I have an infinite potential well with length L. The first task was to calculate the eigenvalues and -functions for the energy of a particle in the well. The requirements were(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\psi(0, L) = 0[/tex] and there is no time-dependence.

I've calculated:

[tex]\hat{H}\psi(x) = E\psi(x)[/tex]

[tex]E = \frac{(\pi\hbar n)^{2}}{2 m L^{2}}[/tex]

[tex]\psi(x) = sin(\frac{\pi\n x}{L})[/tex]

Now the question. We add a small potential "rectangle" V(x) at the center of the potential well. It has length a and height q, a << L. What are the new eigenvalues and -functions for the perturbed case? Im supposed to use first order perturbation theory.

I've done like this:

There's a formula saying:

[tex]E_{n_p} = \int \overline{\psi_{n}} H' \psi_{n} dV =[/tex]

[tex]\int \overline{\psi_{n}} H' \psi_{n} x^{2} dx[/tex]

Which gives the new eigenvalues. I tried just using H' = V(x) = q at first. But the dimension of E didn't match. If you substitute V(x) (which is an energy) into the above equation you see that the dimension will be Energy*distance^4 after integration. So what to do with H' ?

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# Homework Help: First order perturbation theory, quantum physics

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