1. The problem statement, all variables and given/known data One possible solution for the wave function ψn for the simple harmonic oscillator is ψn = A (2*αx2 -1 ) e-αx2/2 where A is a constant. What is the value of the energy level En? 2. Relevant equations The time independent Schrodinger wave equation d2ψ / dx2 = (α2 * x2 - β )ψ β = 2mE/ [tex]\hbar[/tex]2 3. The attempt at a solution So I took the solution that we were given ψn and found the second derivative of this. (I found A[3α + 3α2x2 -2α3x4]e-αx2/2, but I haven't triple checked my algebra yet, I'll do that later). So I have d2ψ / dx2 = (α2 * x2 - β )ψ A[3α + 3α2x2 -2α3x4]e-αx2/2 = (α2 * x2 - β ) * A (2*αx2 -1 ) e-αx2/2 (In a nutshell I plugged the solution we were given into the time independent Schrodinger wave equation). Then I did some tedious algebra, observed that my A's cancelled, and my exponentials cancelled, leaving me with 3α + 3α2 x2 - 2α3x4 = (α2 x2 - β) (2αx2 -1) So some tedious algebra and polynomial division (oh and subsitituting β = 2mE/ [tex]\hbar[/tex]2) yields E = [2 α x2 - α - 4α/[2αx2 -1] ] Which would all be well and good, except I have approximately 0 confidence in this answer. I'm finding this material really confusing, and I don't really know what an answer should look like, is it okay to have α's in it? I don't know... It seems too convenient that my A's just cooperatively went away, that part just seems too good to be true. Anyway, if anybody could give me any encouragement, or pointers towards another way of looking at this, I'd really appreciate it! Thanks!