How to Find Orthonormal Kets for Commuting Degenerate Operators?

[ausdreamer]

Homework Statement

I've solved my problem now. I was trying to show that LHS=RHS:

(|+><-| + |-><+|)^2 = (|+><+| + |-><-|)

this can be done by using <-|->=1 (normalization) and <x|->=0 (orthogonal).

LHS:

(|+><-||+><-|) + (|+><-||-><+|) + (|-><+||+><-|) + (|-><+||-><+|) = 0 + |+><+| + |-><-| + 0 = RHS

One last question I have (though not related to the above question) is how to find a new set of orthonormal kets which both operators A and B have in common, given that operators A and B commute and are both degenerate. If anyone can explain how I'd go about finding these eigenkets that'd be great, thanks :)

Homework Equations

There's isn't one for simplifying these expressions.

The Attempt at a Solution

 

[Berko]

Remember, The + ket and - ket are orthonormal, so what does <+|-> =?

Rewrite the LHS as (...)^2 = (...)(...) and use orthonormality.

 

[ausdreamer]

Yep I've used that to solve the first question (thanks for the help though!) Still would like some pointers on second question:

"One last question I have (though not related to the above question) is how to find a new set of orthonormal kets which both operators A and B have in common, given that operators A and B commute and are both degenerate. If anyone can explain how I'd go about finding these eigenkets that'd be great, thanks :)"

 

[Berko]

Are you looking for help for a specific problem or in general?

 

[ausdreamer]

I have a specific example I'm working on currently...

Operator A is represented by A=[a 0 0;0 -a 0;0 0 -a], B=[b 0 0;0 0 -ib;0 ib 0].

Both A and B are degenerate and I've also shown that A and B commute [A,B]=0.

The question I'm struggling with is in Sakuria, and it is:

c) Find a new set of orthonormal kets which are simultaneous eigenkets of both A and B. Specify the eigenvalues of A and B for each of the 3 eigenket. Does your specification of eigenvalues completely characterize each eigenket?

All I've done on c) so far is find eigenvalues for A to be -a,-a,a with eigenvectors [0;1;0] and [0;0;1] for eigenvalue -a and -a, and eigenvector [1;0;0] for eigenvalue a...and for B, eigenvalues -b,b,b with corresponding eigenvectors [0;i;1] for -b, and [1;0;0] and [0;-i;1] for eigenvalues b and b.

This question is my last outstanding question on an assignment worth ~ 6% of my mark, making this question worth ~0.5% of my mark...And it's due in ~ 7 hours, so any help will be appreciated! :)
voxel has written out the question in full on these forums: https://www.physicsforums.com/archive/index.php/t-340930.html

 

[Berko]

Diagonalize B and find it's eigenvectors. Are they also eigenvectors of A?

 

[vela]

Remember that any linear combination of the degenerate eigenvectors of A is also an eigenvector of A. You want to find those combinations which are eigenvectors of B.