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

 

[denian]

The Non-Linear Schroedinger equation is a complex equation that describes the behavior of quantum systems, particularly those involving particles with mass. It is a non-linear version of the original Schroedinger equation, which is a fundamental equation in quantum mechanics. The Non-Linear Schroedinger equation is often used to study the dynamics of Bose-Einstein condensates and other quantum systems.

Solving the Non-Linear Schroedinger equation for specific boundary conditions can be a challenging task, as the equation is non-linear and does not allow for the application of the superposition principle. This means that the wavefunction cannot be broken down into simpler parts, making it difficult to solve for specific boundary conditions.

To express the probability of finding a particle in a certain state with energy E_a, one can use the Born rule. This rule states that the probability of finding a particle in a particular state is proportional to the square of the magnitude of the wavefunction at that point in space. In other words, the probability is given by |ψ|^2, where ψ is the wavefunction.

In order to determine the wavefunction and thus the probability, one would need to solve the Non-Linear Schroedinger equation for the given boundary conditions and then calculate the magnitude squared of the resulting wavefunction. This can be a complex and time-consuming process, but it is essential for understanding the behavior of quantum systems.

In summary, the Non-Linear Schroedinger equation is a fundamental equation in quantum mechanics that can be used to study the behavior of quantum systems. Solving it for specific boundary conditions can be challenging, and the probability of finding a particle in a certain state can be determined using the Born rule.