The ground states of small atoms and molecules, namely those of the hydrogen, helium, and lithium atoms and the hydrogen molecular ion and hydrogen molecule, are studied in the presence of an external magnetic field. For the one-electron systems the ground state is of the same symmetry for arbitrary field strengths, and the corresponding binding energy shows a monotonic increase with increasing field strength. More interesting, the two- and three-electron systems show ground-state transitions which involve both transitions in the spin multiplicity and spatial quantum numbers. An outlook on the expected cascade of transitions in many electron atoms and molecules is provided. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 789–795, 1998
We have calculated the rates for the processes that result from collisions of positrons with Rydberg atoms in strong magnetic fields. This is the first step in a two-stage charge transfer, which is being explored as a mechanism for the formation of cold anti-hydrogen. Unlike previous theoretical explorations we have also investigated the cases when the energy of the positrons can be comparable to or larger than the binding energy of the atom. We have also examined how big an electric field is needed to destroy the resulting positronium. We find that the charge transfer does not scale as for n = 40 states for positron temperatures above T ∼ 25 K. Also, we find that the positronium atoms are more easily destroyed by electric fields if they emerge from the charge transfer along the magnetic field lines, contrary to what might be expected for the effective electric field. Both results have implications for the formation of anti-hydrogen atoms.
A simple and accurate variational method is formulated and applied to the ground and excited states of the hydrogen atom as well as the hydrogen molecule ion in a homogeneous magnetic field of arbitrary strength. For the ground state, the electronic wave function o is assumed to be a function of a single, appropriately chosen variable R which contains a variational parameter. The functional form of o, and the corresponding energy, are determined by solving a Schr6dinger- like equation, in which the variational parameter appears explicitly in the kinetic and potential energy terms. The variational parameter is optimized by minimizing the lowest eigenvalue of the equation. Excited states may also be obtained by a slight modification of the method, and explicit calculations are carried out for the m1 = - 1 first excited state of hydrogen and the ( )-state of H2 +. Numerical comparisons are made with some recent papers in the literature. Subject headings: atomic processes - magnetic fields - molecular processes - quantum mechanics
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.
We investigate the hydrogen molecule in a strong parallel magnetic field using a fully numerical Hartree-Fock approach. We find that for magnetic fields below $4.2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ T the ground state of
H2 is the strongly bound singlet state ${}^{1}{\ensuremath{\Sigma}}_{g},$ for magnetic fields stronger than $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the ground state of the molecule is the strongly bound triplet ${}^{3}{\ensuremath{\Pi}}_{u},$ and for magnetic fields between $4.2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ T and $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the symmetry of the ground state is the triplet state ${}^{3}{\ensuremath{\Sigma}}_{u},$ which is characterized by repulsion at intermediate internuclear distances and by a weak quadrupole-quadrupole interaction between atoms at large internuclear separation. In this region of magnetic field strength the hydrogen molecule is bound weakly, if at all; the hydrogen atoms behave like a weakly nonideal gas of Bose particles and can form a superfluid phase predicted in earlier works [Korolev and Liberman, Phys. Rev. Lett. 72, 270 (1994)]. For magnetic fields between $\ensuremath{\approx}3\ifmmode\times\else\texttimes\fi{}{10}^{5}$ T and $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the triplet state ${}^{3}{\ensuremath{\Pi}}_{u}$ is found to be metastable. This state may be responsible for an unknown excitonic line observed experimentally [Timofeev and Chernenko, JETP Lett. 61, 617 (1995)].