An $\ell=1$ System in a Magnetic Field

We will derive the Hamiltonian terms added when an atom is put in a magnetic field in section 2. For now, we can be satisfied with the classical explanation that the circulating current associated with nonzero angular momentum generates a magnetic moment, as does a classical current loop. This magnetic moment has the same interaction as in classical EM,

$$\bgroup\color{black}H=-\vec{\mu}\cdot \vec{B}.\egroup$$

For the orbital angular momentum in a normal atom, the magnetic moment is

$$\bgroup\color{black}\vec{\mu}={-e\over 2mc}\vec{L}.\egroup$$

For the electron mass, in normal atoms, the magnitude of $\vec{\mu}$ is one Bohr magneton,

$$\bgroup\color{black}\mu_B={e\hbar\over 2m_ec}.\egroup$$

If we choose the direction of $B$ to be the $z$ direction, then the magnetic moment term in the Hamiltonian becomes

$$\bgroup\color{black}H={\mu_B B\over\hbar} L_z =\mu_BB\left(\matrix{1&0&0\cr 0&0&0\cr 0&0&-1}\right).\egroup$$

So the eigenstates of this magnetic interaction are the eigenstates of $L_z$ and the energy eigenvalues are $+\mu_BB$, 0, and $-\mu_BB$.

Example: The energy eigenstates of an $\ell =1$ system in a B-field.
Example: Time development of a state in a B field.