Spin Orbit Coupling: Commutation Relations?

In summary, the following commutation relations still hold when spin orbit coupling occurs: [Sx,Lx] = 0, [Sx,Ly] = 0, [S^2, Lx] = 0, [L^2, Sx] = 0.
  • #1
sachi
75
1
I'm a little confused as to whether the following commutation relations still hold when spin orbit coupling occurs:
[Sx,Lx] = 0
[Sx,Ly] = 0
[S^2, Lx] = 0
[L^2, Sx] = 0
etc.

thanks very much for your help
 
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  • #2
Err,

What have you done to solve this problem yourself ?

Helping you out does not imply that we will just "spoon feed" you the solution.

How do you think you can tackle this problem ?


marlon
 
  • #3
sachi said:
I'm a little confused as to whether the following commutation relations still hold when spin orbit coupling occurs:
[Sx,Lx] = 0
[Sx,Ly] = 0
[S^2, Lx] = 0
[L^2, Sx] = 0
etc.

thanks very much for your help
The CR's between the ops are not changed by the coupling.
 
  • #4
It doesn't matter how the hamiltonian looks, for a particle in quantum mechanics the Hilbert space has the structure

[tex] \mathcal{H} =L^{2}\left(\mathbb{R}^{3}\right) \otimes \mathbb{C}^{2n+1} [/tex]

,where "n" is the spin of the particle.

Daniel.
 
  • #5
sachi said:
I'm a little confused as to whether the following commutation relations still hold when spin orbit coupling occurs:
[Sx,Lx] = 0
[Sx,Ly] = 0
[S^2, Lx] = 0
[L^2, Sx] = 0
etc.

thanks very much for your help

They are still valid. But that's not the point. The point is whether the *perturbation* hamiltonian commutes with these operators. One finds that the spin orbit hamiltonian commutes with L^2, S^2, J^2 and J_z, but not with L_z and S_z. Therefore, m_s and m_l are not good quantum numbers but must be replaced by m_j and j. So the states of definite energy when the spin-orbit interaction are taken into account are the states labelled by the quantum numbers l,s,j, m_j (instead of the usual l,m_l,s,m_s that one uses to label the unperturbed hydrogenic wavefunctions).

Hope this helps. If it's not clear, write again.

Pat
 

1. What is spin-orbit coupling?

Spin-orbit coupling is a phenomenon in which the spin and orbital motion of an electron are coupled together, resulting in a change in energy levels and splitting of spectral lines. It is caused by the interaction between the electron's magnetic moment and the magnetic field created by its own orbital motion.

2. What are commutation relations in spin-orbit coupling?

Commutation relations refer to the mathematical relationships between different observables in a system. In spin-orbit coupling, these relations describe how the spin and orbital angular momentum operators commute with each other, and how their commutation affects the energy levels of the system.

3. How is spin-orbit coupling measured in experiments?

Spin-orbit coupling can be measured using various experimental techniques, such as spectroscopy, electron spin resonance, and nuclear magnetic resonance. These methods involve inducing a change in the spin or orbital motion of electrons and observing the resulting energy level shifts or spectral line splittings.

4. What are the applications of spin-orbit coupling in research and technology?

Spin-orbit coupling is an important phenomenon in many fields of research, including condensed matter physics, quantum computing, and materials science. It also has practical applications, such as in spintronics devices and quantum information processing.

5. How does spin-orbit coupling affect the properties of materials?

The presence of spin-orbit coupling can significantly influence the electronic, magnetic, and optical properties of materials. It can lead to the formation of topological states, affect the behavior of magnetic materials, and play a role in the absorption and emission of light. Understanding spin-orbit coupling is crucial for predicting and engineering the properties of materials.

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