values are possible which in the preceding formula correspond to integral values of \(n\).
These individual states of the atomic system will be characterized by definite values of the energy \(W_1, W_2, W_3\), etc. As is known, the first postulate of Bohr’s theory of the atom precisely asserts the existence of such discrete stationary states characterized by definite values of the energy. Since the nearest stationary state is a state whose energy differs from that of the first by a finite amount, a transition from one state to another is possible only with the emission of a finite portion of energy. The radiation of an infinitely small fraction of energy, which according to classical electrodynamics must continuously accompany the nonuniform motion of the electron, evidently cannot take place, for it would compel the system to pass successively through a series of states impossible from the point of view of quantum theory. Therefore Bohr’s first postulate contains a rejection of classical radiation. It is replaced by quantum radiation, occurring from time to time and, moreover, in finite portions expressing the difference of energy in the initial and final stationary states.1 The frequency of this radiation is determined by Bohr’s second postulate (the frequency condition), which asserts that the emitted energy is each time one quantum of energy, i.e. is a monochromatic ray of frequency \(\nu\) such that
\[ h \cdot \nu_{m,n}=W_m-W_n \quad \text{or} \quad \nu_{m,n}=\frac{W_m}{h}-\frac{W_n}{h} \tag{1} \]
In connection with the existence of individual stationary states, to which different values of the energy correspond, we shall have a series of possible transitions of the system and, consequently, a series of separate monochromatic lines; they constitute the spectrum of our system, and the formula written above thus represents a spectral formula obtained by theoretical means. We see that it is characterized by the presence of the difference of two terms, i.e. precisely by that feature which is the distinctive trait of all empirical spectral formulas, beginning with the famous Balmer formula.
Introducing for \(W_m\) and \(W_n\) their expressions for that simple case amenable to elementary calculation, when the system consists of a positively charged nucleus and one electron revolving around it, Bohr, as is known, was able to give his spectral formula such a form that it proved to be in complete numerical agreement with the formulae expressing the spectrum of hydrogen (the Balmer and Lyman series) and the spectrum of ionized helium (the Pickering series1). Bohr’s arguments therefore invested the empirical spectral formulae with a precise physical meaning: each of the two members (terms) of the spectral formula denotes an energy store characterizing one of the two states of the atomic system (initial or final), the transition between which is accompanied by an act of emission or absorption of the corresponding wavelength.
However, simple periodic motion can occur only in a system consisting merely of two bodies (nucleus and electron) interacting exactly according to Coulomb’s (or Newton’s) law. Any change in such a system will become a source of perturbing actions, and in the presence of a perturbing force the motion will cease to be simple periodic motion along a Keplerian ellipse (or circle). Any distortion of the central force acting according to the law \(f=\frac{k}{r^2}\) will lead to the appearance of a perturbed motion which will no longer be simple periodic, but will present itself as an aggregate of at least two periodic motions, each with its own period, these periods being incommensurable with one another. Otherwise, after an interval of time containing an integral number of periods of the first and second kind, our system would return to its original state, the cycle of changes would begin to repeat itself, i.e. the motion would degenerate into a periodic one, and the indicated interval of time would be its period.
Such a complex motion, representing a superposition of two independent periodic motions, will take place, for example, if the central Coulomb force of the nucleus is distorted by the action of a suite of electrons which revolves between the nucleus and the outer electron. The screening effect of the inner electrons acts on the elliptical orbit in such a way that it begins to rotate more or less rapidly in its own plane, so that the direction of the major axis will occupy ever new positions (rotation of the perihelion). The form of the orbit will be similar to that shown in Fig. 1, and the orbit will not close if the periods are incommensurable. Such a motion is an example of conditionally periodic motion. In the case of commensurability of the periods it degenerates into periodic motion.
The very same effect is also produced by that small perturbing force which arises as a result of the change of the electron’s mass with velocity—an circumstance used by Sommerfeld1 to explain the fine structure of the hydrogen spectrum.
