# QM - Perturbation to SHO

1. Nov 26, 2011

### NeedPhysHelp8

1. The problem statement, all variables and given/known data
The potential of a simple harmonic oscillator of HF has the following form
$$\frac{1}{2}kx^2 + bx^3 + cx^4$$
The first part of the problem involved finding expressions for the first-order energy corrections for the first three states, which I found below. Basically the x3 term does not contribute at all only x4:
$$E_{0}^{'} = \frac{3}{4}c(\frac{\hbar}{m\omega_0})^2$$
$$E_{1}^{'} = \frac{15}{4}c(\frac{\hbar}{m\omega_0})^2$$
$$E_{2}^{'} = \frac{39}{4}c(\frac{\hbar}{m\omega_0})^2$$
Now the next part of the problem supplied the known transition energies from the 1st and 2nd excited state to the ground state. I was then able to figure out the value for the natural frequency of the unperturbed harmonic oscillator by constructing the following equations knowing that the energy is equal to the unperturbed (E=1/2(n+1)hw) + the perturbed found previously :

$$\Delta E_{1,0} = \hbar\omega_0 + c \frac{9}{8}(\frac{\hbar}{m\omega_0})^2$$
$$\Delta E_{2,0} = 2\hbar\omega_0 + c 9(\frac{\hbar}{m\omega_0})^2$$

I solved these equations and found ω0 but the next part involves calculating the constants k and c . I know the relation that:
$$k = m\omega_0^2$$
But I can't figure out a way to solve for each individually only for c/k2

Thanks!

edit: should I just assume I need to look up a reduced mass for HF? or can it be solved another way

Last edited: Nov 26, 2011