Normal Zeeman Effect and Anamolous Zeeman Effect

The Zeeman effect refers to the splitting of spectral lines observed when the light emitted or absorbed by an atom is subjected to a magnetic field. The normal Zeeman effect occurs when the magnetic field is weak enough that the interaction between the magnetic field and the electron's intrinsic magnetic moment (associated with its spin) dominates over the interaction with the electron's orbital motion. In other words, in normal Zeeman effect we ignore the interaction of dipole moment of electron due to spin with external magnetic field. So, we consider the interaction of dipole moment of electron due to orbital motion with external magnetic field. Interaction energy is given by

For spin g =2

For orbital g =1

If we consider two direction along B

The corresponding Hamiltonian operator

In the absence of magnetic field, the Hamiltonian operator of electron in the field of positively charged nucleus.

It's eigen kets l n l m > eigen function 𝜓 (r, 𝜃, 𝜙)

Energy depends upon n and l as well. It doesn't depend on m. So for the given n and l for different values of m , energy is same. But for the different values of m, the eigen kets l n l m > ( or wavefunction) 𝜓_n l m are different. So the energy value E_nl is degenerate. In the presence of magnetic field

It is obviously that the eigen kets of Ho are also eigen kets of H'.

So, we can use non degenerate perturbation theory. Energy correction

Since m takes the values m = - l .....+l ; (2m +1) different values. So E_nl also takes (m+1) different values.

Thus we have (2l +1) different values of energy so degeneracy is removed.

Anamolous Zeeman effect

In contrast to normal Zeeman effect, the anomalous Zeeman effect occurs at higher magnetic field strengths when the external magnetic field becomes comparable to or greater than the internal atomic fields. The anomalous Zeeman effect leads to more complex line splitting patterns. The Zeeman effect was first observed by the Dutch physicist Pieter Zeeman in 1896 and provided early experimental evidence for the quantization of angular momentum in atoms. In anamolous zeeman effect we consider both interaction of magnetic moment due to orbital motion and that due to spin motion with external magnetic field.

Their simultaneous eigen states l n l s j m j > they constitute basis,

The eigen energy value Enj is degenerate

mj = - j to +j so (2j +1) different values

For (2j+1) different state energy is same (degeneracy)

The operator H' can be considered as perturbation. Here we can use non degenerate perturbation theory to calculate energy correction.

First order correction to energy

Here we use Weigner Eckart theorem

So energy correction

Since mj = -j to +j

There are (2j +1) different corrections so there are (2j +1) different values of total energy and also there are (2j +1) different states. So degeneracy is completed removed.

For S state, l = 0, j = s, s = 1/2

mj = ± 1/2

S state

For P state

l = 1, j=3/2, 1/2

s =1/2

1. j = 1/2 , mj = ± 1/2

2. for j = 3/2, mj = 3/2, -1/2, 1/2, 3/2

g = 2/3 , g = 4/3

This note is a part of the Physics Repository.