The Value ofemby Deflection Experiments

To clarify the discussion as to whether or not a quantum mechanical treatment of the motion of electrons in magnetic fields will lead to a formula for $\frac{e}{m}$, differing from that obtained by classical electro-dynamics the following problem was solved. A uniform magnetic field in the $z$-direction exists in the half-space $x>\mathrm{}0\mathrm{}$. A plane monochromatic de Broglie wave, travelling in the positive $x$-direction representing electrons of arbitrary energy, impinges normally on the plane $x=0\mathrm{}$. Solutions of the wave equation were found fulfilling appropriate boundary conditions at the plane $x=0\mathrm{}$. Currents are calculated quantum mechanically and compared with the corresponding classical expressions. It was found that for electrons possessing energies of the order of magnitude used in deflection experiments, no observable deviations from classical results are predicted. Another quantum mechanical effect is diffraction at slits. Simple approximate calculations show that this effect can produce a fractional error in $\frac{e}{m}$ of the order of the de Broglie wave-length divided by the slit width. These results are opposite to the conclusions reached by Page. We may remark that he solved a problem of "stationary states" which does not represent the actual experiments.