Quantum mechanics

Angular momentum and spin

Quantum angular-momentum algebra determines measurement outcomes; Stern–Gerlach experiments reveal quantization and angular-momentum coupling.

Orbital angular momentum $\mathbf L=\mathbf r\times\mathbf p$ and intrinsic spin $\mathbf S$ obey the same commutation algebra, although spin is intrinsic. The total operator $\mathbf J=\mathbf L+\mathbf S$ governs conserved quantities and atomic level structure.

[J_i,J_j]=i\hbar\epsilon_{ijk}J_k,\quad J^2= j(j+1)\hbar^2,\quad J_z=m\hbar

Definition: Quantized values

For quantum number $j=0,1/2,1,\ldots$ , the component $J_z$ takes values $m\hbar$ , with $m=-j,-j+1,\ldots,j$ . Different components cannot generally be sharp simultaneously because $[J_x,J_y]=i\hbar J_z$ .

Set the spin direction and analyzer axis to explore the two output probabilities and beam separation.

Total angular momentum and coupling

Coupling angular momenta $j_1,j_2$ permits total values $j=|j_1-j_2|,\ldots,j_1+j_2$ . For an electron with $s=1/2$ , spin–orbit interaction splits levels labeled by $j=l\pm1/2$ when $l>0$ . Clebsch–Gordan coefficients relate the coupled and uncoupled bases.

Example: Example: spin-1/2 beam spots

A silver-atom beam through a magnet with a z-gradient forms two spots, corresponding to $m_s=+1/2$ and $-1/2$ . A fully z-polarized input selects one branch; an x-polarized input gives equal probabilities.

Solution

Since $|+x\rangle=(|+z\rangle+|-z\rangle)/\sqrt2$ , measuring $S_z$ gives each outcome probability $1/2$ .

Angular-momentum operators obey $[L_i,L_j]=i\hbar\epsilon_{ijk}L_k$ , so all components cannot be simultaneously sharp. One commonly specifies $L^2$ and $L_z$ , with eigenvalues $\hbar^2 l(l+1)$ and $\hbar m$ , where $m=-l,\ldots,l$ . Spin follows the same algebra but is not orbital motion; for spin one-half, $S^2=3\hbar^2/4$ and $S_z=\pm\hbar/2$ . When adding two angular momenta, allowed $j$ values run from $|j_1-j_2|$ to $j_1+j_2$ in integer steps, useful for atomic spectra and spin-orbit coupling.

Quick check

How many values of m can J_z take for spin j?

For j=1/2, what values can m in J_z take?

References

J. J. Sakurai, Jim Napolitano (2020). Modern Quantum Mechanics