# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# Time independent perturbation.

## The setup

To recap, we were covering the time independent perturbation methods from section 16.1 of the text [1]. We start with a known Hamiltonian , and alter it with the addition of a “small” perturbation

For the original operator, we assume that a complete set of eigenvectors and eigenkets is known

We seek the perturbed eigensolution

and assumed a perturbative series representation for the energy eigenvalues in the new system

Given an assumed representation for the new eigenkets in terms of the known basis

and a pertubative series representation for the probability coefficients

so that

Setting requires

for

We rescaled our kets

where

The normalization of the rescaled kets is then

One can then construct a renormalized ket if desired

so that

## The meat.

That’s as far as we got last time. We continue by renaming terms in 2.10

where

Now we act on this with the Hamiltonian

or

Expanding this, we have

We want to write this as

This is

So we form

and so forth.

**Zeroth order in **

Since , this first condition on is not much more than a statement that .

**First order in **

How about ? For this to be zero we require that both of the following are simultaneously zero

This first condition is

With

or

From the second condition we have

Utilizing the Hermitian nature of we can act backwards on

We note that . We can also expand the , which is

I found that reducing this sum wasn’t obvious until some actual integers were plugged in. Suppose that , and , then this is

More generally that is

Utilizing this gives us

And summarizing what we learn from our conditions we have

**Second order in **

Doing the same thing for we form (or assume)

We need to know what the is, and find that it is zero

Again, suppose that . Our sum ranges over all , so all the brakets are zero. Utilizing that we have

From 2.34 we have

We can now summarize by forming the first order terms of the perturbed energy and the corresponding kets

We can continue calculating, but are hopeful that we can stop the calculation without doing more work, even if . If one supposes that the

term is “small”, then we can hope that truncating the sum will be reasonable for . This would be the case if

however, to put some mathematical rigor into making a statement of such smallness takes a lot of work. We are referred to [2]. Incidentally, these are loosely referred to as the first and second testaments, because of the author’s name, and the fact that they came as two volumes historically.

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

[2] A. Messiah, G.M. Temmer, and J. Potter. *Quantum mechanics: two volumes bound as one*. Dover Publications New York, 1999.