Electronic Properties of Materials/Quantum Mechanics for Engineers/Perturbation Methods

This is the ninth chapter of the first section of the book Electronic Properties of Materials.


Most operators (Hamiltonians) are not simple. Fortunately, with a bit of effort, we can sometimes rewrite the operator () as , where is a Hamiltonian for which we know the solution.

Here, are non degeneratory orthogonal eigenfunctions, and is a small perturbation to the . Additionally, is a real arbitrary parameter, and when , we have:

The problem we want to solve is:

The perturbation is small and in the limit goes to zero.

We will assume that and can be written as powers of .


Multiply through & collect common properties to form equations for each power of :

The powers of are just our unperturbed . We will begin by looking at powers of .


multiply left by and integrate

Begin with left term. These operators are Hermitian. They have special properties, namely that they obey the postulates of quantum mechanics, including a few revations that are useful for proofs. One such property is:

Which we will use here:

Thus, our entire term equals zero. As a result:

Therefore, the first order correction to the eigenvalue is:

Following the same steps we can find the higher order perturbations:

Most simple theories do not require these higher order corrections, but how do we get the wavefunctions in the first place? Lets assume that where coefficient is the projection of onto . Returning to our original term, gather:

Rearranging and substituting gives us:

Multiplying the right side by and integrating gets us:

When , we loose all terms, giving us:

However, when we get:

Since does not seem to be determined from these equations, there is an uncomfortable degree of arbitrariness in selecting . Require normalization:



which is a projection of onto .

is a complex number.

Complex number formula:

What is ? Here, we choose .

<FIGURE> "Title" (Description)

In quantum mechanics usually, but not always, can have arbitrary phase , so long as magnitude of is correct. Here we choose . Therefore:

This dictates that all of is orthogonal to .

As an example, consider adding a correction to the hydrogen atom, what is actually a fairly common occurrence.

This last equation is the influence of gravitational attraction between the positive ion and the negative ion. This is a first order energy correction.

When working with degenerate wavefunctions, the problem becomes slightly more complicated because the interactions amongst the degenerate wavefunctions must be carefully accounted for. That said, this is just bookkeeping. The general procedure for Rayleigh-Schrodinger perturbation theory is as outlined here.

Electronic Properties of Materials/Quantum Mechanics for Engineers
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