guyvsdcsniper
- 264
- 37
- Homework Statement
- Find the non leading vanishing term to the ground state
- Relevant Equations
- Perturbation theory
The eigenvalue for this ##H_0## is given by ##\hbar \omega(n+1) ; (n_x+n_y = n)##
At the ground state, ##nx = ny = 0## so the eigenvalue is simply ##\hbar\omega##
Next we turn the perturbation potential on and I know that the first order shift in the energy is the expectation value of the perturbing Hamiltonian in the unperturbed state corresponding to that energy.
##E_n^1 = \braket{nx,ny|\hat H_1|nx,ny} = \frac {\lambda \hbar}{2m\omega}[\braket{0,0|a_xa_y+a_xa_y^\dagger + a_x^\dagger a_y + a_x^\dagger a_y^\dagger|0,0} = 0 ##
From here, I am to calculate the second order energy shift of the ground state.
I am having trouble applying the formula,
##\sum_{k\neq n} \frac{|\braket{k|\hat H_1|n}|^2}{E_n^0-E_k^0}##
For this problem nx and ny = 0 in the ground state
##\sum_{k\neq 0,0} \frac{|\braket{k|\hat H_1|0,0}|^2}{E_{0,0}^0-E_k^0}##
I can express ##\hat H_1##
##\sum_{k\neq 0,0}(\frac {\lambda \hbar}{2m\omega})^2 \frac{|\braket{k|\hat x \hat y|0,0}|^2}{E_{0,0}^0-E_k^0}##
I have trouble understand what to do next in this problem. Im not really sure what K would be, I know it just be 0,0 which is what n is.
At the ground state, ##nx = ny = 0## so the eigenvalue is simply ##\hbar\omega##
Next we turn the perturbation potential on and I know that the first order shift in the energy is the expectation value of the perturbing Hamiltonian in the unperturbed state corresponding to that energy.
##E_n^1 = \braket{nx,ny|\hat H_1|nx,ny} = \frac {\lambda \hbar}{2m\omega}[\braket{0,0|a_xa_y+a_xa_y^\dagger + a_x^\dagger a_y + a_x^\dagger a_y^\dagger|0,0} = 0 ##
From here, I am to calculate the second order energy shift of the ground state.
I am having trouble applying the formula,
##\sum_{k\neq n} \frac{|\braket{k|\hat H_1|n}|^2}{E_n^0-E_k^0}##
For this problem nx and ny = 0 in the ground state
##\sum_{k\neq 0,0} \frac{|\braket{k|\hat H_1|0,0}|^2}{E_{0,0}^0-E_k^0}##
I can express ##\hat H_1##
##\sum_{k\neq 0,0}(\frac {\lambda \hbar}{2m\omega})^2 \frac{|\braket{k|\hat x \hat y|0,0}|^2}{E_{0,0}^0-E_k^0}##
I have trouble understand what to do next in this problem. Im not really sure what K would be, I know it just be 0,0 which is what n is.