Time-Dependent Degenerate Pertubation Theory for Spin System

In summary, the conversation discusses the use of the Spin Hamiltonian for a system of spin 1 and the application of degenerate perturbation theory to find the eigenvalues of the Hamiltonian. The Spin Pauli Matrices are used to represent the Hamiltonian, which is shown to be a 3x3 matrix in the SZ basis. The correct method for finding the eigenvalues using degenerate perturbation theory is to diagonalize the perturbation with respect to the degenerate subspace of the unperturbed Hamiltonian.
  • #1
jcharles513
22
0

Homework Statement


Consider the so-called Spin Hamiltonian
H=AS2Z+B(S2X-S2Y)

for a system of spin 1. Show that the Hamiltonian in the SZ basis is the 3x3 matrix:

\hbar2*[(A,0,B; 0,0,0; B,0,A)].Find the eigenvalues using degenerate pertubation theory.

Homework Equations


Spin Pauli Matrices

The Attempt at a Solution



I don't know where to start with this since the Pauli Matrices I know are 2x2 matrices. How did the Hamiltonian become a 3x3. What am I missing? Can someone get me started?
 
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  • #2
See here where ##\hbar## has been set equal to 1.
 
  • #3
Thank you. That makes sense. I guess I wasn't searching for the right thing in google. I may have more questions after that but this gets me started.
 
  • #4
Degen. Perturb.

TSny said:
See here where ##\hbar## has been set equal to 1.


I'm confused about how to use degenerate perturbation on this system. My understanding of degenerate perturbation is shaky. I know that I need to take a subspace of this matrix that has the degeneracy and then transform the perturbation of that subspace to this new diagonal basis. But I'm not sure what my subspace of this matrix is?
 
  • #5
I used the 3x3 subspace and got eigenvalues 0,A+B, and A-B. Is this correct? And the correct procedure?
 
  • #6
I believe those are the correct eigenvalues for the Hamiltonian. Did you get them by finding the exact eigenvalues of the complete Hamiltonian?

I think the exercise is to find eigenvalues using degenerate perturbation theory where you treat B as a small parameter so that the unperturbed Hamiltonian is Ho=ASz2 and the perturbation is V = B(Sx2 - Sy2).
 
  • #7
I diagonalized the pertubation (B-part) and then added it the A-part. This made it diagonal. I'm not sure if that's the right way to do it or not? If not, can you get me started on the correct method?
 
  • #8
jcharles513 said:
I diagonalized the pertubation (B-part) and then added it the A-part. This made it diagonal. I'm not sure if that's the right way to do it or not?

In degenerate perturbation theory, you would not diagonalize the B-part in the full 3-dimensional vector space of the hamiltonian. Rather, if you want the perturbation theory correction to the degenerate eigenvalue, A, of the unperturbed Hamiltonian (A-part hamiltonian) you diagonalize the B-part relative to the 2-dimensional space spanned by the degenerate eigenfunctions corresponding to the eigenvalue A of the unperturbed hamiltonian, as explained in standard texts.

In this problem, it turns out that degenerate perturbation theory yields the exact result, and diagonalizing the B-part with respect to the degenerate subspace is equivalent to diagonalizing the entire hamiltonian. But in general that wouldn't be true.

Unless you still have specific questions, I would recommend reviewing degenerate perturbation theory and then coming back with specific questions if you still have any. This is not the place to try to explain the general formalism of degenerate perturbation theory.
 
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Related to Time-Dependent Degenerate Pertubation Theory for Spin System

1. What is time-dependent degenerate perturbation theory for spin system?

Time-dependent degenerate perturbation theory for spin system is a mathematical tool used to describe the behavior of quantum spin systems that are subjected to external perturbations over time. It is a branch of quantum mechanics that allows for the analysis of systems with multiple degenerate (equivalent) energy levels.

2. How does time-dependent degenerate perturbation theory for spin system differ from time-independent perturbation theory?

The main difference between time-dependent and time-independent perturbation theory is that the former takes into account the time evolution of the system, while the latter assumes that the perturbation is constant over time. Time-dependent degenerate perturbation theory is used when the perturbation varies with time, making it more applicable to real-world systems.

3. What are some applications of time-dependent degenerate perturbation theory for spin system?

Time-dependent degenerate perturbation theory has many practical applications in various fields, including atomic and molecular physics, quantum chemistry, and nuclear physics. It is used to study the dynamics of spin systems under the influence of external fields, such as magnetic fields, and to predict the behavior of quantum systems in the presence of time-varying perturbations.

4. What are the limitations of time-dependent degenerate perturbation theory for spin system?

One of the main limitations of time-dependent degenerate perturbation theory is that it relies on certain assumptions about the system, such as the perturbation being small and the energy levels being degenerate. It also does not take into account the effects of non-linear perturbations or higher order terms, which can be significant in some systems.

5. How is time-dependent degenerate perturbation theory for spin system related to other quantum mechanical theories?

Time-dependent degenerate perturbation theory is closely related to other quantum mechanical theories, such as time-independent perturbation theory, time-dependent Schrödinger equation, and time-dependent variational principle. These theories all aim to describe the behavior of quantum systems under the influence of external perturbations, with time-dependent degenerate perturbation theory being the most applicable to systems with degenerate energy levels.

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