On Dielectric Constants and Magnetic Susceptibilities in the new Quantum Mechanics Part I. A General Proof of the Langevin-Debye Formula

Abstract

In contradistinction to the old quantum theory, the new quantum mechanics yields very generally the Langevin and Debye formulas $\ensuremath{\chi}=N\ensuremath{\alpha}+\frac{N{\ensuremath{\mu}}^{2}}{3kT}$ for the magnetic and dielectric susceptibilities respectively. It is believed that our proof is considerably more comprehensive than previous ones, for it assumes only that the atom or molecule has a "permanent" vector moment of constant magnitude $\ensuremath{\mu}$ whose precession frequencies are small compared to $\frac{\mathrm{kT}}{h}$. There is no limit to the allowable number of superposed precessions. There can, for instance, be simultaneous precessions due to internal spins of the electron and to "temperature rotation" of the nuclei. The presence of other simultaneous external fields in addition to the applied electric or magnetic field introduces no difficulty. Besides the effect due to the permanent moment, there is the term $N\ensuremath{\alpha}$ which arises from "high frequency" matrix elements associated with transitions from normal to excited states. This term is proved independent of the temperature. The susceptibility formula contains the factor $\frac{1}{3}$ in the temperature term as generally as in the classical theory because of the high spectroscopic stability characteristic of the new quantum mechanics. It is shown that the mean squares of the $x$, $y$, and $z$ components of a vector are equal in the new quantum dynamics just as in the classical theory, the only difference being that in the new quantum theory we take the average by summing over a discrete distribution of quantum-allowed orientations instead of by integrating over a uniform continuous distribution.