We study some aspects of the quantum theory of a charged particle moving in a time-independent, unidirectional magnetic field. When the field is uniform, we make a few clarifying remarks on the use of angular momentum eigenstates and momentum eigenstates with the diamagnetism of a free electron gas as an example. When the field is nonuniform but weakly varying, we discuss both perturbative and nonperturbative methods for studying a quantum mechanical system. As an application, we derive the quantized energy levels of a charged particle in a Helmholtz coil, which go over to the usual Landau levels in the limit of a uniform field.
A thorough unified treatment is given of the quantum-mechanical operators and wave functions for a molecular system (composed of
N moving charged particles) in static uniform electric and magnetic fields
E and
B. The treatment is rigorous within the nonrelativistic approximation. The system may either be neutral or charged. The fields may have arbitrary intensities and orientations. Close correspondence is maintained between the classical and quantum-mechanical treatments. The wave functions are expressed both in the time-independent energy representation and in time-dependent wave packets. Three types of momentum play important roles. For single-particle systems they are the canonical momentum
P, the mechanical momentum $\mathit{\ensuremath{\Pi}}=\mathit{P}\ensuremath{-}(\frac{e}{c})\mathit{A}=M\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}$, and the pseudomomentum $\mathcal{K}=\mathit{\ensuremath{\Pi}}\ensuremath{-}(\frac{e}{c})\mathit{R}\ifmmode\times\else\texttimes\fi{}\mathit{B}\ensuremath{-}e\mathit{E}t$. The pseudomomentum is a constant of the motion. Except in the absence of magnetic fields, not all of the components of either $\mathit{\ensuremath{\Pi}}$ or
K commute. This complicates the quantum-mechanical formalism. The components of the pseudomomentum have simple classical interpretations, and in quantum mechanics they are related to the operator which performs a boost to a reference frame moving with constant velocity
v. In this moving frame, the electric field intensity is ${\mathit{E}}^{\ensuremath{'}}=\mathit{E}+(\frac{\mathit{v}}{c})\ifmmode\times\else\texttimes\fi{}\mathit{B}$. Thus, in a frame moving with the drift velocity ${\mathit{v}}_{d}=(\frac{c}{{B}^{2}})\mathit{E}\ifmmode\times\else\texttimes\fi{}\mathit{B}$, the components of the electric field intensity perpendicular to
B vanish. In this paper, the velocity boost operators are used to show the relationships between wave functions expressed in reference frames moving with repsect to each other. The dynamics of the
N-particle systems are simplified by making the Power-Zienau-Woolley transformation (which reduces as a special case to a unitary transformation used by Lamb) and by using center-of-mass and internal coordinates. Generalizing previous works, it is shown how the pseudomomentum is involved in separating these degrees of freedom for
N particles in both
E and
B. For neutral molecules, the Schr\"odinger equation is "pseudoseparated" and the internal degrees of freedom are coupled to the center of mass motion only by the "motional Stark Effect," which involves the constant of motion
K. For ionic systems, only one component of the center of mass is coupled to the internal motion. Quantitative estimates of the weak center-of-mass coupling are made for both neutral and ionic
n=1 and
n=2 quantum states of two-body systems by perturbation expansions in powers of the field strengths. In the usual nonrigorous treatments of systems in magnetic fields, no distinction is made between $\mathit{\ensuremath{\Pi}}$ and
K and both are approximated by $M\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}$, where
M is the mass of the molecule and
R is taken to be the classical orbital position of a single particle having the same mass and charge. This is an excellent approximation for ground-state molecules and ions in weak fields. However, the resulting errors can be much larger if the system is in either intense fields or high-lying states (e.g., Rydberg levels). These are the conditions under which the rigorous formalism should be most useful. The determination of the wave functions and energy levels for many-particle ionic systems is complicated due to the coupling with the center of mass. The suggestion is made that it might be useful to boost the reference frame so that it moves with the velocity $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}$ of a single classical particle. The Schr\"odinger equation in this frame could serve as a more suitable basis for perturbative or variational treatments.
Abstract We have calculated the effect of a magnetic field on the evolution of angular momentum eigenfunctions of a charged particle. An additional harmonic potential is supplemented to trap the wave packet. We find the probability density of the wave function is oscillating in the radial direction with a time period determined by the strength of the effective harmonic potential. When the magnetic field is along the z direction, if the initial wave function is an eigenfunction of L ˆ z , the probability density of the particle remains axis-symmetric. While for the case of an eigenfunction of L ˆ x , it is anisotropic in the x−y plane and rotates with a time period inverse proportional to the strength of the external magnetic field. We also extend the results in a phenomenological way to the case with an external magnetic field that varies harmonically in time.
We consider the orbital angular momentum of a free electron vortex moving in a uniform magnetic field. We identify three contributions to this angular momentum: the canonical orbital angular momentum associated with the vortex, the angular momentum of the cyclotron orbit of the wavefunction, and a diamagnetic angular momentum. The cyclotron and diamagnetic angular momenta are found to be separable according to the parallel axis theorem. This means that rotations can occur with respect to two or more axes simultaneously, which can be observed with superpositions of vortex states.
In the introductory remarks to Chapter 5 mention was made of the 1925 proposal of Uhlenbeck and Goudsmit that to an electron there should be assigned an intrinsic angular momentum or spin S = (S x , S y , S z )—this in addition to its quantum angular momentum (or orbital angular momentum) M = (M x , M y , M z ) defined in (5.4.4). This is not to be taken to mean that an electron actually spins about an axis in some physical way. Experiments (in particular the noted Stern-Gerlach experiment) have provided adequate justification for the proposed concept of spin, a proposal also made by R. Bichowsky and H. Urey. Dirac has shown that the notion of spin is necessary if quantum mechanics and Einstein relativity are to be consistent theories. Chemical properties of atoms, and their ability to combine and form molecules, the splitting of spectral lines by a magnetic field applied to an atom (i.e., the Zeeman effect, which we discuss later), all depend on this important notion.
Different classical limits of quantum mechanics are discussed in the setting of a simple example, the two-dimensional motion of a spinless electron in a uniform magnetic field. Both the time-independent energy eigenstates and the time-dependent states developing from initial localized states are exactly solvable. In the first example, an ℏ→0 limit with fixed energy leads to a simple ensemble of classical orbits. In the second, ℏ→0 with fixed initial mean position and momentum leads to a unique classical orbit.