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
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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
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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?
 
##|\Psi|^2=\frac{1}{\sqrt{\pi b^2}}\exp(\frac{-(x-x_0)^2}{b^2}).## ##\braket{x}=\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dx\,x\exp(-\frac{(x-x_0)^2}{b^2}).## ##y=x-x_0 \quad x=y+x_0 \quad dy=dx.## The boundaries remain infinite, I believe. ##\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dy(y+x_0)\exp(\frac{-y^2}{b^2}).## ##\frac{2}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,y\exp(\frac{-y^2}{b^2})+\frac{2x_0}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,\exp(-\frac{y^2}{b^2}).## I then resolved the two...
It's given a gas of particles all identical which has T fixed and spin S. Let's ##g(\epsilon)## the density of orbital states and ##g(\epsilon) = g_0## for ##\forall \epsilon \in [\epsilon_0, \epsilon_1]##, zero otherwise. How to compute the number of accessible quantum states of one particle? This is my attempt, and I suspect that is not good. Let S=0 and then bosons in a system. Simply, if we have the density of orbitals we have to integrate ##g(\epsilon)## and we have...
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