# Spin-Orbit Coupling in Hydrogen Atom: Understanding the Calculation

• Viona
In summary, the spin-orbit coupling in Hydrogen atom arises from the interaction between the electron's spin magnetic moment and the proton's orbital magnetic field. The magnetic field of the proton is calculated using the angular momentum of the electron because the current is cast in terms of the electron's angular momentum, and the period of the electron around the proton is the same as the period of the proton around the electron. This is non-relativistic quantum mechanics, and the direction of the electron's orbital momentum is parallel to the direction of the magnetic field.
Viona
Homework Statement
Why we calculate the magnetic field of the proton (B) using the angular momentum of the electron (L)?
Relevant Equations
Why we had to do this calculations in the rest frame
of the electron?
I was reading in the Book: Introduction to Quantum Mechanics by David J. Griffiths. In chapter Time-independent Perturbation Theory, Section: Spin -Orbit Coupling. I understood that the spin–orbit coupling in Hydrogen atom arises from the interaction between the electron’s spin magnetic moment, and the proton’s orbital magnetic field B. But I did not understand why He calculated the magnetic field of the proton (B) using the angular momentum of the electron (L)! Please see the attached pictures. Thank you!

Delta2
You can answer your question by looking at the derivation. The expression ##B=\dfrac{\mu_0I}{2r}## gives the B-field at the center of a loop of radius ##r##. That is why one starts in the reference frame of the electron. The question then is, what do we do with the current ##I## which is not a quantum mechanical operator? The answer to that is, cast the current in terms of the angular momentum of the electron which is a quantum mechanical operator. One can use the angular momentum of the electron ##L=m_evr## because the period ##T## of the electron around the proton is the same as the period of the proton around the electron.

Viona, Delta2 and Twigg
kuruman said:
You can answer your question by looking at the derivation. The expression ##B=\dfrac{\mu_0I}{2r}## gives the B-field at the center of a loop of radius ##r##. That is why one starts in the reference frame of the electron. The question then is, what do we do with the current ##I## which is not a quantum mechanical operator? The answer to that is, cast the current in terms of the angular momentum of the electron which is a quantum mechanical operator. One can use the angular momentum of the electron ##L=m_evr## because the period ##T## of the electron around the proton is the same as the period of the proton around the electron.
I though the periodic time and the radius of the rotation are affected by the relativistic motion, so the periodic time from the point view of the electron will be different from the point view of the proton.

Viona said:
I though the periodic time and the radius of the rotation are affected by the relativistic motion, so the periodic time from the point view of the electron will be different from the point view of the proton.
This is non-relativistic quantum mechanics.

Viona
kuruman said:
You can answer your question by looking at the derivation. The expression ##B=\dfrac{\mu_0I}{2r}## gives the B-field at the center of a loop of radius ##r##. That is why one starts in the reference frame of the electron. The question then is, what do we do with the current ##I## which is not a quantum mechanical operator? The answer to that is, cast the current in terms of the angular momentum of the electron which is a quantum mechanical operator. One can use the angular momentum of the electron ##L=m_evr## because the period ##T## of the electron around the proton is the same as the period of the proton around the electron.
Yes the period the same. But The direction of the angular momentum is parallel to the direction of the magnetic field, then the electron orbital momentum vector should point to the opposite direction?

Viona said:
then the electron orbital momentum vector should point to the opposite direction?
Why would it point in the opposite direction?

Viona
DrClaude said:
Why would it point in the opposite direction?
Sorry, the electron's orbital momentum points in the same direction of the magnetic field. Thanks!

## 1. What is spin-orbit coupling in a hydrogen atom?

Spin-orbit coupling is a phenomenon that occurs in atoms where the spin of an electron interacts with its orbital motion around the nucleus. In a hydrogen atom, the electron's spin and orbital angular momentum are coupled together, resulting in a splitting of energy levels.

## 2. How is spin-orbit coupling calculated in a hydrogen atom?

The calculation of spin-orbit coupling in a hydrogen atom involves using the Schrödinger equation, which describes the behavior of quantum particles. The spin-orbit coupling term is added to the Hamiltonian operator, which represents the total energy of the system.

## 3. What factors affect the strength of spin-orbit coupling in a hydrogen atom?

The strength of spin-orbit coupling in a hydrogen atom is affected by the nuclear charge, the distance between the electron and the nucleus, and the orbital angular momentum of the electron. These factors determine the magnitude of the spin-orbit coupling term in the Hamiltonian operator.

## 4. Why is understanding spin-orbit coupling important in studying hydrogen atoms?

Understanding spin-orbit coupling in hydrogen atoms is important because it can help explain the fine structure of energy levels in the atom. This is crucial for understanding the behavior of atoms and molecules in various chemical reactions and physical processes.

## 5. Can spin-orbit coupling be observed experimentally in a hydrogen atom?

Yes, spin-orbit coupling in a hydrogen atom can be observed experimentally through various spectroscopic techniques, such as the Zeeman effect. This effect is the splitting of spectral lines due to the interaction between the electron's spin and its orbital motion, which is caused by spin-orbit coupling.

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