Ulrich Mutze
Separated quantum dynamics
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ABSTRACT. An approximation to the time-dependent Schr\"{o}dinger equation
for n distinguishable particles is proposed, for which the computational
solution is an O(n^2) problem.
The Hamiltonian of the system is assumed to be a sum of single-particle
operators and of pair-operators.
The proposed modification of Hamiltonian dynamics is given by a set of n one-particle Schr\"{o}dinger equations,in which the potential felt by each of the particles gets an additive contribution from each of the other particles. This contribution depends on the contributing particle only through the probability density associated with its one-particle wave function.
These equations are similar to time-dependent Hartree-Fock equations,
when the latter are formulated for wave functions instead of density operators. The difference reflects the fact that Hartree-Fock assumes identical particles, whereas the present work is about distinguishable particles.
It is shown that the dynamical evolution of these one-particle wave functions changes neither the norm of these functions, nor the expectation value of the Hamiltonian with respect to the product of the one-particle wave functions.
For the time step evolution operator of this modified dynamics, a simple
definition is given in terms of the Hamiltonian evolution operators
of the whole system and its subsystems.
In a computational model of a three-particle system, the time-discrete
trajectory starting from a product state was computed for the
Schr\"{o}dinger equation and its approximation. Very good agreement of the expectation values of particle positions was found over the whole course of the simulation. For the correlations between two or three particle positions, the agreement was found to decrease considerably with time.