Conventional description of the matter wave

[George444fg]

TL;DR

Conventional description of the matter wave

I have been working on a relatively simple problem. Just take a quantum wave function for which a physical requirement is that an arbitrary displacement of x or an arbitrary shift of t should not alter the character of the wave, and I want to find the state function solution. A possible guess that works is sin(kx-wt)+acos(kx-wt). I found out that a=±i, and then I have to say which one corresponds to the convention. I said that it must be that γγ=i, because if it was -i, then the time derivative of the state function would have been negative, and using Schrödinger equation that would imply negative energy states. Am I right?

 

[azntoon]

Yes, you are correct. The convention that is usually used is that the wave function should have a positive energy, and so the time derivative of the wave function should be positive. Therefore, the coefficient of the cosine term must be +i in order for the wave function to satisfy this requirement.