3. General Angular Momentum
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Unified treatment of orbital and spin angular momentum: general theory for any j.
Abstract Definition
Any set of operators satisfying the angular momentum algebra:
$$[\hat{J}_i, \hat{J}_j] = i\hbar\epsilon_{ijk}\hat{J}_k$$
Can be orbital $\vec{L}$, spin $\vec{S}$, or total $\vec{J} = \vec{L} + \vec{S}$
General Eigenvalue Problem
$$\hat{J}^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle$$
$$\hat{J}_z|j,m\rangle = \hbar m|j,m\rangle$$
where:
- $j = 0, 1/2, 1, 3/2, 2, \ldots$ (non-negative integer or half-integer)
- $m = -j, -j+1, \ldots, j-1, j$
- $2j+1$ possible values of $m$ for each $j$
Ladder Operators
$$\hat{J}_\pm = \hat{J}_x \pm i\hat{J}_y$$
Commutation relations:
$$[\hat{J}_z, \hat{J}_\pm] = \pm\hbar\hat{J}_\pm, \quad [\hat{J}_+, \hat{J}_-] = 2\hbar\hat{J}_z, \quad [\hat{J}^2, \hat{J}_\pm] = 0$$
Action on States
$$\hat{J}_\pm|j,m\rangle = \hbar\sqrt{j(j+1) - m(m\pm1)}|j,m\pm1\rangle$$
Special cases:
- $\hat{J}_+|j,j\rangle = 0$ (top of ladder)
- $\hat{J}_-|j,-j\rangle = 0$ (bottom of ladder)
Matrix Representation
In $|j,m\rangle$ basis, angular momentum operators are $(2j+1) \times (2j+1)$ matrices
Example: $j=1$
$$J_z = \hbar\left(\begin{array}{ccc}1&0&0\\0&0&0\\0&0&-1\end{array}\right)$$
Orbital vs Spin
Orbital angular momentum $\vec{L}$:
- Only integer values: $\ell = 0, 1, 2, \ldots$
- Derived from spatial wave functions
- Classical analog exists
Spin angular momentum $\vec{S}$:
- Can be half-integer: $s = 0, 1/2, 1, 3/2, \ldots$
- Intrinsic property
- No classical analog
Selection Rules
For transitions between angular momentum states:
- Electric dipole: $\Delta j = 0, \pm 1$ (but $j=0 \not\leftrightarrow j'=0$)
- Magnetic dipole: $\Delta j = 0, \pm 1$
- Electric quadrupole: $\Delta j = 0, \pm 1, \pm 2$
Rotations
Rotation by angle $\alpha$ around axis $\hat{n}$:
$$\hat{R}(\alpha,\hat{n}) = e^{-i\alpha\vec{J}\cdot\hat{n}/\hbar}$$
Rotations form SO(3) group (for integer $j$) or SU(2) (for half-integer $j$)