The case described corresponds to the superposition of two periodic motions: one is the motion of the electron in an elliptical orbit (if there were no perturbation); the second is the rotation of the orbit itself.
Fig. 1.
Each such motion is characterized by its own period, during which there is completed a cycle of full changes of the coordinate characterizing the motion. Thus, the radius vector runs through all the values corresponding to it in the time \(\tau_1\), which represents the period of revolution along the ellipse; in the time \(\tau_2\), on the other hand (the period of revolution of the perihelion), the angle characterizing the position of the major axis of the ellipse (the line of apsides) changes by \(2\pi\). In accordance with the two periodic motions of which the process consists, it will be necessary to add to the general mechanical and electrodynamical conditions two quantum conditions of the same type as above, i.e.
\[ \oint p_1\,dq_1 = nh \]
\[ \oint p_2\,dq_2 = kh, \]
where \(q_1\) and \(q_2\) are two coordinates determining the two simple periodic motions; \(p_1\) and \(p_2\) are the corresponding momenta, and \(n\) and \(k\) are two quantum (integer) numbers. Each possible state of the system is therefore characterized by a pair of quantum numbers \(n\) and \(k\), and the energy of this state is a function of two numbers
\[ W_1 = W(n_1,k_1);\quad W_2 = W(n_2,k_2). \]
In accordance with this, Bohr’s spectral formula takes the form:
\[ \nu=\frac{W(n_1,k_1)}{h}-\frac{W(n_2,k_2)}{h}. \tag{2} \]
Of the two quantum numbers, the first, \(n\), characterizes, from the kinetic side, as we have seen, the periodic motion of the electron in the orbit; it is called the total quantum number; its magnitude
has predominant significance for the determination of energy: for a given \(n\), the energy changes comparatively little in passing from one value of \(k\) to another.
The number \(k\) is usually called the azimuthal quantum number; it determines the kinetically periodic rotation of the orbit in its plane (the rotation of the perihelion); its magnitude has, generally speaking, only secondary significance for the determination of the energy of the system: it introduces only a correction to the spectral term, which is determined chiefly by the value of the principal azimuthal number \(n\).
If one compares these results, obtained from theoretical considerations, with the empirical data of spectral symbolism, then the internal similarity of formula (2) to the spectral formula of Rydberg, describing the regularities in spectra more complex than the spectrum of hydrogen, is immediately striking.
It is known that the spectrum of hydrogen—the simplest of all spectra—consists of separate lines arranged in such a way that the distance between them gradually decreases in passing from the red end to the violet—the lines accumulate toward the violet boundary of the spectrum. Balmer was the first to point out that the frequencies corresponding to all these lines can be embraced by one simple formula; in other words, all these lines constitute a certain connected whole, which received from Balmer the name of a series. Balmer’s formula has the form:
\[ N = R\left(\frac{1}{2^2} - \frac{1}{m^2}\right), \quad m = 3, 4, 5\ldots \]
Here \(N\) is the wave number, i.e. the number indicating how many waves fit within the length of one centimeter. Thus
\[ N = \frac{1}{\lambda}, \]
if \(\lambda\) is the wavelength. This number is simply connected with what is usually called the frequency of radiation \((\nu)\), and which indicates the number of waves emitted by the source in one second. If the velocity of light is denoted by \(C\), then
\[ C = \nu \cdot \lambda \]
or
\[ \nu = \frac{C}{\lambda}. \]
Hence it is clear that \(\nu = C \cdot N\), and, consequently, Balmer’s formula makes it easy to calculate the ordinary frequencies of spectral lines.
\(R\) is a constant equal to 109 678. Giving \(m\) integral values beginning with 3 \((3, 4, 5\ldots)\), we obtain the wave numbers, and therefore also the frequencies, of all the lines making up the Balmer spectral series.
