How Do You Compute the 1st Order Wave Function Correction in Quantum Mechanics?

[cscott]

Homework Statement

A have a bit of a general question regarding 1st order wave function corrections using perturbation theory.

In a problem like the infinite potential well where you have states numbered like n = 1, 2, 3, ..., how do you compute the sum for the 1st order correction when you have infinite terms?:

$$\psi_n^{(1)} = \Sigma_{l \ne n} \frac{<\psi_n^{(0)}|H'|\psi_l^{(0)}>}{E_n^{(0)} - E_l^{(0)}} \psi_l^{(0)}$$

I guess I don't know how to get <n|H'|l> so I can evaluate the sum

 

[kuruman]

Do you know what your perturbation H' looks like? Sometimes the non-zero terms in the summation result in something that can be summed analytically.

 

[cscott]

This was the thinking I was missing!

So for H' = constant there is no first-order correction because $l \ne n$, yes?

 

[kuruman]

Correct. If you add a constant to your Hamiltonian, you shift the zero of energy but you do not change its eigenstates.

 

[paranormal]

I would first clarify that the problem is referring to the first-order correction to the wave function using perturbation theory. This method is commonly used in quantum mechanics to approximate the energy levels and wave functions of a system by considering a small perturbation to the original Hamiltonian.

To compute the first-order correction for the wave function, we need to consider the perturbation term, H', and its effect on the unperturbed states, which are represented by the wave functions \psi_n^{(0)}. The formula provided is the correct expression for the first-order correction, where the sum is taken over all states (l) except for the unperturbed state (n).

In order to evaluate the sum, we need to calculate the matrix element <\psi_n^{(0)}|H'|\psi_l^{(0)}>, which represents the overlap between the perturbed and unperturbed states. This can be done by solving the Schrödinger equation with the perturbation term included, and then taking the inner product between the resulting wave functions.

While the sum may seem infinite, in practice we can truncate it after a certain number of terms, as the contribution from higher energy states is usually small. In summary, the first-order correction to the wave function can be computed by evaluating the matrix element and summing over all states except the unperturbed state.