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Bohr’s Principle of Analogy in Quantum Theory
Yu. A. Krutkov.
1. When Planck, almost 21 years ago, in order to obtain the formula for the distribution of energy in the spectrum of a black body, put forward his hypothesis of quanta, he did not need a new assumption about the frequency of the light emitted by his model of matter. This model was a linear vibrator; it is obvious that the light emitted by it has a frequency equal to the frequency of the proper oscillations of the vibrator.
The properties of Planck’s vibrator contradicted classical electrodynamics, but nevertheless its connection with radiation was conceived as in the old “quasi-elastic” optics. Thus, for example, the polarization of the light emitted by the vibrator obviously had to be determined by the direction (and character) of its oscillations.
Bohr took as the model of the emitting matter Rutherford’s model of the atom, endowing it with quantum properties. Planck’s quantum hypothesis acquired with him the following form: an atomic system assumes only a definite series of states, to which there corresponds a discontinuous series of energy values; with every change of the energy of the system, including here emission and absorption of energy, the system passes from one of these states into another¹). Since the systems under consideration by no means perform harmonic oscillatory motion, one more hypothesis is required, namely, about the frequency of the light emitted by the model. In Bohr this is his second rule: the radiation emitted (or absorbed) in the transition from one state to another is monochromatic with frequency \(\nu\), given by the equation:
\[ E' - E'' = h\nu, \qquad \ldots\ldots\ldots (1) \]
where \(h\) is Planck’s constant, and \(E'\), \(E''\) are the energies of the two states.
At first it seemed that this second rule, perhaps, ought to be replaced by another, but the derivation of Planck’s spectral formula given by Einstein²) shows with great convincingness that it must be preserved.
It is obvious that from a rule determining the frequency from the energy balance one cannot extract anything concerning the polarization and intensity of the emitted light. The intensity must depend on the number of atoms in the given state and on the probability
¹) N. Bohr. On the Quantum-Theory of Line-Spectra I. Acad. Copenhague 8 série, t. IV, nº 1, fasc. 1 (1918).
²) A. Einstein. Verh. d. D. phys. Ges. 18, p. 318, (1916).
transition from this state to another given one. Nothing can be said about the polarization.
In 1918 Bohr gave a rule which to some extent removes this gap.^1)
2. For simplicity let us consider a system with one degree of freedom. If its generalized coordinate is denoted by \(q\), and the corresponding momentum by \(p\), then, under certain general assumptions about the character of the motion, we may state the quantum hypothesis for our system as follows: the system assumes a series of states given by the equation:
\[ J=2\int p\,dq=nh,\ . . . . . . . . . . . . \tag{2} \]
where the integral is extended over the entire range of variation of \(q\), and \(n\) is an integer: \(0, 1, 2, 3, \ldots\); \(J\) is a function of the energy \(E\) of the state under consideration; the limits of the integral depend on \(E\), and the integrand is \(p\); at the limits the integrand vanishes, therefore
\[ \frac{\partial J}{\partial E}=2\int \frac{\partial p}{\partial E}\,dq; \]
but \(\partial p/\partial E=1/\dot q\); consequently,
\[ \frac{\partial J}{\partial E} =2\int \frac{1}{\dot q}\,dq =2\int dt =T, \]
where \(T\) is the period of the motion. Denoting by \(\omega=\frac{1}{T}\) the frequency of the motion, we have
\[ \frac{\partial E}{\partial J}=\omega\ . . . . . . . . . . . . \tag{3} \]
Let us consider a transition between states given by the quantum numbers \(n'\) and \(n''\). According to Bohr’s second rule we have:
\[ \nu=\frac{1}{h}(E'-E''). \]
Let now \(n'\), \(n''\) be large, and their difference \(n'-n''\) small in comparison with \(n'\) and \(n''\).
Then, using (3), we may write \(E'-E''\) (approximately) as \(\omega(J'-J'')\).
\[ \nu=\frac{\omega}{h}(J'-J''). \]
But \(J'=n'h\) and \(J''=n''h\), consequently
\[ \nu=\omega(n'-n'')\ . . . . . . . . . . . . \tag{4} \]
Thus, for large \(n'\), \(n''\) and a small difference \(n'-n''\), the system emits, according to quantum theory, the frequency \(\omega\) and its overtones.
What frequencies does our system emit according to classical electrodynamics?
^1) N. Bohr, loc. cit.
