On a Remarkable Case of Quantization.

P. Ehrenfest and G. Breit. Ein bemerkenswerter Fall von Quantisierung. Zeitschr. für Physik. 1922. Vol. 9. Issue 4.

In recent years the quantum theory has been advancing irresistibly; to a considerable extent the cause of this is Bohr’s “principle of analogy.” Mystical at its foundation, it has, in all examples where there has been one possibility or another of applying it, led to correct results. In a paradoxical example from the standpoint of the method of quantization, to which Ehrenfest and Breit point, the principle of analogy gives an exhaustive answer.

Let us imagine a plane system of rectangular coordinate axes \((xy)\), with whose origin the center of a rigid dipole coincides. Let this dipole rotate about its center in the coordinate plane. The simple quantization of this circular motion leads to the following formula for the angular momentum:

\[ p = n \cdot \frac{h}{2\pi}. \qquad\qquad\qquad\qquad\qquad (1) \]

where \(n\) is an integer; \(h\) is Planck’s constant; we see that the angular momentum changes in steps; the size of the step is \( \frac{h}{2\pi} \).

Let now some perturbing force act on the dipole, say, in the form of periodic impulses—one asks: what will the motion of the dipole be? It is easy to understand that it will first describe an angle in the positive (or negative) direction equal to \(f\cdot 2\pi\), where \(f\), generally speaking, is an irrational number; then it is acted upon by an impulse, and it will go in the opposite direction and describe an angle \(=-f\cdot 2\pi\). This will continue as long as the periodic impulses act. If this motion is quantized, the following result is obtained:

\[ \int_{-f2\pi}^{+f\cdot 2\pi} p\,d\varphi = nh;\quad p=\frac{nh}{8\pi\cdot f}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (2) \]

In this case the moment of momentum also changes by steps; the magnitude of the step is \(\frac{h}{8\pi f}\); it obviously depends on the period with which the impulses follow one another. Suppose now that this period is infinitely large; then, obviously, our motion turns simply into circular motion. If we compute the magnitude of the step for the moment of momentum in this case by formula (2), we obtain zero, i.e. we fall into contradiction with formula (1); thus, the more general result does not give the particular one. How is this paradox to be resolved? Obviously, the method of quantization cannot be applied to isolated motions. Quantization is conceivable where the isolated motion can be represented as the result of statistics. If our motion is not individual, then we have the right to apply the principle of analogy. This is what is done. The motion of the dipole can quite well be characterized by the mathematical expression for the projection of the electric moment on one of the coordinate axes; we expand this expression in a Fourier series; then the amplitude of the term with overtone \(s=n_1-n_2\) gives us the probability of transition from the quantum state \(n_1\) to the quantum state \(n_2\). It turns out that, for very large \(f\), the amplitude of all overtones will be very small, except those for which the relation holds:

\[ S=4f,\quad \text{or}\quad n_2-n_1\leq 4p. \]

Consequently, in the case of infinitely large \(f\), the most probable transitions are those for which

\[ p_2-p_1=(n_2-n_1)\frac{h}{8\pi f} =\frac{(n_2-n_1)h}{2\pi\cdot n_2-n_1} \leq \frac{h}{2\pi}. \]

Thus, the principle of analogy gives a result coinciding with the quantum condition for simple circular motion. The paradox is resolved.

A. Predvoditelev.