CONTEMPORARY ADVANCES IN PHYSICS 693 



field — a conclusion from the earlier atomic theory, which for magnetic 

 fields has become a fact of experience through the experiments of 

 Gerlach and Stern and others. To calculate the polarization of a gas, 

 it is necessary to make a further assumption concerning the relative 

 probabilities of these various orientations in a gas in thermal equi- 

 librium; having done so, one obtains a formula for the polarization, 

 or the dielectric constant, or the susceptibility of the gas as function 

 of applied field and temperature. The customary assumption leads 

 to a formula which, in the limiting case of high temperature and low 

 field, agrees with the celebrated equation of Langevin for the polariza- 

 tion of a paramagnetic gas by a magnetic field :-^ 



Susceptibility = IIH = NAP/SkT. (194) 



Interpretation of the Free Electron in Wave- Mechanics 



We now depart from the calculation of Eigenwerte and Stationary 

 States, and return to the original ideas of de Broglie. 



For a free electron moving in a field-free region — or any particle 

 moving in a region where no force acts upon it — with a constant speed 

 V along a direction which I will take as the x-direction, the (non- 

 relativistic) wave-equation assumes the form: 



TT2 H JJ- lA = (E = imv^-). (195) 



This equation admits a sine-function as its solution whatever the 

 value of the constant E and consequently does not restrict the energy- 

 values which the electron is allowed to take (a contrary result would 

 have been hard to swallow!). Assigning the value Ejh to the fre- 

 quency of the sine-wave and the value Ej xlmE to its speed, we obtain 

 for the wave-length of the wave-train, "associated with " a free electron 

 (or free particle) of mass m and speed v, this value : 



^ ^ El -V2m£ ^ h ^ h_ _ 



For electrons of such speeds as ordinarily occur in discharge-tubes, 

 these wave-lengths are of the magnitude of X-ray wave-lengths; for 

 instance, a 150- volt electron is associated with a wave-length of very 

 nearly one Angstrom unit, 



25 C. Manneback, Phys. ZS., 28, pp. 72-84 (1927); J. H. Van Vleck, Phys. Rev. 

 (2), 29, pp. 727-744; 30, pp. 31-54 (1927). For the classical derivation of formula 

 (194) and meaning of the symbols cf. my article "Ferromagnetism," this Journal, 

 6 (1927), pp. 351-353. 



