Non-Linear Schroedinguer equation

[eljose]

Let be the NOn-linear Schroedinguer equation:

i\hbar \frac{\partial \psi}{\partial t}=-\hbar^{2}(2m)^{-1} \nabla ^{2} \psi + |\psi|^{3}

for example..the question is..how the hell do you solve it for certain boundary conditions that the Wavefunction must satisfy if you can,t apply superposition principle?...How do you express the probability of finding a particle in state a with energy E_a?..

 

[Careful]

Let be the NOn-linear Schroedinguer equation:

i\hbar \frac{\partial \psi}{\partial t}=-\hbar^{2}(2m)^{-1} \nabla ^{2} \psi + |\psi|^{3}

for example..the question is..how the hell do you solve it for certain boundary conditions that the Wavefunction must satisfy if you can,t apply superposition principle?...How do you express the probability of finding a particle in state a with energy E_a?..

A few comments :
(a) you usually find good approximations through an iteration procedure starting from a solution of the linear equation.
(b) the interesting part about non-linear equations is that soliton, ie. particle-like, solutions might exist.
(c) One should not interpret the \psi wave associated to the NLSE as a probability density - there is however a ``clever'' way to get to the standard probablity interpretation; see ``combining relativity and quantum mechanics : Schroedingers interpretation'' A.O. Barut 1987.
(d) energy eigenstates are never observed and neither does a full dynamical treatment of the radiation degrees of freedom allow for such thing *theoretically*.

Careful