The projection \(\xi\) of the displacement of the system onto any direction, under the assumptions about the character of the motion that have already been mentioned, can be represented by the following Fourier series \({}^{1}\):
\[ \xi=\sum C_{\tau}\cos 2\pi(\tau\omega t+c_{\tau}). \qquad\qquad (5) \]
Thus, according to classical electrodynamics, the system emits the frequency \(\omega\) and its overtones \(\tau\omega\).
A comparison of this result with formula (4), in which we give the difference \(n'-n''\) various values, shows that for large quantum numbers and small differences between them, quantum theory and classical electrodynamics give coincident results for the emitted frequency \(\nu\). But it should be remembered that the processes by which emission occurs are different in the two theories: according to electrodynamics, all \(\tau\omega\) are emitted simultaneously; according to quantum theory, to each \((n'-n'')\omega\) there corresponds its own transition from the state \(n'\) to the state \(n''\).
According to electrodynamics, the coefficients (“amplitudes”) \(C_{\tau}\) in the expansion (5) determine the intensities of the various lines of the emitted spectrum. Bohr assumes that for large \(n\) these coefficients measure the probability of transition from the state \(n=n'\) to the neighboring state \(n=n''=n'-\tau\) \((n'-n''=\tau)\); knowing \(C_{\tau}\), we can draw conclusions about the intensity of the emitted lines.
Such an assertion—we may call it the principle of correspondence—is quite natural: for large quantum numbers all the formulas of quantum theory pass over into the classical formulas.
Further, Bohr assumes that even for small quantum numbers the value of the “amplitude” \(C_{\tau}\) in the expansion (5), corresponding to the frequency \(\tau\omega\), permits one to estimate the probability of the transition for which \(n'-n''=\tau\). Thus, for example, if for all motions a certain \(C_{\tau}\) is equal to zero, then one must assert that there is no transition \(n'-n''=\tau\). This is Bohr’s principle of analogy.
For Planck’s linear vibrator the expansion (5) reduces to a single term \(\tau=1\). Hence we conclude that for it \(n'-n''\) is always equal to unity: the emitted frequency is equal to its frequency \(\omega\).
3. Let us pass to a system with \(s\) degrees of freedom. If the system belongs to the so-called “conditionally periodic” \({}^{2}\) systems, then (in the absence of so-called “commensurabilities” between the motions) the quantum hypothesis states:
\[ J_k=2\int p_k\,dq_k=n_k h \quad (k=1,2,\ldots,s). \qquad (6) \]
The energy \(E\) of the system is a function of all the \(J_k\). It is not difficult to show that, instead of (3), we shall now have the formulas
\[ \frac{\partial E}{\partial J'_k}=\omega_k \quad (k=1,2,\ldots,s). \qquad (7) \]
\({}^{1}\) See Charlier, Mechanik d. Himmels, I.
\({}^{2}\) l. c.
where \(\omega_k\) are the “mean frequencies” of the oscillations of the separate degrees of freedom.
For \(\nu\) we have formula (1). Let us again suppose that all \(n'_k, n''_k\) are large and that the differences \(n'_k-n''_k\) are small in comparison with these numbers. Then, using (7), we obtain:
\[ \nu=\frac{1}{h}(E'-E'')=\frac{1}{h}\sum_{k=1}^{s}\omega_k(J'_k-J''_k) =\sum_{1}^{s}\omega_k(n'_k-n''_k)\ .\ .\ .\ .\ (8) \]
The system emits the frequencies \(\omega_k\), overtones \((n'_k-n''_k)\omega_k\), and combination tones
\((n'_1-n''_1)\omega_1+(n'_2-n''_2)\omega_2+\cdots\).
What frequencies does the system emit according to electrodynamics? It can be shown that the “coordinates” \(q_k\), and consequently also the displacement \(\xi\) in any direction, are represented by an \(s\)-fold Fourier series\(^3\):
\[ \xi=\sum\cdots\sum C_{\tau_1\ldots\tau_s} \cos 2\pi\bigl[(\tau_1\omega_1+\cdots+\tau_s\omega_s)t+c_{\tau_1\ldots\tau_s}\bigr];\ .\ (9) \]
the emitted spectrum has the frequencies \(\tau_1\omega_1+\cdots+\tau_s\omega_s\).
For large quantum numbers the two spectra coincide; for small ones they are entirely different.
The coefficients \(C_{\tau_1\ldots\tau_s}\) in the expansion (9) determine the intensity and polarization of the light of frequency \(\tau_1\omega_1+\cdots+\tau_s\omega_s\). Bohr concludes that, in the limiting case of large \(n_k\), these coefficients measure the probabilities of spontaneous transitions for which \(n'_k-n''_k=\tau_k\), and then extends this rule to the case of finite \(n_k\).
