The Equation for the Wave Function of an N-Particle System in the Many-Body Problem

Mohsen Razavy

A classical many-body problem composed of an infinite number of mass points coupled together by springs is quantized. The masses and the spring constants in this system are chosen in such a way that the motion of each particle is exponentially damped. Because of the quadratic form of the Hamiltonian, the many-body wave function of the system can be written as a product of two terms: a time-dependent phase factor which contains correlations between the classical motions of the particles, and a stationary state solution of the Schrödinger equation. By assuming a Hartree type wave function for the many-particle Schrödinger equation, the contribution of the time-dependent part to the single particle wave function is determined, and it is shown that the time-dependent wave function of each mass point satisfies the nonlinear Schrödinger–Langevin equation. The characteristic decay time of any part of the subsystem, in this model, is related to the stiffness of the springs, and is the same for all particles.

L. Szasz

В.Г. Соловьев

S>In previous work N. N. Bogolyabov (Doklady Akad. Nauk 1l9, 244, 1958) suggested a generalization for the Fock method in which the energy minimum is found in the individual particle quasi-independent functions. The suggested variation utilizes a wider span of functions which consider the wave functions of particle pairs in addition to the quasi-independent functions of individual particles. The new variation is used for deriving the equations for a wave function of N particle systems in the many-body problem. The new principle and mathematical method permit both particle pair and N particle system correlations. It is pointed out that while the number of particles in spatially uniform problems is conserved, in spatially non-uniform problems it is not. (R.V.J.)

T. D. Lee, C. N. Yang

Richard L. Hall

A system of identical nonrelativistic particles is considered. It is shown that the wave functions for relative motion, which have the correct permutation symmetry, must satisfy two functional equations. In the case of bosons these equations are solved for those bound states where the wave function is also in a single-product form. The only solutions are Gaussian functions. Consequently these are the only functions which can reduce the N-body energy expectation to an integral over a single variable. Furthermore, we show that our reduced two-body Hamiltonian which in general gives energy lower bounds yields the exact energy of the entire system only for the Hooke's law interaction. Neither possibility is allowed by fermions.

A. P. Machil'skii