# Zeeman effect and defining the g_F Factor

• TFM
In summary, the conversation involves someone asking for help deriving the equation for the g_F-factor of the hyperfine states for a hydrogen atom interacting with an external magnetic field. The suggested approach is to treat the external magnetic field as a perturbation to the internal magnetic field and use first order perturbation theory to find the orbital and spin angular momentum eigenstates of the Hamiltonian. This will allow for the calculation of the g factor.

## Homework Statement

A hydrogen atom is interacting with an external magnetic field.
1. Derive the equation for the $$g_F$$-factor of the hyperfine states.

## The Attempt at a Solution

Okay, so the question asks to define the gF factor, however, I am not quite sure where to start.

I know firstly that it is based on the diagram of the vector arrows, as (crudely drawn) attached:

I also know the answer I need to get (it is mentioned in the notes):

$$g_F = g_J\frac{F(F + 1) + j(j + 1) - I(I +1)}{2F(F + 1)} + \frac{\mu_N}{\mu_B}g_I \frac{F(F + 1) + I(I + 1) - j(j +1)}{2F(F + 1)}$$

Also, for the gJ, it starts with:

$$H_J = \frac{\mu_N}{\hbar}(\hat{L} + 2\vec{S})\cdot \vec{B} = \frac{\mu_N}{\hbar}(\hat{J} + \vec{S})\cdot \vec{B}$$

and I know for gF, we have:

$$H_J = \frac{\mu_B}{\hbar}(\hat{L} + 2\vec{S})\cdot \vec{B} - g_I ({\frac{\mu_N}{\hbar} \vec{I}\cdot \vec{B})$$

Anyone got any suggestions about what I should do first?

TFM

#### Attachments

• gF diag.jpg
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I suspect the approach would be to treat the external magnetic field as a perturbation to the internal magnetic field of the Hydrogen atom.

In other words, $$\vec{B} = \vec{B}_{Internal} + \vec{B}_{External}$$.

Considering $$\vec{B}_{Internal}$$ first, we can find the orbital angular momentum (l), spin angular momentum (s),ml,ms (or J,m?) eigenstate of the Hamiltonian (possibly after some simplifying assumptions?).

We then apply first order perturbation theory for $$\vec{B}_{External}$$.

Since you know the energy, you should be able to get the g factor.

Hopefully, this is the way to go.

## 1. What is the Zeeman effect?

The Zeeman effect is the splitting of spectral lines in the presence of an external magnetic field. It was discovered by Dutch physicist Pieter Zeeman in 1896 and is a result of the interaction between the magnetic field and the magnetic moments of atoms or molecules.

## 2. How does the Zeeman effect work?

The Zeeman effect occurs when the energy levels of the electrons in an atom or molecule are split into different sublevels due to the presence of an external magnetic field. This splitting is caused by the interaction between the magnetic field and the magnetic moment of the electron, resulting in multiple spectral lines instead of a single line.

## 3. What is the gF factor in relation to the Zeeman effect?

The gF factor, also known as the Landé g-factor, is a dimensionless quantity that describes the strength of the interaction between the magnetic field and the magnetic moment of an electron. It is a crucial parameter in understanding the Zeeman effect and is used to calculate the energy splitting between the different sublevels.

## 4. How is the gF factor determined?

The gF factor can be determined experimentally by measuring the energy splitting between the spectral lines in the presence of a known magnetic field. It can also be calculated using the formula gF = (Eupper - Elower)/μBB, where Eupper and Elower are the energies of the upper and lower sublevels, μB is the Bohr magneton, and B is the magnetic field strength.

## 5. What are some applications of the Zeeman effect?

The Zeeman effect has numerous applications in various fields of science and technology. It is used in spectroscopy to study the energy levels of atoms and molecules, in astronomy to determine the strength and direction of magnetic fields in stars and planets, and in medical imaging techniques such as magnetic resonance imaging (MRI). It also plays a crucial role in the development of quantum technologies, such as quantum computing and quantum cryptography.