The Zeeman effect is the phenomenon of splitting energy levels with different angular momentum due to the presence of a weak external magnetic field, named after Pieter Zeeman who first observed it.10 The weak magnetic field breaks 10 P. Zeeman,“Over de invloed eener magnetisatie op den aard van het door een stof uitgezonden licht; Over de invloed eener magnetisatie op den aard van het door een stof uitgezonden licht; On the influence of magnetism on the nature of the light emitted by a substance,” Verslagen Meded. Afd. Natuurk. Kon. Akad.

Wetensch. 5, 181 (1896); P. Zeeman, “XXXII. On the influence of magnetism on the nature of the light emitted by a substance,” Lond. Edinb. Dublin Philos. Mag. J. Sci. 43(262), 226–239 (1897); P. Zeeman,

“The effect of magnetisation on the nature of light emitted by a substance,” Nature 55, 347 (1897).

the degeneracy of the energy eigenstates because states of different angular momentum have different magnetic dipole moments. In this example, we’ll see how this works and we will assume throughout that the external magnetic field is uniform and pointed in the z direction:

⃗B = B0 ˆz , (9.163)

where B0 is the magnitude of the magnetic field. You will further study this effect in Exercise 9.7.

Let’s apply this magnetic field to the ground state of hydrogen. What is the difference between the energies when the electron has spin-up vs. spin-down in the ground state?

For what value of the magnetic field B0 does this separation equal that between the ground state and the first excited state of hydrogen?