Spin orbit coupling and hydrogen problem

In summary, the conversation discusses the concept of good quantum numbers in perturbation theory for hydrogen. It is mentioned that the good quantum numbers are n, l, j, and mj, and these refer to the quantum numbers of the eigenvectors that diagonalize the perturbation. The speaker is confused as to why l is considered a good quantum number but not s, as they believe these should be treated equally as they represent the orbital and spin angular momentum respectively. It is also mentioned that j and mj are good quantum numbers as they refer to the total angular momentum in an isotropic system. However, it is noted that l and s are only approximately good quantum numbers in the case of weak perturbations. The reason for not mentioning s
  • #1
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I don't know if you are familiar with it, but in pertubationt theory for hydrogen it is handy to look for eigenstates of J = L + S since this commutes with the hamiltonian (L and S are not separately conserved).
My book then says that the good quantum numbers are: n,l, j, mj
I must admit I'm not used to this idea of good quantum numbers - my book hasn't introduced the term properly (I'm guessing it is just the quantum numbers belongning to the set of eigenvectors that diagonalizes the pertubation) and the thing that bothers me the most is: Why is l a good quantum number but not s? Surely these should be treated on eqaul footing since they are just the length of the total orbital and spin angular momentum respectively. And why would l be a good quantum number when L is not conserved separately.
 
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  • #2
Well, j and mj are certainly good quantum numbers as they refer to the total angular momentum which is conserved in an isotropic system.
l and s are only approximately good in so far as the perturbation is weak and will lift in lowest order the splitting of the degenerate states with different j but same s and l.
s is probably not mentioned as it is always 1/2 and this doesn't have to be written down all the time.
 

1. What is spin-orbit coupling?

Spin-orbit coupling is a phenomenon in quantum mechanics where the spin and orbital angular momentum of an electron are coupled together. This coupling leads to a splitting of energy levels in atoms and molecules, and plays an important role in various physical processes.

2. How does spin-orbit coupling affect the properties of hydrogen?

In hydrogen, spin-orbit coupling leads to a splitting of the energy levels of the 1s orbital, resulting in two energy levels: the spin-up state and the spin-down state. This splitting is known as the fine structure of hydrogen, and it affects various properties of the atom, such as its spectral lines and magnetic moment.

3. What is the hydrogen problem in quantum mechanics?

The hydrogen problem refers to the difficulty in accurately predicting certain properties of the hydrogen atom, such as its energy levels and spectral lines, using the Schrödinger equation. This is due to the neglect of spin-orbit coupling in the original equation, which becomes important for high-energy states of the atom.

4. How is the hydrogen problem solved?

The hydrogen problem can be solved by using the relativistic correction to the Schrödinger equation, which takes into account the effects of spin-orbit coupling. This leads to a more accurate prediction of the energy levels and properties of hydrogen, and is known as the Dirac equation.

5. What are the applications of spin-orbit coupling and the hydrogen problem?

Spin-orbit coupling and the hydrogen problem have various applications in physics, chemistry, and materials science. They are important for understanding the behavior of atoms and molecules, and play a crucial role in fields such as quantum computing, spectroscopy, and spintronics.

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