- #1
dwdoyle
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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.