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Full Article | PDF Article**Journal of the Optical Society of America**- Vol. 9,
- Issue 1,
- pp. 27-30
- (1924)
- •doi: 10.1364/JOSA.9.000027

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- We shall for simplicity assume in the present discussion that the system is non-degenerate.

- Since the writing of the present article, H. A. Kramers has published in Nature (May10, 1924) a very interesting formula for dispersion, in which the polarization is imagined as coming not from actual orbits, but from “virtual oscillators” such as have been suggested by Slater and advocated by Bohr. Kramers states that his formula merges asymptotically into the classical dispersion. To verify this in the general case, the writer has computed the classical polarization formula for an arbitrary non-degenerate multiply periodic orbit. This formula is the analogue of Eq. (3) of the present paper, and is more complicated than the ordinary Eq. for dispersion by linear oscillators. By pairing together positive and negative terms in the Kramers formula, a differential dispersion may be defined resembling the differential absorption of the present article. It is found that this differential quantum theory dispersion approaches asymptotically the classical dispersion by the general multiply periodic orbit, the behavior being very similar to that in the correspondence principle for absorption. This must be regarded as an important argument for the Kramers formula.

Since the writing of the present article, H. A. Kramers has published in Nature (May10, 1924) a very interesting formula for dispersion, in which the polarization is imagined as coming not from actual orbits, but from “virtual oscillators” such as have been suggested by Slater and advocated by Bohr. Kramers states that his formula merges asymptotically into the classical dispersion. To verify this in the general case, the writer has computed the classical polarization formula for an arbitrary non-degenerate multiply periodic orbit. This formula is the analogue of Eq. (3) of the present paper, and is more complicated than the ordinary Eq. for dispersion by linear oscillators. By pairing together positive and negative terms in the Kramers formula, a differential dispersion may be defined resembling the differential absorption of the present article. It is found that this differential quantum theory dispersion approaches asymptotically the classical dispersion by the general multiply periodic orbit, the behavior being very similar to that in the correspondence principle for absorption. This must be regarded as an important argument for the Kramers formula.

We shall for simplicity assume in the present discussion that the system is non-degenerate.

Since the writing of the present article, H. A. Kramers has published in Nature (May10, 1924) a very interesting formula for dispersion, in which the polarization is imagined as coming not from actual orbits, but from “virtual oscillators” such as have been suggested by Slater and advocated by Bohr. Kramers states that his formula merges asymptotically into the classical dispersion. To verify this in the general case, the writer has computed the classical polarization formula for an arbitrary non-degenerate multiply periodic orbit. This formula is the analogue of Eq. (3) of the present paper, and is more complicated than the ordinary Eq. for dispersion by linear oscillators. By pairing together positive and negative terms in the Kramers formula, a differential dispersion may be defined resembling the differential absorption of the present article. It is found that this differential quantum theory dispersion approaches asymptotically the classical dispersion by the general multiply periodic orbit, the behavior being very similar to that in the correspondence principle for absorption. This must be regarded as an important argument for the Kramers formula.

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$${B}_{s}^{r}={B}_{r}^{s}={A}_{r}^{s}{c}^{3}/8\pi h{{\nu}_{rs}}^{3}.$$

$$q={\mathrm{\Sigma}}_{{\tau}_{1}{\tau}_{2}{\tau}_{3}}{A}_{{\tau}_{1}{\tau}_{2}{\tau}_{3}}^{(q)}\hspace{0.17em}\text{cos}\hspace{0.17em}[2\pi ({\tau}_{1}{\omega}_{1}+{\tau}_{2}{\omega}_{2}+{\tau}_{3}{\omega}_{3})\hspace{0.17em}t+{\u220a}_{{\tau}_{1}{\tau}_{2}{\tau}_{3}}^{(q)}]$$

$${A}_{r}^{s}={(2\pi )}^{4}(1/3{c}^{3}){e}^{2}{\nu}_{rs}^{3}{A}_{{\tau}_{1}{\tau}_{2}{\tau}_{3}}^{2}{h}^{-1}$$

$$dW/dt={\mathrm{\Sigma}}_{{\tau}_{1}{\tau}_{2}{\tau}_{3}}{K}_{{\tau}_{1}{\tau}_{2}{\tau}_{3}}$$

$${K}_{{\tau}_{1}{\tau}_{2}{\tau}_{3}}=(2{\pi}^{3}/3){e}^{2}\left\{\rho (\nu )\left({\tau}_{1}\frac{\partial {D}_{{\tau}_{1}{\tau}_{2}{\tau}_{3}}}{\partial {J}_{1}}+{\tau}_{2}\frac{\partial {D}_{{\tau}_{1}{\tau}_{2}{\tau}_{3}}}{\partial {J}_{2}}+{\tau}_{3}\frac{\partial {D}_{{\tau}_{1}{\tau}_{2}{\tau}_{3}}}{\partial {J}_{3}}\right)+{D}_{{\tau}_{1}{\tau}_{2}{\tau}_{3}}\frac{\partial \rho}{\partial \nu}\left({\tau}_{1}\frac{\partial}{\partial {J}_{1}}+{\tau}_{2}\frac{\partial}{\partial {J}_{2}}+{\tau}_{3}\frac{\partial}{\partial {J}_{3}}\right)({\tau}_{1}{\omega}_{1}+{\tau}_{2}{\omega}_{2}+{\tau}_{3}{\omega}_{3})\right\}$$

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