Modern Physics · Upper Division

Quantum Mechanics: Spin & Angular Momentum

Spin is a purely quantum mechanical property with no classical analogue. It determines the magnetic properties of atoms, the structure of the periodic table, and the statistical behavior of identical particles.

PrerequisitesQuantummechanics(Ch.20)⋅Atomicstructure(Ch.21)⋅Linearalgebrabasics⋅ComplexnQuantum mechanics (Ch. 20) \cdot Atomic structure (Ch. 21) \cdot Linear algebra basics \cdot Complex numbers
Learning Goals
  • UsetheangularmomentumcommutationrelationstoderivethespectrumofL^2andL^zeigenUse the angular momentum commutation relations to derive the spectrum of L̂^{2} and L̂z eigenvalues.
  • Represent spin-½ states as spinors and compute measurement probabilities using Pauli matrices.
  • Add two angular momenta using Clebsch-Gordan coefficients and identify singlet and triplet states.
  • Calculate the fine-structure splitting of hydrogen levels due to spin-orbit coupling.
  • Apply the Wigner-Eckart theorem to determine selection rules for spectral transitions.

SP.1 Orbital Angular Momentum

In quantum mechanics, the angular momentum operator L̂ = r̂ × p̂ has components L̂x, L̂y, L̂z. Their fundamental commutation relation is:

[L̂x, L̂y] = i\hbarL̂z \qquad (and cyclic permutations)(SP.1)

This algebraic structure determines everything: you cannot simultaneously specify two components of angular momentum. The commuting observables are L̂² and L̂z:

L^2∣l,m⟩=ℏ2l(l+1)∣l,m⟩L^z∣l,m⟩=\hbarm∣l,m⟩L̂^{2} |l, m⟩ = \hbar^{2} l(l+1) |l, m⟩ \qquad L̂z |l, m⟩ = \hbarm |l, m⟩(SP.2)

Here l = 0, 1, 2, ... (orbital quantum number) and m = −l, −l+1, ..., +l (magnetic quantum number, 2l+1 values). The magnitude of angular momentum is ℏ√(l(l+1)), not ℏl — a purely quantum result with no classical analogue.

SP.2 Spin-½ and the Pauli Matrices

The Stern-Gerlach experiment (1922) revealed that electrons have an intrinsic angular momentum — spin — with s = ½. The spin operators satisfy the same algebra as orbital angular momentum, but with s = ½:

S^2∣s,ms⟩=ℏ2s(s+1)∣s,ms⟩=(3/4)ℏ2∣s,ms⟩S^z=±ℏ/2Ŝ^{2} |s, m_{s}⟩ = \hbar^{2} s(s+1) |s, m_{s}⟩ = (3/4)\hbar^{2} |s, m_{s}⟩ \qquad Ŝz = \pm\hbar/2(SP.3)

The two spin-½ states are |↑⟩ (mₛ = +½) and |↓⟩ (mₛ = −½). The spin operators in this two-dimensional Hilbert space are:

S^=(ℏ/2)σwhere\sigmax=[[0,1],[1,0]],\sigmay=[[0,−i],[i,0]],\sigmaz=[[1,0],[0,−1]]Ŝ = (\hbar/2) \sigma \qquad where \qquad \sigmax = [[0,1],[1,0]], \qquad \sigmay = [[0,-i],[i,0]], \qquad \sigmaz = [[1,0],[0,-1]](SP.4)

These are the Pauli matrices. They satisfy σᵢσⱼ = δᵢⱼI + iεᵢⱼₖσₖ. An arbitrary spin-½ state is a two-component spinor:

∣χ⟩=α∣↑⟩+β∣↓⟩=(α,β)T∣α∣2+∣β∣2=1|\chi⟩ = \alpha|↑⟩ + \beta|↓⟩ = (\alpha, \beta)^T \qquad |\alpha|^{2} + |\beta|^{2} = 1(SP.5)
Example SP.1 — Spin Measurement in the x-direction

Anelectronisinstate∣↑⟩(spin−upalongz).Whatistheprobabilityofmeasuring+ℏ/2aAn electron is in state |↑⟩ (spin-up along z). What is the probability of measuring +\hbar/2 along the x-axis?

