[tex]\psi(x,0) = N exp[-\alpha(x-a)^2][/tex]
(1):This wavefunction is a solution to the time dependent schrödinger equation for a harmonic oscillator, but not to the time independent one. How is that possible?
(2):Explain without calculating how would you find the time dependent wave function [tex]\psi(x,t)[/tex]?
The Attempt at a Solution
(1)According to my book it seems to explain that any solution to the schrödinger equation can be explained by a linear combination of the basis states.
I would assume that if [tex]\psi(x,t)[/tex] is a linear combination of time dependent basis states, that [tex]\psi(x,0)[/tex] would also be a linear combination of time independent basis states, and that [tex]\psi(x,t)[/tex] consist of the same basis states as [tex]\psi(x,0)[/tex] except that each basis state is multiplied with the time dependent factor [tex]exp[-iEn*t/h][/tex]. In that case, [tex]\psi(x,0)[/tex] would be a solution, but it is not. So im stuck in this contradiction.
(2)To find the time dependent state, i would try to identify the basis time independent states describing the wave function. Then multiply each basis state with the corresponding [tex]exp[-iEn*t/h][/tex] factor. However i cant identify any basis time independent states if any in [tex]\psi(x,0)[/tex].