- #1
Antti
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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
[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? I am 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' ?
[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? I am 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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