Eigenstates of Sx:|+x⟩=(1/2)(∣↑⟩+∣↓⟩)∣−x⟩=(1/2)(∣↑⟩−∣↓⟩x⟩ = (1/\sqrt2)(|↑⟩ + |↓⟩) \qquad |-x⟩ = (1/\sqrt2)(|↑⟩ - |↓⟩
Express state:|↑⟩ = (1/2)∣+x⟩+(1/2)∣−x⟩1/\sqrt2)|+x⟩ + (1/\sqrt2)|-x⟩
Probability:P(+x)=∣⟨+x∣↑⟩∣2=∣1/2∣2=12P(+x) = |⟨+x|↑⟩|^{2} = |1/\sqrt2|^{2} = \frac{1}{2}
Expectation value:⟨Sx⟩=⟨↑∣Sx∣↑⟩=(ℏ/2)⟨↑∣\sigmax∣↑⟩=(ℏ/2)(1,0)[[0,1],[1,0]](1,0)T=0Sx⟩ = ⟨↑|Sx|↑⟩ = (\hbar/2)⟨↑|\sigmax|↑⟩ = (\hbar/2)(1,0)[[0,1],[1,0]](1,0)^T = 0
Interpretation:50/50chanceof±ℏ/2alongx,withzeroaverage—x−spiniscompletelyundeterminedwhenz50/50 chance of \pm\hbar/2 along x, with zero average — x-spin is completely undetermined when z-spin is definite. This is the Heisenberg uncertainty principle for non-commuting spin components.

SP.3 Addition of Angular Momenta

When two particles with angular momenta j₁ and j₂ are combined, the total angular momentum J = J₁ + J₂ has quantum numbers j = |j₁−j₂|, |j₁−j₂|+1, ..., j₁+j₂. The combined states are expressed in the Clebsch-Gordan basis:

∣j,m⟩=∑(m1,m2)⟨j1m1j2m2∣jm⟩∣j1m1⟩∣j2m2⟩|j, m⟩ = \sum(m_{1}, m_{2}) ⟨j_{1} m_{1} j_{2} m_{2} | j m⟩ |j_{1} m_{1}⟩ |j_{2} m_{2}⟩(SP.6)

The coefficients ⟨j₁m₁j₂m₂|jm⟩ are Clebsch-Gordan coefficients, tabulated for small j. For two spin-½ particles:

∣1,1⟩=∣↑↑⟩∣1,0⟩=(∣↑↓⟩+∣↓↑⟩)/2∣1,−1⟩=∣↓↓⟩∣0,0⟩=(∣↑↓⟩−∣↓↑⟩)/2|1,1⟩ = |↑↑⟩ \qquad |1,0⟩ = (|↑↓⟩ + |↓↑⟩)/\sqrt2 \qquad |1,-1⟩ = |↓↓⟩ \qquad |0,0⟩ = (|↑↓⟩ - |↓↑⟩)/\sqrt2(SP.7)

The j=1 triplet is symmetric under particle exchange; the j=0 singlet is antisymmetric. This decomposition determines the spectral terms of helium (para- vs orthohelium), deuteron binding, and exchange interactions in ferromagnetism.

SP.4 Spin-Orbit Coupling and Fine Structure

An electron moving in a Coulomb field sees a magnetic field in its rest frame (from the moving nucleus). This couples orbital and spin angular momentum:

HSO=(1/2m2c2)(1/r)(dV/dr)L\cdotS(spin−orbitHamiltonian)H_{SO} = (1/2m^{2}c^{2}) (1/r)(dV/dr) L\cdotS \qquad (spin-orbit Hamiltonian)(SP.8)

Using J = L + S → L·S = (J²−L²−S²)/2 = ℏ²(j(j+1) − l(l+1) − s(s+1))/2. The energy splitting between j = l+½ and j = l−½ states is the fine-structure splitting, of order α⁴m_ec² (α = 1/137 is the fine structure constant). This lifts the degeneracy of the 2p₁/₂ and 2p₃/₂ levels of hydrogen by 0.365 cm⁻¹ (observed as the sodium D-line doublet).

