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raintrek
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[SOLVED] Integration change of variables
An electron is confined in an infinite one dimensional well where 0 < x < L with L = 2 x 10^-10m. Use lowest order perturbation theory to determine the shift in the third level due to the perturbation:
[tex]V(x) = V_{0}\left(\frac{x}{L}\right)^{2}[/tex]
where [tex]V_{0} = 0.01eV[/tex].
After a change of variables, the following integral will be useful:
[tex]\int^{3\pi}_{0}\phi^{2}sin^{2}\phi d\phi = \frac{9}{2}\pi^{3} - \frac{3}{4}\pi[/tex]
I've evaluated this question into the following integral:
[tex]\Delta E_{3}^{(3)} = \frac{2}{L}\frac{V_{0}}{L^{2}}\int^{L}_{0} x^{2} sin^{2}\left(\frac{3\pi x}{L}\right) dx[/tex]
However I have no idea how to "change variables" with an integral like this, let alone how to get the limits change from 0-L to 0-3pi. Can anyone offer assistance? Many thanks in advance...
Homework Statement
An electron is confined in an infinite one dimensional well where 0 < x < L with L = 2 x 10^-10m. Use lowest order perturbation theory to determine the shift in the third level due to the perturbation:
[tex]V(x) = V_{0}\left(\frac{x}{L}\right)^{2}[/tex]
where [tex]V_{0} = 0.01eV[/tex].
After a change of variables, the following integral will be useful:
[tex]\int^{3\pi}_{0}\phi^{2}sin^{2}\phi d\phi = \frac{9}{2}\pi^{3} - \frac{3}{4}\pi[/tex]
The Attempt at a Solution
I've evaluated this question into the following integral:
[tex]\Delta E_{3}^{(3)} = \frac{2}{L}\frac{V_{0}}{L^{2}}\int^{L}_{0} x^{2} sin^{2}\left(\frac{3\pi x}{L}\right) dx[/tex]
However I have no idea how to "change variables" with an integral like this, let alone how to get the limits change from 0-L to 0-3pi. Can anyone offer assistance? Many thanks in advance...