Perturbation Theory: Calculating 1st-Order Correction

ooleonardoo
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Homework Statement
Calculate the second-order corrections to energy for the following Hamiltonian matrix.
Use the degenerate perturbation theory. Consider 'b' as perturbation.
Relevant Equations
...
Of course, this question consisted of two parts. In the first part, we needed to calculate the first-order correction. It was easy. In all the books on quantum mechanics I saw, only first-order examples have been solved. So I really do not know how to solve it. Please explain the solution method to me. Thankful
2021-11-23 11_09_58-Untitled - Paint.png
 
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What are those books you mention? I would say Sakurai and probably a lot of other books do actually explain how to do PT to arbitrary orders.
 
Gaussian97 said:
What are those books you mention? I would say Sakurai and probably a lot of other books do actually explain how to do PT to arbitrary orders.
gasiorowicz, zettili , griffiths
The level of the book you mentioned is high for me. Do you know of any other book that explains this with an example?
 
Actually, Griffiths does have a discussion on second-order PT.
Anyway, can you show us how did you compute the first-order correction?
 
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This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
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The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
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