So I'm trying to solve old qualifying exam problems, one of which is a particle on a ring with a constant electric field perturbation. The un-perturbed problem is straightforward, and we then add a constant electric field in the x-direction (the ring lies in the xy-plane) of magnitude E. Therefore, we add [itex] -ReE \cos \phi [/itex] as a perturbing term to the Hamiltonian. R is the radius of the ring and phi is the angular coordinate on the ring.
We then ask for the first (non-zero) energy correction to the original ground state. Because of the cosine term, the first order correction vanishes, I believe, so we need to look to second order or beyond.
However, the second order correction involved a sum of terms of the form:
[tex] \langle \psi_{k} | H_{1} | \psi_{0} \rangle =\alpha \int_{0}^{2 \pi} d \phi e^{ik \phi} \cos \phi [/tex]
Unless I'm screwing up badly, I believe the above terms are also zero. So am I making a mistake here? Or am I correct and I need to look for third order corrections to get a non-zero term?
We then ask for the first (non-zero) energy correction to the original ground state. Because of the cosine term, the first order correction vanishes, I believe, so we need to look to second order or beyond.
However, the second order correction involved a sum of terms of the form:
[tex] \langle \psi_{k} | H_{1} | \psi_{0} \rangle =\alpha \int_{0}^{2 \pi} d \phi e^{ik \phi} \cos \phi [/tex]
Unless I'm screwing up badly, I believe the above terms are also zero. So am I making a mistake here? Or am I correct and I need to look for third order corrections to get a non-zero term?
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