Conventional description of the matter wave

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George444fg
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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 Schrodinger equation that would imply negative energy states. Am I right?
 

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