Exact centre-of-mass separation is performed for the problem of a charged two-body system in a homogeneous magnetic field. Common eigenstates of three exact and approximate constants of motion are constructed in the creation and annihilation operator formalism. Their expression as an infinite series in coordinate representation is obtained using two real linear canonical transformations. An expansion of the wavefunction in this eigenstate basis provides an infinite system of differential equations for the relative motion. This system of equations differs from the infinite nucleus mass approximation by the existence of small terms which couple states with different values of the magnetic quantum number. The usual limitation of the number of electron Landau states leads to a finite number of coupled equations.
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.
The techniques of center of mass (c.m.) separation in a homogeneous magnetic field are tested on an exactly solvable problem: a bound system of arbitrary particles interacting with a general quadratic potential. An effective mass increasing with the field strength appears naturally in the collective energy and shows striking similarities for charged and neutral systems. In the charged case, the internal motion cannot be separated from the c.m. motion. These properties and several techniques discussed in the present paper are also valid for a Coulomb interaction. The present exact model provides a test of approximate separation methods employed in the Coulomb case.
For neutral molecules 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 molecular ions a complete set of constants of motion is not known, neither exactly nor approximately. In this paper we investigate the structure an approximate constant of motion for a molecular ion can have in general and arrive at an interesting formula. An explicit approximate constant of motion that obeys this structure formula is derived from a physically intuitive picture that says that an approximately conserved quantity can be obtained as the pseudomomentum relative to the instantaneously fixed-nuclei center of charge of the system. Some special examples are discussed in detail.
The motion of a charged particle in a constant magnetic field is treated in both relativistic and non-relativistic quantum theory. Operators representing the center of the orbit, which obey the commutation law for conjugate variables, are introduced and their connections with energy, angular momentum, and magnetic moment studied. Energy eigenfunctions in an operator form are obtained by factorization. Previously derived eigenfunctions in coordinate space are obtained and are shown to be eigenfunctions for the operators for the center of the orbit as well as for the energy. Corresponding relativistic eigenfunctions are derived by a simple device which enables one to construct solutions of the Dirac equation from solutions of the Schr\"odinger equation.
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.