Using the test functionnbspphi c cosalphax defined in the

Question

Consider the linear harmonic oscillator Hamiltonian:

H^ = -h²d²/2mdx² + 1/2(kx²)

Using the test function φ = C cos(αx) defined in the range -Π/2 <= αx <= Π/2

a) find a relationship between C and α to normalize the function.

b) find the expectation value (W(α)) = ∫φ*Hφdx

c) minimize the value of (W(α)) to find the variational function.