A question on Schroedinguer equation

[eljose]

let be the SE equation in the form:

i\bar\frac{d\psi}{dt}=-\frac{\hbar^{2}}{2m}D^{2}\psi+V(x)\psi+NV_{0}\psi

where N is a big big number N>>1 then what would be the solution?..thanks.

 

[Tzar]

That depends on V(x) doesn't it??

 

[Physics Monkey]

eljose,

You will find that physical probabilities are independent of N and V_0. The only thing that will depend on N V_0 is the average total energy. This is a physical consequence of the fact that the reference point of potential energy is arbitrary in quantum mechanics.

This is mathematically evident in that any position independent potential can be absorbed into the wave function as a phase. You can check for yourself that the substitution \psi = e^{-i N V_0 \,t/\hbar} \psi' yields a Schrodinger equation for \psi' given by

<br /> i \hbar \frac{\partial \psi&#039;}{\partial t} = - \frac{\hbar^2}{2 m} \nabla^2 \psi &#039; + V(x) \psi&#039;<br />

However, since \psi and \psi&#039; only differ by an overall phase, albeit a time dependent one, they produce the same physics.