First Order Perturbation Theory

exercise 3_4928

Given a 1D infinte potentail well, calculate the correction to the eigenenergies using first order perturbation theory.

The perturbed hamiltonian is given by:

\(\hat{H}=\hat{H_0}+\lambda\hat{ V} \ ; \ \lambda <<1\)

\(\hat{H_0}\) is the potential well hamiltonian, the eigenfunction and eigenenergies are:

\(\psi_n(x) = \sqrt{\frac{2}{L}}sin(\frac{\pi n x}{L})\ ; \ E_n = \frac{\hbar^2\pi^2n^2}{2mL^2} \)

The corrections to the eigenenergies of \(\hat{H}\) are:

\(E_n = E_n^{[0]} + \lambda E_n^{[1]} \\ E_n^{[1]} = \langle\psi_n|\hat{V}|\psi_n\rangle \ ; \ E_n^{[0]} \text{ is the unperturbed eigenenergies}\)

find the corrections for:

(a) \(\hat{V} = V_0\)

(b) \(\hat{V} =\delta(x-L/2)\)

(c) \(\hat{V} = \begin{cases} V_0 & \quad x \in [0,L/2] \\ 0 & \quad x \in [L/2,0] \end{cases} \)

In your answers pay attention if there's a different correction for even and odd n's.

Solve (b) and (c) again but for \(\hat{H_0}=\frac{\hat{P}^2}{2m}\), the hamiltonian of a free particle. Did you get a different results?