Subsequently it was discovered that hydrogen possesses two more series, situated chiefly in the infrared and ultra-
in the violet parts of the spectrum. These series are expressed by entirely analogous formulas:
\[ N=R\left(\frac{1}{1^{2}}-\frac{1}{m^{2}}\right),\quad m=2,3,4\ldots \quad \text{Lyman series;} \]
\[ N=R\left(\frac{1}{3^{2}}-\frac{1}{m^{2}}\right),\quad m=4,5,6\ldots \quad \text{Paschen series,} \]
where \(R\) has its previous value.
Obviously, all three hydrogen series can be covered by a formula of the form
\[ N=R\left(\frac{1}{n^{2}}-\frac{1}{m^{2}}\right)=\frac{R}{n^{2}}-\frac{R}{m^{2}}. \]
The first term is a constant term for each series, while the second is a variable one. Choosing \(n\) equal to 1, 2, or 3 and giving \(m\) all possible integral values, we obtain any of the three hydrogen series.
Rydberg showed that more complex spectra, for example the spectra of the alkali or alkaline-earth metals, can be decomposed into several independent series, each of which has a resemblance to the hydrogen series described above. Like the latter, the series established by Rydberg can be represented by formulas composed of the difference of two terms: one constant for the whole series and one variable, i.e. changing in passing from one line of the series to another line of the same series. But, unlike the hydrogen formula, each term is a function of two variables, the integer \(m\) and a certain parameter \(a\), characteristic for each atom. For each atom one can count several terms with definite values of the parameters. Depending on whether the integer entering into the expression of the term is constant or variable, the term plays the role of the constant or variable member of the serial formula. Combinations of the various terms give all possible series of the given element. The structure of each term is also entirely analogous to the terms of the hydrogen series: they have the form
\[ \frac{R}{(m+a)^{2}} \]
where \(R\) is again the same constant as above, usually called the Rydberg constant1.
According to Paschen, in what follows we shall use the abbreviated symbolic notation, by the convention:
\[ \frac{R}{(m+a)^{2}}=m,a. \]
Naturally the question arises as to how the whole complex aggregate of lines that make up, for example, the spectrum of calcium is to be broken down into separate integral series. Indeed, this is a problem requiring a very great deal of combinatorial skill and ingenuity. It is not for nothing that Paschen says that it is hopeless to try to find series by examining tables of spectra “unless one is a Rydberg.” But, of course, there are a number of physical features that facilitate this work. Since the lines making up one series have some common roots, there must naturally be some resemblance among them. Indeed, some spectral lines are sharply defined, others are blurred on one side or another, and sometimes on both sides. Many lines occur only in the form of groups—doublets, triplets, etc. (in general, multiplets), and the distances between the members of the group either remain unchanged throughout the spectrum or change in a regular manner. Finally, those characteristic changes to which spectral lines are subjected in an electric or magnetic field (splitting in the Stark or Zeeman effect) prove to be different for different series and similar for lines of one and the same series. All these external signs of internal kinship are extremely useful in assigning lines to series. In recent times experimentalists, guided by the theory of atoms, have learned sometimes to realize conditions that make it possible to call forth, gradually, one line of a given series after another (Ray, Fochtbauer, Wood), or to establish which line is to be regarded as the fundamental line of a series (Franck, Grotrian), which of course at once solves the problem of finding the series. Finally, the very latest theoretical and experimental investigations concerning the character of the distribution of intensity among the lines of complex multiplets have put into our hands, in some cases, methods of establishing serial relations not by guesswork, but in a quite rational way (Sommerfeld, Ornstein, Burger, Dorgelo, Laporte).
In one way or another, for a rather large number of elements—although far from all—these series dependences have been established and, consequently, the parameters entering into the terms have been established.
Thus, for example, the variable term of the so-called principal series (Prinzipal Serie) of the alkali metals can be written in the form:
\[ \text{Principal series P.S.} \ . \ . \ . \ . \ m,p, \]
where \(m\) symbolizes an integer, successively taking the values \(2,3,4\ldots\), and \(p\) is the value of the parameter \(a\), repeated in all the variable terms corresponding to the different lines of the principal series.
