Probability and Flux Densities in the Center-of-Mass Frame

Abstract

For an arbitrary nonstationary wave function of a nonrelativistic closed many-body system consisting of arbitrary interacting particles, the general expressions for the time-dependent one-particle probability and flux densities in the center-of-mass frame without applying Born-Oppenheimer approximation are obtained. Even the wave function for the translation is additionally introduced; it disappears in the center-of-mass frame automatically. It is shown that for the rotational ground state the time-dependent probability and flux densities of an arbitrary particle in the center-of-mass frame are isotropic. It means that the angular dependence is absent but these densities depend on radius and time. More importantly, it is shown that the angular components of the time-dependent flux density vanish. With these statements, one can calculate the radial component of the radius- and time-dependent electronic flux density within the Born-Oppenheimer approximation via the continuity equation. Application of this theory to the pulsating or exploding "quantum bubble" of the vibrating or dissociating Na molecule in the rotational ground state, respectively, is found elsewhere in this issue.

Na2 Vibrating in the Double-Well Potential of State 2 1Σu+ (JM = 00): A Pulsating “Quantum Bubble” with Antagonistic Electronic Flux

“…In particular, the symmetry of the state implies that the EPD, NPD, and NFD are isotropic. A theorem due to Barth, based in part on ref , permits us to deduce that the EFD is also isotropic. The reduced radial continuity equation yields an expression for the EFD in terms of the EPD.…”

“…Elsewhere in this issue Barth proves the following theorem: If the molecular wave function satisfies the SE iℏ ∂|Ψ(t)⟩/∂t = H|Ψ(t)⟩ and the total angular momentum and its z-component vanish (i.e., J = 0 and M = 0), then the electronic and nuclear flux densities are spherically symmetric. 4 Though the BOA wave function does not obey the SE, it does satisfy the constraints of the theorem approximately. Therefore, we assume that the EFD for our state of interest is spherically symmetric.…”

“…We show that the rotational quantum numbers JM = 00 imply isotropy of the nuclear probability and flux densities, as well as that of the electronic population density and we infer from this agreement with the one-electron case that the electronic flux density is also isotropic. (A rigorous proof of this property is presented in this volume by Ingo Barth …”

“…(A rigorous proof of this property is presented in this volume by Ingo Barth. 4 ) The isotropy of all densities and fluxes means that the nonrotating multielectron homonuclear diatomic molecule evolves as a pulsating "quantum bubble", irrespective of the number of electrons.…”