Thermodynamics, Statistical Physics, and Quantum Mechanics

5.28 Constant Matrix Perturbation (Stony Brook)

Define where is the eigenvalue. We wish to diagonalize

the matrix by finding the determinant of

When confronted by a cubic eigenvalueequation, it is best first to try to

guess an eigenvalue. The obvious guesses are The one that works

is so we factor this out to get

We call these eigenvalues respectively. When we construct the
eigenfunctions, only the one for is unique. Since the first two have degen-
erate eigenvalues, their eigenvectors can be constructed in many different
ways. One choice is

b) Since the three states form a complete set over this space, we can
expand the initial state as

a)

the matrix by finding the determinant of

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