- #1
CAF123
Gold Member
- 2,948
- 88
Homework Statement
[/B]
The isotropic harmonic oscillator in 2 dimensions is described by the Hamiltonian $$\hat H_0 = \sum_i \left\{\frac{\hat{p_i}^2}{ 2m} + \frac{1}{2} m\omega^2 \hat{q_i}^2 \right\} ,$$ for ##i = 1, 2 ## and has energy eigenvalues ##E_n = (n + 1)\hbar \omega \equiv (n_1 + n_2 + 1)\hbar \omega, n = 0, 1, 2, \dots##. What is the degeneracy of the first excited level? Use degenerate perturbation theory to determine the splitting induced by the perturbation ##\hat H' = K\hat q_1 \hat q_2## where ##K## is a constant.
Homework Equations
[/B]
Raising and lowering operators
The Attempt at a Solution
The degeneracy of the first excited state is 2 so the splitting of the energy to first order is given by the expression $$\sum_{n=1}^2 b_n (\hat H'_{kn} - E^{(1)} \delta_{kn}) = 0$$. The matrix element I need to compute is $$\hat H'_{kn} = \langle E_k^{(0)} | K \hat q_1 \hat q_2 | E_n^{(0)} \rangle = \frac{K \hbar}{2 m \omega} \langle E_k^{(0)} | (\hat a_1 + a_1^{\dagger}) (\hat a_2 + a_2^{\dagger})| E_n^{(0)} \rangle$$ I then thought if the ##\hat a_i## raise the ##n_i## or ##k_i##, I could write this like $$\frac{K \hbar}{2 m \omega} \langle k_1, k_2^{(0)}| (\hat a_1 + a_1^{\dagger}) (\hat a_2 + a_2^{\dagger})| n_1, n_2^{(0)} \rangle$$ and then for example, ##\hat a_1 \hat a_2 |n_1, n_2 \rangle = \hat a_1 \sqrt{n_2} | n_1, n_2 - 1 \rangle = \sqrt{n_1} \sqrt{n_2} | n_1 - 1, n_2 - 1 \rangle?##. Would this be right? Many thanks.