Interaction of atoms, molecules, and ions with constant electric and magnetic fields

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 $\mathit{E}$ and $\mathit{B}$. 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 $\mathit{P}$, the mechanical momentum $\mathit{\Pi }=\mathit{P}-\left(\frac{e}{c}\right)\mathit{A}=M\dot{R}$, and the pseudomomentum $\mathcal{K}=\mathit{\Pi }-\left(\frac{e}{c}\right)\mathit{R}\times \mathit{B}-e\mathit{E}t$. The pseudomomentum is a constant of the motion. Except in the absence of magnetic fields, not all of the components of either $\mathit{\Pi }$ or $\mathcal{K}$ 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 $\mathit{v}$. In this moving frame, the electric field intensity is ${\mathit{E}}^{\prime }=\mathit{E}+\left(\frac{\mathit{v}}{c}\right)\times \mathit{B}$. Thus, in a frame moving with the drift velocity ${\mathit{v}}_d=\left(\frac{c}{B^2}\right)\mathit{E}\times \mathit{B}$, the components of the electric field intensity perpendicular to $\mathit{B}$ 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 $\mathit{E}$ and $\mathit{B}$. For neutral molecules, the Schrödinger 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 $\mathcal{K}$. 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 $n=1\mathrm{}$ and $n=2\mathrm{}$ 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 $\mathit{\Pi }$ and $\mathcal{K}$ and both are approximated by $M\dot{R}$, where