In exactly the same way, for the series consisting of diffuse lines and called the diffuse or first subordinate series (I Neben Serie), the variable term has the form:
First subordinate series I N.S. . . . . . \(m,d\)
For the second subordinate series II N.S. . . . . . \(m,s\)
For the Bergmann series B.S. . . . . . \(m,b\), etc.
The structure of the constant terms has the form:
\[ \begin{array}{rcl} \text{For } \mathrm{P.S.} & \ldots & 1,s\\ \text{” } \mathrm{I\,N.S.} & & 2,p\\ \text{” } \mathrm{II\,N.S.} & & 2,p\\ \text{” } \mathrm{B.S.} & & 3,d\ \text{etc.} \end{array} \]
The complete symbols of the series are:
\[ \begin{array}{rcl} \mathrm{P.S.} & \ldots & 1,s — m,p,\quad \text{where } m=2,3,4\ldots,\\ \mathrm{I\,N.S.} & & 2,p — m,d,\quad \text{” } m=3,4,5\ldots,\\ \mathrm{II\,N.S.} & & 2,p — m,s,\quad \text{” } m=3,4,5\ldots,\\ \mathrm{B.S.} & & 3,d — m,b,\quad \text{” } m=4,5,6\ldots\ \text{etc.} \end{array} \]
Thus the wave number (and consequently also the frequency) of any line of the principal series, for example, can be calculated by the formula:
\[ N=R\left(\frac{1}{(1+s)^2}-\frac{1}{(m+p)^2}\right),\quad \text{where } m=2,3,4\ldots, \]
if for \(R\) one substitutes its numerical value given above, and for \(s\) and \(p\) takes the values characteristic of the element under study. Comparing this formula with formula (2), we see that of the two quantum numbers of the latter the principal quantum number plays the role of the integer appearing in the constant or variable terms of the Rydberg formula, while the azimuthal quantum number determines the parameters whose combination makes it possible to distinguish the principal, 1st subordinate, 2nd subordinate, etc., series of the elements. We thus arrive at the idea that to each such parameter there corresponds one or another value of the azimuthal quantum number. It turns out that Bohr’s symbolism and Rydberg’s symbolism give coinciding results if we establish that
\[ \begin{array}{rcl} \text{to the parameter } s & \text{there corresponds} & k=1,\\ \text{” } p & \text{”} & k=2,\\ \text{” } d & \text{”} & k=3,\\ \text{” } b & \text{”} & k=4\ \text{etc.} \end{array} \]
However, the presence of several series in the spectrum of complex elements does not yet exhaust the complexity of these spectra. Some of these series consist, as has already been mentioned, not of simple lines, but of groups: doublets, triplets, etc. In other words, in Rydberg’s formula each term may have not one, but several values close to one another, so that the formula determines several lines differing little in frequency and therefore constituting a group.
with different values of the parameter, e.g. \(p_1, p_2, p_3\). In the sense of formula (2), this would mean that for our atom, for given \(n\) and \(k\), several different states are possible with slightly differing values of the energy.
This suggests that the conditions of motion in such atoms are more complicated, and that, besides the two described types of periodic motions, new ones are possible, corresponding to new degrees of freedom and, consequently, to new quantum numbers. Indeed, the atom is a complex system consisting of a nucleus and several inner groups of electrons rotating about it, arranged, of course, not in a single plane; around this entire system the outer (optical) electron moves in orbits which, as we have seen, undergo a rotation of the perihelion owing to perturbing forces. But the complex grouping of the inner electrons moving about the nucleus entails still another kind of perturbation. If we draw a line coinciding with the vector of the angular momentum of that aggregate of gyroscopes which our atom represents, then this vector, generally speaking, will not be perpendicular to the plane of the orbit of the outer electron. Owing to this circumstance, the possibility opens up of yet another perturbed motion, namely a precessional motion, in which the normal to the plane of the electron’s orbit will describe a cone about the unchanged direction of the angular-momentum vector (Fig. 2). This new motion will take place with its own period, again incommensurable with the first two (otherwise degeneration occurs, i.e. two motions combine to form one simple periodic motion). To this periodic motion the quantum conditions likewise prescribe certain restrictions in the form of a phase integral \(^{1}\)
Fig. 2.
