Time Independent Perturbation Theory

Consider the time independent Schrödinger equation,R2(453)

$$\hat{H} \phi_n(x) = E_n \phi_n(x),$$ (10)

for a system described by the Hamiltonian $\hat{H}=\hat{p}^2/2m +\hat{V}$, and assume that all the eigenfunctions $\phi_n(x)$ are known. The goal of this section is to show that these eigenfunctions $\phi_n(x)$ can be used to solve the time independent Schrödinger equation of a slightly different problem: a problem described by the Hamiltonian $\hat{H'}=\hat{H}+\lambda \hat{\omega}$. This is accomplished by implementing the equations of Perturbation Theory derived in this section.

Consider the equation

$$(\hat{H} + \lambda \hat{\omega}) \tilde{\Phi}_n(\lambda ,x) = \tilde{E}_n(\lambda) \tilde{\Phi}_n(\lambda , x),$$ (11)

where $\lambda$ is a small parameter, so that both $\tilde{\Phi}_n(\lambda)$ and $\tilde{E}_n(\lambda)$ are well approximated by rapidly convergent expansions in powers of $\lambda$ (i.e., expansions where only the first few terms are important).

Expanding $\tilde{\Phi}_n(\lambda)$ we obtain,

$$\tilde{\Phi}_n(\lambda , x) = \sum_j {C_{jn}(\lambda) \phi_j(x)}.$$

Substituting this expression in the time independent Schrödinger equation we obtain,

$$\sum_j {C_{jn}(\lambda) [\hat{H} \phi_j(x) + \lambda \hat{\omega} \phi_j (x)]} = \tilde{E_n}(\lambda) \sum_k {C_{kn}{(\lambda)} \phi_k(x)}.,$$

therefore,

$$C_{ln}(\lambda) E_l + \lambda \sum_j C_{jn}(\lambda)<\phi_l\vert\hat{\omega}\vert\phi_j> = \tilde{E_n}(\lambda) C_{ln}(\lambda).$$ (12)

Expanding $C_{kj}$ and $E_n$ in powers of $\lambda$ we obtain,

$C_{kj}(\lambda)= C_{kj}^{(0)} + C_{kj}^{(1)}\lambda + C_{kj}^{(2)} \lambda^2 + ...,$

and

$\tilde{E}_n(\lambda) = E_n^{(0)} + E_n^{(1)}\lambda + E_n^{(2)} \lambda^2 + ...$

Substituting these expansions into Eq. (12) we obtain,

$(C_{ln}^{(0)}E_l^{(0)} -E_n C_{ln}^{(0)}) + \lambda(C_{ln}^{(1)}E_l + \sum_j C... ... - E_n^{(2)} C_{ln}^{(0)} -E_n C_{ln}^{(2)} -E_n^{(1)}C_{ln}^{(1)} ) + ... = 0.$

This equation must be valid for any $\lambda$. Therefore, each of the terms in between parenthesis must be equal to zero.

Zeroth order in $\lambda$ $$\begin{cases} C_{ln}^{(0)} (E_l^{(0)} -E_n) =0, \\ \text{if... ...C_{nn}^{(0)} =1, \ \text{and} \ E_l^{(0)} =E_n. \\ \end{cases}$$

First order in $\lambda$ $$\begin{cases} C_{ln}^{(1)}(E_l -E_n) = E_n^{(1)}C_{ln}^{(0)... ..._{nn}^{(0)}<\phi_n\vert\hat{\omega}\vert\phi_n>.\\ \end{cases}$$

Note that $C_{nn}^{(1)}$ is not specified by the equations listed above. $C_{nn}^{(1)}$ is obtained by normalizing the wave function written to first order in $\lambda$.

2nd order in $\lambda$ $$\begin{cases} C_{ln}^{(2)}(E_l -En) + \sum_j C_{jn}^{(1)}<\p... ...phi_l\vert\hat{\omega}\vert\phi_n>}{(E_l -E_n)}.\\ \end{cases}$$

Exercise 9: Calculate the energy shifts to first order in $\lambda$ for all excited states of the perturbed particle in the box described by the following potential:

Assume that the potential is described by perturbation $W(x)=$Sin$(\frac{\pi x}{a})$ to the particle in the box.