On the basis of the rule, various conclusions are possible. For example, since in the expansion (9) \(\tau\) assumes all values from \(-\infty\) to \(+\infty\), one must also allow such transitions in which not all \(n_k\) decrease (contrary to Sommerfeld’s proposal), which is confirmed by observations of the Stark effect and of the fine structure of hydrogen lines. Further: if some coefficient \(C_{\tau_1\ldots\tau_s}\) is equal to zero for all motions of the system and for any direction, then the transition \(n'_k-n''_k=\tau_k\) \((k=1,2,\ldots,s)\) must be regarded as impossible. If \(C_{\tau_1\ldots\tau_s}\) is equal to zero only for displacement in a definite direction, then the transition \(n'_k-n''_k=\tau_k\), \(k=1,2,\ldots,s\), gives radiation polarized in a plane perpendicular to this direction.
4. Let us consider a system with three degrees of freedom, for which one of the “coordinates” is cyclic. Let this be, for example, an angle. Then the corresponding angular momentum is a constant quantity.
We take as coordinates this angle \(\vartheta\), the distance of the particle from the axis of rotation \(\rho\), and \(z\), measured along the axis of rotation, i.e. cylindrical coordinates.
Then
\[ \left. \begin{aligned} z&=\sum_{-\infty}^{+\infty}\sum C_{\tau_1\tau_2} \cos 2\pi\bigl[(\tau_1\omega_1+\tau_2\omega_2)t+c_{\tau_1\tau_2}\bigr], \\[4pt] \rho&=\sum_{-\infty}^{+\infty}\sum C'_{\tau_1\tau_2} \cos 2\pi\bigl[(\tau_1\omega_1+\tau_2\omega_2)t+c'_{\tau_1\tau_2}\bigr]; \end{aligned} \right\} \ .\ .\ .\ .\ (10) \]
further, for \(\vartheta\), it is easy to obtain
\[ \pm \vartheta = 2\pi \omega_3 t + \sum \sum C''_{\tau_1 \tau_2} \cos 2\pi \left[ (\tau_1\omega_1+\tau_2\omega_2)t + c''_{\tau_1 \tau_2} \right]. \tag{10'} \]
where \(\omega_3\) is the mean frequency of rotation about the axis. Introducing rectangular coordinates \(x,y\) instead of \(\rho,\vartheta\), we easily obtain:
\[ \left. \begin{aligned} x=\rho \cos \vartheta &= \sum_{-\infty}^{+\infty}\sum D_{\tau_1\tau_2} \cos 2\pi \left[ (\tau_1\omega_1+\tau_2\omega_2+\omega_3)t + d_{\tau_1\tau_2} \right], \\[6pt] y=\rho \sin \vartheta &= \pm \sum_{-\infty}^{+\infty}\sum D_{\tau_1\tau_2} \sin 2\pi \left[ (\tau_1\omega_1+\tau_2\omega_2+\omega_3)t + d_{\tau_1\tau_2} \right]. \end{aligned} \right\} \tag{10''} \]
We see that, according to (10) and (10″), the motion is composed of harmonic oscillations along the \(z\)-axis with frequencies \(|\tau_1\omega_1+\tau_2\omega_2|\), and of circular motions perpendicular to the axis with frequencies \(|\tau_1\omega_1+\tau_2\omega_2+\omega_3|\): the radiation will have components polarized linearly and circularly.
By the principle of analogy one must conclude that \(n_3\) may either remain constant; then, when \(n_1,n_2\) change, on which no restrictions are imposed, the radiation is linearly polarized along the \(z\)-axis; or \(n_3\) changes by \(\pm 1\), and the radiation is circularly polarized. Observations on the effect of electric and magnetic fields on the hydrogen atom confirm these conclusions. The conclusion that \(n_3\) changes only by \(\pm 1\) receives further confirmation from the law of conservation of the sum of the mechanical and electromagnetic angular momenta1.
The number of examples elucidating the significance of Bohr’s new rule could be increased. A very remarkable application to the question of the intensities of the spectral lines of hydrogen is contained in Kramers’ memoir2; the intensities of the lines of ionized helium (the Fowler series) in the Stark phenomenon were considered by Epstein3. Bohr’s rule, while remaining theoretically obscure, gives excellent agreement with the data of experiment.