The exact wavefunction of interacting N degrees of freedom as a product of N single-degree-of-freedom wavefunctions

摘要

Solving quantum systems with many or even with only several coupled degrees of freedom is a notoriously hard problem of central interest in quantum mechanics. We propose a new direction to approach this problem. The exact solution of the Schrodinger equation for N coupled degrees of freedom can be represented as a product of N single-degree-of-freedom functions phi(n), each normalized in the space of its own variable. The N equations determining the phi's are derived. Each of these equations has the appearance of a Schrodinger equation for a single degree of freedom. The equation for phi(1) is particularly interesting as the eigenvalue is the exact energy and the density is an exact density of the full Hamiltonian. The ordering of the coordinates can be chosen freely. In general, the N equations determining the phi's are coupled and have to be solved self-consistently. Implications are briefly discussed. (C) 2015 Elsevier B.V. All rights reserved.