Exact solutions of nonrelativistic classical and quantum field theory with harmonic forces (Q1821272)

The nonlinear field equation \[ i\hslash \partial_ t \psi (x,t)=[\frac{p^ 2}{2m}+\frac{m\omega^ 2}{2}\int dx'\psi^+(x',t)\psi (x',t)(x'-x)^ 2]\psi (x,t);\quad (p=-i\hslash \partial x) \] describing nonrelativistic particles interacting via harmonic forces is studied. The evolution of identified collective variables of the system is obtained, and the fully quantum solution of the initial problem for the field operator \(\psi\) (x,t) is found. It is compared with the classical solution of the same problem. The classical formula is obtained from the quantum one, taking the limit of large number of particles \(N=\int | \psi (x,t)|^ 2 dx\) and desregarding the noncommutativity of the \[ u(t)=()\int dx \psi^+ x^ 2 \psi -(1/2N)(\int dx \psi^+ x\psi)^ 2 \] operators at different times. However, the reconstruction of the quantum field at the time from the classical solution is impossible.