Theorem SP.1 — Wigner-Eckart Theorem (Conceptual Statement)
Matrix elements of tensor operators between angular momentum eigenstates factor into a geometric part (Clebsch-Gordan coefficient) and a reduced matrix element:⟨j′m′∣Tqk∣jm⟩=⟨jmkq∣j′m′⟩⟨j′∣∣Tk∣∣j⟩j' m' | T^{k}_q | j m⟩ = ⟨j m k q | j' m'⟩ ⟨j' || T^{k} || j⟩This theorem states that selection rules and relative intensities of spectral lines are determined entirely by angular momentum algebra, independentofthedetaileddynamics.ItexplainswhyelectricdipoletransitionsrequireΔndependent of the detailed dynamics. It explains why electric dipole transitions require \Delta
Definition SP.1 — Common Traps
  • Spin is intrinsic angular momentum: it is not a classical ball spinning in space.
  • Components do not commute: measuring one spin axis changes predictions for another.
  • Add angular momenta with Clebsch-Gordan rules: quantum vector addition is not ordinary component addition.
  • Magnetic moments follow spin and orbital contributions: signs and g-factors matter.
Exercises — SP.1–SP.4 Spin and Angular Momentum
1.
Anelectronisinstate∣↑⟩.Findtheprobabilityofobtaining+ℏ/2inanx−directionspinAn electron is in state |↑⟩. Find the probability of obtaining +\hbar/2 in an x-direction spin measurement.
Straightforward
2.
Countthenumberofdistinct∣j,mj⟩statesforahydrogen2pelectron(ℓ=1,s=1/2Count the number of distinct |j, m_{j}⟩ states for a hydrogen 2p electron (ℓ=1, s=1/2.
Straightforward
3.Anelectronspininitiallyinthe+xdirectionisplacedinmagneticfieldB=Bz^.FindhoAn electron spin initially in the +x direction is placed in magnetic field B = Bẑ. Find how the expectation values ⟨Sx⟩, ⟨Sy⟩, ⟨Sz⟩ evolve in time.
Straightforward
4.Two electrons are in the singlet state. Compute the CHSH parameter S for the optimal angles. Does quantum mechanics violate Bell's inequality?
Intermediate
5.Calculatethespin−orbitfinestructuresplittingforthen=2hydrogenlevels.WhichlevelCalculate the spin-orbit fine structure splitting for the n=2 hydrogen levels. Which levels are split, and by how much?
Intermediate
6.Explain the density matrix formalism for spin-½. What is the Bloch vector? How does the density matrix distinguish pure states from mixed states and separable from entangled states?
Challenging
Key Takeaways
  • [Lx,Ly]=i\hbarLz—angularmomentumcomponentsdon′tcommute;onlyL2andLzaresimultaneouLx, Ly] = i\hbarLz — angular momentum components don't commute; only L^{2} and Lz are simultaneouly definable.
  • Quantumnumbers:l=0,1,2,⋯ ;m=−l,⋯ ,+l.Magnitude∣L∣=ℏ(l(l+1))≠\hbarlQuantum numbers: l = 0,1,2,\cdots; m = -l,\cdots,+l. Magnitude |L| = \hbar\sqrt(l(l+1)) ≠ \hbarl.
  • Spin−12:s=12;eigenstates∣↑⟩,∣↓⟩;Paulimatrices\sigmax,\sigmay,\sigmazrepresentspinoperatorsSpin-\frac{1}{2}: s = \frac{1}{2}; eigenstates |↑⟩, |↓⟩; Pauli matrices \sigmax, \sigmay, \sigmaz represent spin operators.
  • Twospin−12:j=0(singlet,antisymmetric)orj=1(triplet,symmetric).Clebsch−GordandTwo spin-\frac{1}{2}: j = 0 (singlet, antisymmetric) or j = 1 (triplet, symmetric). Clebsch-Gordan decomposition.
  • Spin−orbitcouplingHSO∝L\cdotS→finestructuresplitting;j=l±12levelshavedifferentSpin-orbit coupling H_{SO} \propto L\cdotS \to fine structure splitting; j = l \pm \frac{1}{2} levels have different energies.
  • Wigner−Eckart:selectionrules(\Deltal=±1,\Deltam=0,±1)followfromangularmomentumalgebraaloneWigner-Eckart: selection rules (\Deltal=\pm1, \Deltam=0,\pm1) follow from angular momentum algebra alone.