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

•The exact wavefunction of N interacting particles can be fully factorized.•Each factor is a single-degree-of-freedom wavefunction.•The equation for each factor has the appearance of a Schrödinger equation.•Understanding of the theory may lead to new kinds of approximations to the problem.

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 Schrödinger equation for N coupled degrees of freedom can be represented as a product of N single-degree-of-freedom functions φn, each normalized in the space of its own variable. The N equations determining the φ's are derived. Each of these equations has the appearance of a Schrödinger equation for a single degree of freedom. The equation for φ1 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 φ's are coupled and have to be solved self-consistently. Implications are briefly discussed.

Graphical abstractDownload high-res image (26KB)Download full-size image