Finding eigenvalues of a Hamiltonian involving Sz, Sz^2 and Sx

[Physicslad78]

I have the Hamiltonian for an S=5/2 particle given by:

H= a.Sz + b.Sz^2 +c.Sx where Sz and Sx are the spins in z and x directions respectively. The resulting matrix is tridiagonal symmetric but I can't find the eigenvalues..Any idea how to diagonalise it.

N.B: a is a variable and must be kept as a in matrix whereas b and c can be assigned values.


Thanks guys.

 

[diazona]

I guess you'd just use the normal eigenvalue equation
$$\left|H - E I\right| = 0$$
where I represents the identity matrix... that, or Mathematica ;-) (seriously, if you have access to some software tool which can find symbolic eigenvalues, it's probably not worth the work to diagonalize a 6x6 matrix)

If you have trouble with that, post the details of the calculations you did. Sometimes it's just as simple as a little math error.

 

[Physicslad78]

Mathematica doesn't solve it..Eigenvalues produces a root solution and solving a 7 equation system with 7 unknowns (if I want to find the eigenvectors) doesn't help either..Maybe I need a similarity transformation and I have tried some but in vain..dont know what to do..:(

 

[diazona]

You can use the Solve function to explicitly find the roots in the solution in terms of a, but it is rather ugly :/ One thing's for sure, I wouldn't want to have to come up with that by hand!

 

[Matterwave]

I would use some kind of program, it's not simple to find the eigenvalue of a 6x6 matrix unless there happens to be some real tricks...and I don't think there are for tridiagonal matrices (I don't know of any). There should be 6 eigenvalues (and eigen-vectors) though, not 7...

 

[Physicslad78]

Thanks guys..I have tried Solve as well but it just runs forever! MatterrWave, yeah I meant 7 unknowns (6 are the elements of spinor and the 7th is the eigenvalue)..I will see if I can meddle with it and get an answer

 

[Matterwave]

Uhm, Maple has a eigenvalue/eigenvector finder, though I'm not sure if it will be any good at a 6x6 matrix. I'm going to try and see. :p

EDIT: wow, even constructing these matrices takes a while...I may do it later <_<

 

[Physicslad78]

Thanks a lot Matter wave...waiting for ur results if u do it or if any..lol..meanwhile will try sorting it out and have got a non separable differential equation waiting for me as well..:)