For neutral particle systems in a homogeneous magnetic field, there exists a constant of motion, the pseudomomentum, which allows a complete pseudoseparation of the center-of-mass motion. For ions such a separation is not possible, and the Landau orbit of the center-of-mass and internal (electronic) motion are intimately coupled. We investigate different physical situations for which this interaction becomes strong at laboratory magnetic-field strengths. In particular, we find that the Landau orbit itself can change substantially upon this interaction.
The approximate constant of motion introduced previously for the description of a charged system in a homogeneous magnetic field is interpreted physically as the kinetic momentum of the collective motion. The algebra satisfied by this operator and by the exact constants of motion describes the behavior in a magnetic field of a single particle possessing the total charge of the system. With this algebra, we generalize the approximate constant of motion and obtain a family of operators depending on arbitrary parameters. Canonical transformations based on these new operators separate the Hamiltonian into collective, internal, and coupling terms. These terms take the same form for charged and neutral systems although the collective energy represents different physical behaviors. The coupling between the internal and collective motions is small for some choices of the parameters if at least one particle is much heavier than the other ones.
The pseudomomentum is conserved for a single charge in any constant (over space and time) electromagnetic field. It is also conserved for a neutral two-body system, not only in a pure (constant) magnetic field, but also in a pure (constant) electric field, and explicit wave equations can be written. A neutral two-body system in a constant electric field is treated in a way that parallels our previous approach to this system in a constant magnetic field. Relative motion is separated, but the relative time now survives, whereas a spacelike degree of freedom gets eliminated. In other cases, there is, in general, no evidence of a closed form of the wave equations, but conservation of the pseudomomentum can be required as a reasonable condition for implicitly determining the so-called ``three-body terms'' in the wave equation.
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.
In a magnetic field anions are known to possess, within the approximation of infinite nuclear mass, an infinite manifold of bound states. This approximation, however, neglects effects due to the coupling of the anionic center-of-mass (c.m.) motion to the electronic motion. Recently these effects have been reported to be crucial for the mere existence and properties of the field-induced anions [Phys. Rev. Lett. 86, 5450 (2001)]. We develop a theoretical approach to study anions in the presence of a magnetic field including the motional c.m. effects. We aim at weak to moderate magnetic-field strengths typical for present-day laboratories. Our approach involves canonical transformations of the original Hamiltonian, implementing the integrals of motion and reasonable adiabatic approximations. As a result we derive a three-dimensional Hamiltonian that can be universally applied to atomic or molecular negative ions in a magnetic field by specifying the properties of the underlying neutral systems. We consider this Hamiltonian to be suitable for rigorous investigations of moving anions in fields. For the classical dynamics a reduction to an effective two-dimensional problem is possible. This can serve as a basis for future studies, for instance, of the autodetachment dynamics of anions due to c.m. effects.
In a magnetic field the center of mass (c.m.) motion of an atom or molecule couples to the electronic motion. It is demonstrated that this coupling dramatically influences the properties of negative ions. Neglecting c.m. effects the external field gives rise to a series of infinitely many bound states of the ion. Center of mass effects terminate this series and turn bound states into short-lived resonance states. Whether bound states exist at all, their number and properties as well as the lifetimes of the resonance states depend on the neutral system to which an electron is attached, and on the magnetic field.