- #1

knowlewj01

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## Homework Statement

An electron is confined to a 1 dimensional infinite well [itex]0 \leq x \leq L[/itex]

Use lowest order pertubation theory to determine the shift in the second level due to a pertubation [itex]V(x) = -V_0 \frac{x}{L}[/itex] where Vo is small (0.1eV).

## Homework Equations

[1]

[itex]E_n \approx E_n^{(0)} + V_{nn}[/itex]

[2]

[itex]V_{nn} = \int_{-\infty}^{\infty} \psi_{n}^{(0) *} (x) V(x) \psi_{n}^{(0)} (x) dx[/itex]

the following integral may be useful:

[3]

[itex]\int_{0}^{2\pi}\phi sin^2 \phi d\phi = \pi^2 [/itex]

## The Attempt at a Solution

From [1] and the known result for E2 of an infinite well

[itex]E_2 = \frac{4\hbar^2 \pi^2}{2mL^2} - \frac{2V_0}{L^2}\int_{0}^{L} x sin^2\left(\frac{2\pi x}{L}\right) dx[/itex]

I can't see a substitution that will get it into the form in [3], anyone have any ideas?

Also, is equation [1] a general result for the time independant case for first order pertubations?

Thanks