Consider the Hamiltonian H=0.5p^2+ 0.5x^2, which at t=0 is described by:
ψ(x,0)= 1/sqrt(8*pi) θ1(x) + 1/sqrt(18pi) θ2(x), where:
θ1= exp(-x^2/2); θ2=(1-2x^2)*exp(-x^2/2)
a) Normalize the eigenfunctions and rewrite the initial state in terms of normalized eigenfunctions. Determine the values of energy corresponding to these eigenfunctions.
In (a), to normalize, take the integral of the function's conjugate times the function.
The Attempt at a Solution
I began by normalizing θ1 and θ2. The first one:
A^2∫exp(-x^2)dx=1, so A=1/pi^(1/4)
For θ2, B^2[∫(exp(-x^2/2) -4∫(exp(-x^2/2)*x^2 + 4∫(exp(-x^2/2)*x^4]=1
So 1=B^2[sqrt(pi)(1 - 4/2 + 4(3/4)] ==> B= 1/(4*pi)^1/4
Now I have to rewrite the initial state in terms of normalized eigenfunctions. My question is, do I simply divide the first term in ψ by A and the second term by B? Or do I have to divide the entire ψ function by some common value, based on A and B? Also, I am assuming the question does not require to normalize ψ, based on the fact that the coefficients there are not normalized.