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md.xavier
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Homework Statement
In a given basis, the eigenvectors A and B are represented by the following matrices:
A = [ 1 0 0 ] B = [ 2 0 0 ]
[ 0 -1 0] [ 0 0 -2i ]
[ 0 0 -1] [ 0 2i 0 ]
What are A and B's eigenvalues?
Determine [A, B].
Obtain a set of eigenvectors common to A and B. Do they form a complete basis?
Homework Equations
(A - λI)x = 0
[A, B] = AB - BA
The Attempt at a Solution
Okay, so, I calculated the eigenvalues and the commutator quite easily.
For A, I got eigenvalues 1 and -1, with -1 having degeneracy 2.
For B, I got eigenvalues 2 and -2, with 2 having degeneracy 2.
The commutator was 0, so they commutate.
Now, as far as common eigenvectors go - I could only find one. [1 0 0] transposed.
Is this due to the eigenvalues having degeneracy? Does the fact that two observables commuting implies that they have a common complete basis of eigenvectors only hold up if they don't come from degenerate eigenvalues?
Thank you for your help -- the material given to me was not very clear regarding this particular case.