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## Homework Statement

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.

## Homework Equations

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.

Thank you!