Given the following potential energy for a molecular vibration:

Given the following potential energy for a molecular vibration:

V(x)=1/2kx2+c1x3+c2x4+c3x5

Use first order perturbation theory to calculate the first three energy levels of an oscillating molecule with this potential. α=mω/ℏ

ψ0(x)=(α/pi)1/4exp(-αx2/2)

ψ1(x)=(4α3/pi)1/4xexp(-αx2/2)

ψ3(x)=(α/4pi)1/4(2αx2-1)exp(-αx2/2)