Trying to piece together a full time dependent wave function, I have solved for the constant (call it "a_n"), which, when squared and summed from 1 to infinity is intended to give us the probability of a specific allowed energy.
The initial wave function at t = 0: Psi[x,0] = a (x-L/2)^2 (x+L/2)
where I found constant a to be Sqrt[105/L^7]
the constant a_n should be equal to the integral of -L/2 to L/2 for initial Psi[x,0] multiplied by the conjugate of the time independent solution (which i think is the same either way): Sqrt[2/L] Sin[n Pi x/L]
When we sum the constant (a_n)^2 from 1 to infinity, we are supposed to find all probabilities of allowed energies, and the sum equals 1.
However, when I perform the sum in mathematica, I get returned an answer of 1/4
The odd numbered n's return 1/8 for the sin solutions, and the even number n's for the cos solutions return the same thing. If n is a multiple of 2, the cos solutions are negative, but if multiple of 4, positive.
I've attached a segment of my mathematica code for reference. mathematica code for work so far
If anyone could double check my work so far and try to explain why I am not getting the correct sum, that'd be great. I don't know if I miscalculated a_n, or the initial constant "a", but I can't see any errors. :(