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

In summary, the conversation discusses the Hamiltonian for an S=5/2 particle, given by H= a.Sz + b.Sz^2 +c.Sx, and the difficulties in finding its eigenvalues and diagonalizing the resulting matrix, even with the use of software tools like Mathematica and Maple. There is a suggestion to use the normal eigenvalue equation and the Solve function, but it is mentioned that the calculations can be complex and time-consuming. There is also a mention of a non-separable differential equation that needs to be solved.
  • #1
Physicslad78
47
0
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.
 
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  • #2
I guess you'd just use the normal eigenvalue equation
[tex]\left|H - E I\right| = 0[/tex]
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.
 
  • #3
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..:(
 
  • #4
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!
 
  • #5
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...o_O
 
  • #6
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
 
  • #7
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 <_<
 
Last edited:
  • #8
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..:)
 

1. What is a Hamiltonian in quantum mechanics?

A Hamiltonian is an operator in quantum mechanics that represents the total energy of a system. It is used to describe the time evolution of a quantum system and is an essential tool for solving the Schrödinger equation.

2. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used to describe the behavior of linear operators, such as the Hamiltonian. Eigenvalues are the possible values that an operator can return when acting on a vector, while eigenvectors are the corresponding vectors that do not change direction when acted upon by the operator.

3. How do I find the eigenvalues of a Hamiltonian involving Sz, Sz^2, and Sx?

To find the eigenvalues of a Hamiltonian involving these operators, you can use the ladder operators technique. This involves using the operators to create a set of simultaneous equations that can be solved to find the eigenvalues. Another method is to use matrix diagonalization techniques.

4. Why are the eigenvalues of a Hamiltonian important?

The eigenvalues of a Hamiltonian represent the possible energies that a system can have. By finding the eigenvalues, we can determine the possible energy states of a system and understand how it will evolve over time. Additionally, the eigenvectors associated with the eigenvalues can provide information about the spatial distribution of the system.

5. Are there any other ways to find the eigenvalues of a Hamiltonian?

Yes, there are other methods that can be used to find the eigenvalues of a Hamiltonian. These include perturbation theory, variational methods, and numerical techniques. The choice of method depends on the complexity of the system and the level of accuracy required.

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