\[ \oint p_3\,dq_3=jh\quad (j=1,2,3,\ldots), \]
where \(q_3\) and \(p_3\) are the coordinate and momentum determining this third kind of motion (the third degree of freedom), and \(j\) is the corresponding quantum number, usually called the inner quantum number. Thus, from the kinematic point of view, this inner quantum number characterizes the third periodic motion accessible to our atom: the precessional rotation of the plane of the orbit.
Of course, a change of this number \(j\) (with \(n\) and \(k\) unchanged) changes the energy of the system somewhat, so that now it must be expressed in the form:
\[ W=W(n,k,j)\quad \text{etc.}, \]
\(^{1}\) The integral extends over the whole range of \(q_3\).
and consequently the spectral formula will take the form:
\[ \nu=\frac{W(n,k,j)-W(n',k',j')}{h}. \]
By varying \(n'\), \(k'\), \(j'\), we obtain all the lines of the corresponding series. Since a change in \(j\) alters the value of the energy only very slightly, for given \(n\) and \(k\) different \(j\)'s will correspond to close values of the energy, i.e. the lines emitted when the system passes into the state \((n',k',j'_1)\) will differ little in frequency from the line obtained if the final state of the system is \((n',k',j'_2)\). We obtain groups: doublets, triplets, and so on, depending on how many combinations are possible for the given \(n\) and \(k\).
Thus the principal quantum numbers (\(n\) and \(n'\)) determine, in essential features, the position of the corresponding line; the azimuthal quantum numbers (\(k\) and \(k'\)) characterize its belonging to one or another series (principal, subordinate, etc.); and the inner quantum numbers (\(j, j'\)) determine whether the given series consists of simple lines, doublets, triplets, etc. Such is the significance of the quantum numbers from the point of view of determining the spectrum of the atom. Kinematically, as we have seen, they characterize respectively the periodic motion of the optical electron along its orbit (\(n\)), the rotation of the orbit in its plane (the rotation of the perihelion \((k)\)), and the rotation of the plane of the orbit about the axis of angular momentum (the precessional rotation of the orbit \((j)\)). A clear knowledge of the kinematic meaning of these quantum numbers is extremely important for understanding those restrictions which are imposed on the possible combinations of these numbers by the third postulate of the theory of the atom, likewise put forward by Bohr and known under the name of the correspondence principle.
The first and second postulates of Bohr, based on the quantum theory, free us from the necessity of obeying the laws of classical electrodynamics, thereby making it possible to preserve Rutherford’s nuclear theory of the atom.
However, this liberation is at the same time a grievous deprivation, for we simultaneously lose all that powerful and elaborated apparatus with whose aid the electrodynamics of the nineteenth century attained a very high degree of perfection. Moreover, the successes of the classical theory clearly show that its laws are not false, but only not fully applicable to those exceedingly rapid motions of charges which characterize the atom. It is, however, beyond doubt that as one passes from states in which the motions occur with a high optical frequency to states in which the emitted energy would have a low frequency, corresponding, for example, to the frequency of radio signals, we approach more and more the region where classical electrodynamics reigns by right: the difference between the results of the new, quantum theory and the old, ordinary theory becomes less and less perceptible.
