- #1
DataGG
Gold Member
- 157
- 22
I've no idea if I should be posting this here or in the general forums.
This is not really an exercise as much as an example. I'm not understanding something though:
1. Homework Statement
Using perturbation theory, find the exact expression for the energy given by the hamiltonian:
$$\hat{H}=\hat{H_0} + \hat{H_p} = - \dfrac{\hbar}{2m}\dfrac{d^2}{dX^2} + \dfrac{1}{2}m\omega \hat{X}^2 + q\epsilon \hat{X}$$
I'm only having problems with the second-order correction.
The formula for the second-order correction is below. My question is: How do we find ##m## and ##n##? Is their inequality the only restriction? Can they both go to infinity? How do I find where the sum stops?
$$E^{(2)}_n=\sum _{m \ne n} \dfrac{|<\phi _m |\hat{W}|\phi _n >^2 |}{E_n ^{(0)} -E_m^{(0)}} $$
[/B]
This is an example. The full exercise is explained in a book. However, in order to compreehend one of the steps, I need to find out how to find out where the sum stops.
This is not really an exercise as much as an example. I'm not understanding something though:
1. Homework Statement
Using perturbation theory, find the exact expression for the energy given by the hamiltonian:
$$\hat{H}=\hat{H_0} + \hat{H_p} = - \dfrac{\hbar}{2m}\dfrac{d^2}{dX^2} + \dfrac{1}{2}m\omega \hat{X}^2 + q\epsilon \hat{X}$$
I'm only having problems with the second-order correction.
The formula for the second-order correction is below. My question is: How do we find ##m## and ##n##? Is their inequality the only restriction? Can they both go to infinity? How do I find where the sum stops?
Homework Equations
$$E^{(2)}_n=\sum _{m \ne n} \dfrac{|<\phi _m |\hat{W}|\phi _n >^2 |}{E_n ^{(0)} -E_m^{(0)}} $$
The Attempt at a Solution
[/B]
This is an example. The full exercise is explained in a book. However, in order to compreehend one of the steps, I need to find out how to find out where the sum stops.
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