### 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:

$$E_n = E_n^{} + \lambda E_n^{} \\ E_n^{} = \langle\psi_n|\hat{V}|\psi_n\rangle \ ; \ E_n^{} \text{ is the unperturbed eigenenergies}$$

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?