For sufficiently slow circular motions of the electron, i.e. motions corresponding to large values of the quantum numbers, the frequency emitted by it according to the classical theory coincides with the frequency determined by Bohr’s second postulate: the individual stationary states prove to differ so little from one another that there is actually no need to speak of “discreteness”; we have a series of states so close to one another that the transition from one to another takes place continuously, and not by finite jumps, as follows from Bohr’s first postulate: finite differences are replaced by infinitely small changes, by differentials, and the laws assume their usual differential form, characteristic of the classical theory. From this circumstance Bohr drew the bold conclusion that even in the region where there is no coincidence between the two theories, there must nevertheless exist a certain correspondence, an analogy that permits one, from the conclusions suggested by classical theory, to infer how processes governed by quantum laws should proceed. Quantum theory, not yet possessing its own developed apparatus, obtains the possibility of borrowing support from its older rival, which in this way is transformed into a faithful ally. This clever diplomatic move of Bohr’s received the name of the “principle of correspondence” and must be regarded as a third postulate, whose purpose is to determine the behavior of the real quantum atom by analogy with the behavior of the imagined classical atom, from which it differs by the features formulated in Bohr’s first and second postulates.
In what way, then, is the correspondence established between classical and quantum processes, as required by Bohr’s third postulate?
Applying both theories to the processes under study, Bohr showed that for those states of an atomic system which are characterized by large values of the quantum numbers, the frequency calculated according to the second postulate coincides with the frequency determined by classical electrodynamics. According to the latter, the frequency of the radiation accompanying the uniform revolution of an electron in a circle corresponds to the frequency of this revolution, i.e. the radiation must be represented as simply monochromatic. In the case where the motion of the electron takes place along an ellipse or along an even more complicated curve, Fourier’s theorem enters the scene. It gives us the possibility of representing this complex periodic motion as the superposition of a series of simple uniform circular revolutions, with suitably chosen amplitudes and with frequencies comprising the fundamental one (equal to the frequency of the periodic motion) and a series of harmonics, i.e. integral multiples of the fundamental. Thus the complex radiation may be regarded as composed of a series
of monochromatic radiations with frequencies forming a harmonic series. Bohr’s investigations showed that the radiation frequencies according to the second postulate, i.e. the frequencies corresponding to different quantum transitions, coincide with the classical frequencies defined above for the case of large quantum numbers. The coefficients in the series mentioned determine the amplitudes, and consequently the energy, of the individual monochromatic lines, i.e. their intensity. From the point of view of quantum theory the intensity of a line is determined by whether the probability of the transition that causes the appearance of the given line is large or small. Therefore the indicated coefficients must be regarded as a measure of the probability of the corresponding quantum jumps. The correspondence principle, or Bohr’s third postulate, consists in the assertion that also for the case of small quantum numbers there corresponds to each frequency computed classically, i.e. by resolving the actual motion into a set of simple rotations, a definite quantum frequency. This correspondence is established by the following requirement: to the classical frequency of the \(S\)-th harmonic overtone there corresponds the frequency obtained in the transition from an orbit with quantum number \(n'\) to an orbit with quantum number \(n''\), such that \(n' - n'' = S\). In other words, the probability of the transition \(n' \to n''\) is measured by the amplitude corresponding to the \(S\)-th overtone of the Fourier expansion, i.e. by the coefficient of the corresponding term of the trigonometric series. If the character of the actual motion is such that in the corresponding expansion the coefficient of the \(S\)-th overtone is equal to zero, this means that the probability of the transition corresponding to a change of the quantum number by \(S\), i.e. the transition
\[ n' - n'' = S \]
does not occur. From this point of view the correspondence principle is a limiting principle, indicating which transitions conceivable from the quantum point of view in fact do not occur.\(^1\)
Let us apply the considerations set forth to those separate types of periodic motions into which the motion of the electron can be decomposed and which are connected with our quantum numbers \(n, k, j\).
