Solutions of the Dirac wave equation representing a free electron moving in a uniform magnetic field are obtained. The functions are similar to those obtained by Landau and by Uhlenbeck and Young as solutions of the Schr\"odinger equation. A wave packet is constructed representing a beam of electrons passing through a slit. The results agree with the classical predictions to terms of the order of the de Broglie wave-length of the electron divided by the radius of curvature of its classical path. For experimental cases this ratio is of the order ${10}^{\ensuremath{-}8}$ to ${10}^{\ensuremath{-}10}$. Hence it is concluded that the difference between magnetic deflection measurements of
me and other determinations cannot be explained as a quantum effect.
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
me, 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>0. A plane monochromatic de Broglie wave, travelling in the positive
x-direction representing electrons of arbitrary energy, impinges normally on the plane
x=0. Solutions of the wave equation were found fulfilling appropriate boundary conditions at the plane
x=0. 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
me 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.
The wave equation for a stream of electrons passing through a uniform magnetic field is solved and the characteristic functions discussed. The motion at right angles to the field is found to be completely quantized.The current density is investigated, and the mean square and mean fourth power radii of curvature found for each quantum state. The mean square radius of curvature is found to be the same for all quantum states of the same energy and identical with that predicted by the classical theory.The mean radius of curvature is determined by interpolation and is found to be less than that predicted by the classical theory. Therefore values of $\frac{e}{\ensuremath{\mu}}$ calculated by means of the classical formula from measured values of the mean radius of curvature should be too large. The error is estimated and found to be of the same general magnitude as the discrepancy between the results of deflection experiments and those of spectroscopic measurements.
It is shown that the relativistic wave equation for electrons in a uniform magnetic field leads to the same wave function as that already deduced by Page from the non-relativistic equation. As in the latter case the motion at right angles to the field is quantized.An expression is found for the current density from the relativistic wave equation. The relativistic expression differs from the non-relativistic only by a constant factor which does not affect the calculation of the mean radii of curvature of the electron current.Hence, for the relativistic case, as for the non-relativistic, the mean radius of curvature is less than that expected on the classical theory. It follows that the classical relativistic relation between $\frac{\ensuremath{\epsilon}}{\ensuremath{\mu}}$ and the mean radius of curvature upon deflection gives a value of $\frac{\ensuremath{\epsilon}}{\ensuremath{\mu}}$ which is too large.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by spinors - coordinate spinors, momentum spinors, spinor wave functions. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Schrödinger equation the wave function is represented by a single complex quantity, in the Dirac equation a step forward is taken and the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors.In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. The Lagrangian densities for the scalar field and for the electron field, along with their corresponding propagators, are presented. An equation of motion for the electron is proposed that is relativistically invariant, in contrast to the Dirac equation, which lacks this invariance. This novel equation permitted the construction of a relativistically invariant procedure for the second quantization of the fermion field in spinor coordinate space.