Question about angular momentum and spin

[rabbit44]

Hi quick question, is there a reason that the eigenvalues of Jz, Lz and Sz are all denoted m? Are they the same? Because I have seen them interchanged such as in addition of angular momenta.

Thanks

 

[malawi_glenn]

One call them M since they was found experimentally by interaction by EM-fields, "Magnetic Quantum Number" :-)

One should use subscript in order to differ between them:

$$m_j, \, m_s, \, m_\ell$$

 

[Entropia]

for your question! The eigenvalues of Jz, Lz, and Sz are typically denoted as m because they represent the component of angular momentum, spin, and total angular momentum along a specific axis (usually the z-axis). However, they are not necessarily the same, as each of these operators (Jz, Lz, Sz) correspond to different physical quantities (angular momentum, orbital angular momentum, and spin, respectively). Therefore, their eigenvalues may differ depending on the system being studied.

In terms of interchanging them, this is typically done when discussing the addition of angular momenta. In this case, we are considering the total angular momentum of a system, which is the sum of the individual angular momenta of its components. In this context, the eigenvalues of Jz, Lz, and Sz may be interchanged because they all contribute to the overall angular momentum of the system.

I hope this helps clarify the relationship between these operators and their eigenvalues. Let me know if you have any other questions!