The motion determined by the number \(n\) is motion along an elliptical orbit, which must be regarded, generally speaking, as capable of being composed of an infinite number of simple harmonic rotations. The corresponding radiation is represented classically by a Fourier series with all possible frequencies. Thus
\(^1\) In the preceding lines the essence of the correspondence principle has been set forth in an extremely schematic way. We do not at all touch upon the question of the significance of the correspondence principle for questions of intensity, polarization of light, etc. For more detail see the article by Yu. Krutkov, Advances in the Physical Sciences, vol. II, issue 2, as well as the book by Buchwald, Korrespondenzprinzip, Vieweg, 1923.
Thus each overtone corresponds to a larger or smaller coefficient, and by the correspondence principle this means that there is a greater or lesser probability for any transition
\[ n' - n'' = S, \]
whatever \(S\) may be. Thus the principal quantum number may change by any value: each of such transitions has one probability or another and, consequently, is possible.
The situation is different with the azimuthal quantum number \(k\). As we have seen, it determines the rotation of the perihelion. This latter is a uniform circular motion. Expanding it in a Fourier series, we find that all terms of the series representing overtones are absent, their coefficients are equal to zero, and consequently the corresponding quantum transitions are impossible. Of all the transitions corresponding to a change of the quantum number \(k\), only transitions of the type
\[ k' - k'' = \pm 1 \]
turn out to be possible (the two signs correspond to the possibility of uniform rotation of the perihelion clockwise and counterclockwise). Thus the correspondence principle leads to the conclusion that under ordinary conditions only those transitions are possible in which the azimuthal quantum number changes by \(\pm 1\). The other transitions do not occur. The correspondence principle appears as a selection principle. Hence, among other things, it is clear that the presence of external forces may play the role of a new perturbing cause, under the action of which the uniform character of the rotation of the perihelion will be distorted; the motion will become more complicated, new terms will appear in its Fourier expansion, i.e. the possibility of transitions will open up
\[ k' - k'' = S, \]
where \(S\) already has other values besides \(\pm 1\); the restrictions imposed on the change of the azimuthal quantum number disappear: the selection principle ceases to operate (is violated) in the presence of external forces (for example, an electric or magnetic field). Similar reasoning leads, for the number \(j\), to the conclusion that it, generally speaking, may change either by 0 or by \(\pm 1\), i.e. it too is subject to the action of the selection principle.
Let us briefly compare the results set forth. The magnitude of the energy of each atomic state is determined by three atomic numbers \((n, k, j)\), with the principal role being played by the value of the number \(n\), a lesser one by the number \(k\), and the least significant by \(j\). The difference of the energies in two states determines the frequency of the monochromatic line emitted in the transition from the first state to the second. All possible transitions
are restricted by the requirement that, in them, the number \(k\) may either increase or decrease by one, that \(j\) changes according to the rule \(\Delta j=0,\pm 1\), and that the number \(n\) may change arbitrarily.
Thus it is clear that, having determined the energy of the system as a function of the three variables \(W(n,k,j)\), and having plotted on a drawing all possible values of \(W\), we shall be able to indicate from our drawing which transitions from one state to another are possible and what lines, consequently, can be emitted by our system. A scheme constructed in this way would be a very visual graphical scheme of the spectrum of the given atom. Since each stationary state and its energy are determined by three quantum numbers, it would be convenient to represent them in a three-dimensional coordinate system. With a suitable choice of the coordinate system, each point corresponding to a stationary state could be associated with some graphically conveniently representable
Fig. 3.
quantity serving as a measure of the energy of this state. The transitions from one point to another that are possible from the standpoint of the correspondence principle should be indicated on the graph by means of a visual rule, while the difference of the energies of two states, likewise determined graphically, would give, on the corresponding scale, the frequency of the emitted or absorbed line. In this way our graph would represent the entire spectrum of the atom, with the individual lines oriented into the corresponding series depending on the value of the number \(k\). Unfortunately, representations in three-dimensional space have very little graphical clarity and are difficult to carry out. In our case the matter is made easier by the fact that the energy of a stationary state is essentially determined by the two quantum numbers \(n\) and \(k\), while the third, inner quantum number changes the value of the energy only to a small extent. Therefore our scheme can be reduced to a two-dimensional one and conveniently constructed.
