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etotheipi
I apologise in advance if this is a silly question! We are aware that the wave function ##\Psi(x, t)## of a particle moving along one dimension will satisfy the differential equation$$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + V(x)\Psi$$We are told that if we assume a general form ##\Psi = Ae^{i(kx - \omega t)}## (Edit, cred. PeroK: For ##V(x) = 0##) then the equation will spit out physically correct results. But what about all of the other possible forms for ##\Psi##, do we just discard them? There are some examples that we can discard immediately, e.g. something like ##\Psi = Ae^{i(x - vt)}## that is not dimensionally homogenous. More generally if we run other forms than ##Ae^{i(kx-\omega t)}## through the equation then whilst we can find mathematically acceptable solutions, the results are silly physically (for starters, they don't agree with with the De Broglie relations).
Are there any other forms for ##\Psi## that give valid results?
Are there any other forms for ##\Psi## that give valid results?
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