Magnetic Quantum Number

In a system where Angular Momentum is preserved ( i.e. where the Hamiltonian is invariant under rotations ), The wave function satisfies the constraint $\psi (r,0)\approx \psi (r,\theta )$, which forces the constraint $1=e^{2\pi \frac{l_z}{\hslash }}$, where $l_z$ is an eigenvalue of the Angular Momentum $L_z$, i.e. $L_z|\psi ⟩=l_z|\psi ⟩$ for an Eigenstate $|\psi ⟩$.

As such $l_z\in \hslash \mathbb{N}$ is defined as the magnetic quantum number of a particle.

• Clarify the statement from Principles of Quantum Mechanics, page 314: “Physically this means that a state is not uniquely specified by just its angular momentum (which only fixes the angular part of the wave function), but it can be specified by its energy and angular momentum in a rotationally invariant problem.”