Past exam of the physics A course of the University of Cambridge / 2012 / Paper 1 / Section C / C10

angular momentum generates rotations. Its components obey $[J_i,J_j]=i\hslash {ϵ}_{ijk}{J}_k$, and $J^2$ commutes with each component. A compatible basis has $J^2={\hslash }^{2}j(j+1)$ and $J_z=m\hslash$, where $m=-j,-j+1,\dots ,j$. ladder operators change $m$ by one, terminate at $\pm j$, and give $2j+1$ magnetic substates. Different Cartesian components generally cannot all be sharp simultaneously.

orbital angular momentum is $\mathbf{L}=\mathbf{r}\times \mathbf{p}=-i\hslash \mathbf{r}\times \nabla$. Its angular eigenfunctions are spherical harmonics, with integer $\ell$ arising from their spatial rotation properties and single-valuedness. spin is intrinsic angular momentum, not literal rotation of a finite charged sphere. It may be half-integer; spin $1/2$ is represented by $\mathbf{S}=\hslash \mathbf{\sigma }/2$. For several contributions, $\mathbf{J}=\mathbf{L}+\mathbf{S}$ obeys the same algebra. Coupling angular momenta $j_1,j_2$ permits $J=|j_1-j_2|,\dots ,j_1+j_2$, with Clebsch-Gordan coefficients relating product and coupled bases. spin-orbit coupling can distinguish these total-angular-momentum sectors.

The Stern-Gerlach apparatus sends a collimated neutral atomic beam through an inhomogeneous magnetic field and records its deflection. A magnetic moment experiences $F_z\simeq {\mu }_z\partial B_z/\partial z$; field uniformity would produce precession but no analogous net separating force. A classically continuous distribution of moment orientations would broaden the beam continuously. Gerlach and Stern instead observed separated silver-beam components. Reversing or removing the field and comparing the undeflected profile controls the magnetic origin; collimation, velocity spread and finite resolution affect the spots. The original experiment demonstrated space quantization; its modern silver interpretation uses the unpaired electron spin with nearly zero orbital angular momentum, rather than a direct measurement of a free electron's trajectory.^[ref]

For two spin-$1/2$ particles, the triplet state sector has $S=1$ and symmetric spin statesThe spin singlet state has $S=0$ and antisymmetric spin stateTheir total $S^2$ values are $2{\hslash }^2$ and zero, while ${\mathbf{S}}_1\cdot {\mathbf{S}}_2$ has values ${\hslash }^2/4$ and $-3{\hslash }^2/4$. The singlet is rotationally invariant and gives anticorrelated spin measurements along a common axis. For identical fermions the complete state is antisymmetric: a symmetric spatial state pairs with a singlet spin state, whereas an antisymmetric spatial state pairs with triplet spin. Orbital, spin and coupled angular momenta share an algebra, but their physical origins and allowed representations differ.