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Two-state perturbation problem (QM)

  • Thread starter Tim87
  • Start date
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1. Homework Statement

Given is a hamiltionan H0:

E1 0
0 E2

with E1 and E2 being eigenvalues of two eigenstates phi1 and phi2

A distortion W, with W12 = W21* (complex conjugate):

0 W12
W21 0

Calculate the eigenvalues and eigenstates of H = H0 + W


2. Homework Equations

See for most info:
http://farside.ph.utexas.edu/teaching/qm/lectures/node50.html

it's a two-state perturbation problem


3. The Attempt at a Solution

I found out most things like how to calculate the eigenvalues. A lot of it is explained at the mentioned site (http://farside.ph.utexas.edu/teaching/qm/lectures/node50.html) The thing I don't get is how they got from the eigenvalues to the eigenstates.

I have tried the following:

( E1 - E1' W12 )(a) (0)
( W21 E2 - E1' )(b) = (0)

with E1' being the first new eigenvalue: E1'=E1 + |W12|^2/(E1-E2)

But I do not get to the answer provided on the site.


I hope I have been clear about my question, so someone can explain it to me. Thanks in advance!

Tim
 

Answers and Replies

979
1
Note that since eigenvectors are only defined up to a scaling, your matrix only represents one equation. So define a=kb, and solve for k. (This is the canonical way to proceed. In practice people just stare at it and then write down the solution.)
 

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