Degenerate Perturbation Theory and Matrix elements

[dwdoyle]

Homework Statement

I did poorly on my exam, which I thought was very fair, and am now trying to understand certain aspects of perturbation theory. There are a total of three, semi related problems which i have questions about. They are mainly qualitative in nature and involve an intuitive understanding of perturbation theory, which I guess i do not have.

I do not require solutions to the problems, but I am asking for help understanding the intuition and physics behind them.

Question 1:

-What does it mean if the perturbation matrix $W$ is diagonal? What does it mean if it is not diagonal? What does it mean if $$W_{aa} =W_{bb}$$? Or if $$W_{ab} = W_{ba}$$ Or if $$W_{ab} \neq W_{ba} $$ ?

-Without just doing the integrals, how can I determine qualitatively what the matrix elements $$W_{ij}$$ will be given my degenerate wave forms? I understand that if the wave interacts with the perturbation such that the perturbation is at a node of the wave or the wave is equally positive and negative along the perturbation that the perturbation does not effect the wave or energy at all.

-What is the relationship between symmetric perturbations and the matrix elements of $$W_{ij}$$ ?

Question 2:

-5.2: Is there a way to qualitatively determine this just by looking at the hamiltonian?
-5.3 I said that H' is not comprised of a "good" eigenbasis. Does this imply $$W_{ij} \neq 0$$ means the degenerate states chosen are not a "good" eigenbasis? What makes one eigenbasis good and another not?

-Does the presence of off diagonal elements in H or H' determine degeneracy lifting?

Question 3:

I feel like this problem reflects all my confusion with degenerate perturbation theory, being that I don't really know how to answer any of it.

Homework Equations

The Attempt at a Solution

Im not really looking for solutions, more just clarifications.

 

[vela]

Do you really not know what a diagonal matrix is?

 

[dwdoyle]

Do you really not know what a diagonal matrix is?

Im asking how to interpret diagonal / non diagonal perturbation hamiltonians. And how Wij being diagonal / not diagonal relates to the symmetry or lack of symmetry of a given perturbation.

I really don't see how the answer to my questions reduces to whether or not I know the definition of a diagonal matrix.

 

[vela]

From your questions, it doesn't seem like you know what a diagonal matrix is. Why else would you ask if it means $W_{aa} = W_{bb}$, $W_{ab} = W_{ba}$, or $W_{ab} \ne W_{ba}$? How do you hope to answer part (a) if you don't even know what you're looking for?

 

[dwdoyle]

A diagonal matrix has only diagonal elements. That doesn't mean $W_{aa} = W_{bb}$. A nondiagonal matrix has off diagonal elements. That doesn't mean $W_{ab} = W_{ba}$.

I don't understand how the given perturbation, it's interaction with the degenerate wave and the matrix elements relate. That is all I am asking. That is what I am asking when I say "what does it mean if $W_{ab} = W_{ba}$"

 

[blue_leaf77]

What does it mean if the perturbation matrix $W$ is diagonal? What does it mean if it is not diagonal?

Think about the eigenstates. Imagine you have a diagonal matrix, and then you add another diagonal matrix to it. Will the resulting matrix be still diagonal? From the answer to this question, how can you conclude about the eigenstates of resulting matrix with those of the first matrix? For now clarify these questions first.

Without just doing the integrals, how can I determine qualitatively what the matrix elements

WijWij​

W_{ij} will be given my degenerate wave forms?

You can't, you have to evaluate the integral of various matrix elements of $W$.

The case of $W_{ab}\neq W_{ba}$ with $W_{ab}$ being real poses a non-trivial physics because then it means $W$ is not Hermitian.

 

[vela]

A diagonal matrix has only diagonal elements. That doesn't mean $W_{aa} = W_{bb}$. A nondiagonal matrix has off diagonal elements. That doesn't mean $W_{ab} = W_{ba}$.

I don't understand how the given perturbation, it's interaction with the degenerate wave and the matrix elements relate. That is all I am asking. That is what I am asking when I say "what does it mean if $W_{ab} = W_{ba}$"

Sorry, I was misinterpreting your questions. I'd just write down integrals for the various matrix elements and analyze whether they will vanish or not. For example, one of the off-diagonal elements is given by
$$\langle 1\ 2 \lvert \hat{W} \rvert 2 \ 1 \rangle = \iint dx\,dy\ \sin(\pi x/a)\sin(2\pi y/a) \hat{W} \sin(2\pi x/a)\sin(\pi y/a).$$ If $W$ only depends on $x$ like in (C), then the integration over $y$ will yield 0, so you know for (C), $W_{ab} = W_{ba}^* = 0$.