We decompose L·S using ladder operators and write the spin-orbit Hamiltonian explicitly as a 2×2 matrix in the Sz eigenbasis.

Given a general spin-1/2 state, we calculate the probabilities of measuring positive spin along y and z, and verify with known eigenstates.

Deriving the explicit matrix form of the time-reversal operator for a spin-1/2 particle by expanding the exponential, and proving the famous result Θ² = −1 for fermions.

Using angular momentum conservation to constrain the final state of a particle decay, enumerating possible states, determining parity, and extracting spin measurement probabilities from Clebsch-Gordan coefficients.

Working through a discrete rotation group, its subgroups, eigenvalues of symmetry operators, non-commutativity of rotations and reflections, and using symmetry to identify vanishing matrix elements.

Deriving the transfer matrix connecting incoming and outgoing WKB amplitudes across a general potential barrier, and extracting the tunneling transmission coefficient.

Deriving the explicit matrix form and normalization of the spin eigenstates |n; +⟩ for a general unit vector n̂ defined by spherical angles θ and φ.