In the accompanying drawing there is given a scheme of graphical representation proposed by Bohr. Along the horizontal axis are laid off the values of \(n\), and along the vertical, \(k\). To each pair of integral values \(n\) and \(k\) there corresponds
some stationary state. The magnitude of the energy in it is measured by the length of the perpendicular dropped from the corresponding point to the vertical axis. Under these conditions the scale for the quantity \(k\) plays no role, for only horizontal distances on our graph have physical meaning. The scale of \(n\), however, must be chosen in accordance with the indicated requirement. It is easy to see that in this case the scale of values of \(n\) cannot be a uniform scale, but is chosen with the special purpose of giving the indicated meaning to the horizontal segments. Therefore the lines of constant \(k\) may be a system of horizontal straight lines (situated at equal or unequal distances from one another), while the lines of constant \(n\) represent
Fig. 4.
a family of more or less complicated curves. Taking into account the fact that all the marked points of any one horizontal row correspond to one and the same \(k\) and to variable \(n\), we see that these points represent the variable terms of one of the Rydberg series. Thus, the line \(k=1\) gives the totality of terms of type \(m,s\), i.e. the variable terms of the second subsidiary series (II N. S., see p. 380). In exactly the same way the line \(k=2\) determines terms of the form \(m,p\), i.e. the variable terms of the principal series (P. S.), and so on. To obtain all members of the principal series, for example, i.e. members of the form \(1s-mp\), one must designate transitions from all points of the second row, beginning with the point \(n=2,\ k=2\), to the point of the first row determined by the data \(n=1,\ k=1\). These transitions are indicated in the drawing by arrows. The lengths of the horizontal distances between the corresponding points, i.e. the lengths of the horizontal projections of these arrows, represent, on a suitable scale, the frequencies of the emitted lines. Since the energy corresponding to any position of the electron
in the atom, is a negative quantity (it is measured by the energy that must be added to the atom in order that its energy become zero, i.e. in order that the electron recede to infinity without acquiring velocity), then the stationary states correspond to the smaller energy the farther from the vertical axis the point at which they are represented. Therefore transitions from right to left will correspond to an increase in the energy of the system, i.e. to absorption of light, while transitions from left to right will correspond to a decrease in the energy of the system, i.e. to emission. The requirement of the correspondence principle, according to which the change in \(k\) must be either \(+1\) or \(-1\), is expressed in our diagram by the condition that transitions are possible only between two adjacent horizontal rows.
As we have already said, a vertical displacement of any row introduces no distortions whatever into our scheme. This circumstance may be used in order, in our two-dimensional diagram, to reflect the influence exerted on the structure of the spectral lines by the third quantum number \(j\). We have seen that the energy levels for all states characterized by the same \(n\) and \(k\) and by different \(j\) differ little from one another. Therefore, depicting each row \(k\) in the form of several rows placed one above another and corresponding to different values of \(j\), we shall have to plot the states corresponding to different \(j\) (with identical \(n\) and \(k\)) as points situated, in horizontal distance, very close to one another. Wishing to preserve the proper scale, we must give the drawing corresponding dimensions, since otherwise the points \(j_1, j_2, j_3\) will become indistinguishable. The preceding drawing represents, strictly speaking, the spectrum of sodium, in which all orbits (with the exception of the group \(S\), i.e. \(k=1\)) are double. But at the scale used, the horizontal displacement of the individual points would be indistinguishable. The use of such a more complicated diagram, an example of which is given by Fig. 4, is carried out quite analogously to what was described above. The correspondence principle, as before, permits one to connect by arrows only those points for which \(\Delta k=\pm 1,\ \Delta j=0,\pm 1\).
The method of representing spectra set forth here offers considerable advantages in clarity and finds wide application in a number of works on atomic theory.