02 Jun 2026

Angular Momentum Eigenvalues and Eigenvectors

Eigenvalues of J squared and Jz, ladder construction, spherical harmonics, and angular momentum eigenstates.

msc semester-i quantum-mechanics angular-momentum eigenvalues

The angular momentum eigenvalue problem is built around two commuting operators: $J^2$ and $J_z$. Since they commute, their eigenstates can be chosen simultaneously.

Simultaneous eigenstates

Let the common eigenstate be written as $\lvert j,m\rangle$. It is defined by

\[\boxed{ J^2\lvert j,m\rangle=\hbar^2j(j+1)\lvert j,m\rangle }\]

and

\[\boxed{ J_z\lvert j,m\rangle=\hbar m\lvert j,m\rangle. }\]

Here $j$ is the angular momentum quantum number and $m$ is the magnetic quantum number.

The allowed values are

\[j=0,\frac12,1,\frac32,\dots\]

and, for fixed $j$,

\[m=-j,-j+1,\dots,j-1,j.\]

Thus for a given $j$, the number of allowed $m$ values is

\[2j+1.\]

Ladder operation

The operators $J_+$ and $J_-$ act as

\[\boxed{ J_\pm\lvert j,m\rangle = \hbar\sqrt{j(j+1)-m(m\pm1)}\, \lvert j,m\pm1\rangle. }\]

The coefficient becomes zero at the top and bottom of the ladder:

\[J_+\lvert j,j\rangle=0, \qquad J_-\lvert j,-j\rangle=0.\]

This termination condition forces the allowed $m$ values to be finite and evenly spaced.

Orbital angular momentum

For orbital angular momentum, the quantum number is usually written as $l$ instead of $j$:

\[L^2\lvert l,m\rangle=\hbar^2l(l+1)\lvert l,m\rangle,\] \[L_z\lvert l,m\rangle=\hbar m\lvert l,m\rangle.\]

For orbital angular momentum,

\[l=0,1,2,3,\dots\]

and

\[m=-l,-l+1,\dots,l.\]

Only integer $l$ values occur for orbital angular momentum because the spatial wavefunction must be single-valued under a rotation by $2\pi$.

Spherical harmonics

In coordinate representation, the simultaneous eigenfunctions of $L^2$ and $L_z$ are the spherical harmonics:

\[L^2Y_l^m(\theta,\phi)=\hbar^2l(l+1)Y_l^m(\theta,\phi),\] \[L_zY_l^m(\theta,\phi)=\hbar mY_l^m(\theta,\phi).\]

Since

\[L_z=-i\hbar\frac{\partial}{\partial\phi},\]

the azimuthal dependence is

\[Y_l^m(\theta,\phi)\propto e^{im\phi}.\]

Single-valuedness requires

\[e^{im(\phi+2\pi)}=e^{im\phi},\]

so $m$ must be an integer for orbital angular momentum.

Normalization and orthogonality

The spherical harmonics are orthonormal:

\[\int_0^{2\pi}\int_0^\pi Y_l^{m*}(\theta,\phi)Y_{l'}^{m'}(\theta,\phi) \sin\theta\,d\theta\,d\phi = \delta_{ll'}\delta_{mm'}.\]

They also form a complete basis for angular functions on the sphere.

Main points

• The state $\lvert j,m\rangle$ is a simultaneous eigenstate of $J^2$ and $J_z$.
• The eigenvalue of $J^2$ is $\hbar^2j(j+1)$.
• The eigenvalue of $J_z$ is $\hbar m$.
• For each $j$, there are $2j+1$ possible $m$ values.
• Orbital angular momentum has integer $l$; spin can have integer or half-integer $j$.

Practice questions

1. Derive the coefficient in $J_\pm\lvert j,m\rangle$.
2. Explain why $m$ runs from $-j$ to $j$.
3. Why is orbital angular momentum restricted to integer $l$?
4. Show that $Y_l^m(\theta,\phi)$ is an eigenfunction of $L_z$.
5. How many states correspond to $l=3$?