Physic Labs

Quantum mechanics

Spin and the Pauli exclusion principle

Spin is intrinsic quantum angular momentum; the Pauli principle connects fermionic antisymmetry to the impossibility of occupying the same quantum state.

Besides position and motion, a quantum particle has spin: an intrinsic degree of freedom measured as angular momentum. For an electron, measuring spin along an axis gives only two outcomes, although the pre-measurement state may be a superposition.

S2=s(s+1)ℏ2,Sz∣s,m⟩=mℏ∣s,m⟩S^2=s(s+1)\hbar^2,\qquad S_z|s,m\rangle=m\hbar|s,m\rangle

Definition: Spin-1/2 and spin states

The electron has s=1/2s=1/2. In the SzS_z eigenbasis, a normalized state is ∣ψ⟩=α∣+z⟩+β∣−z⟩|\psi\rangle=\alpha|+z\rangle+\beta|-z\rangle, with ∣α∣2+∣β∣2=1|\alpha|^2+|\beta|^2=1. The probabilities for +ℏ/2+\hbar/2 and −ℏ/2-\hbar/2 are ∣α∣2|\alpha|^2 and ∣β∣2|\beta|^2.

Rotate the state vector on the Bloch sphere, then see an inhomogeneous magnetic field split the beam into two Born-rule-weighted spots.

The Bloch sphere: a state is a direction

Up to a global phase, every pure spin-1/2 state is a point on the unit sphere: ∣ψ⟩=cos⁡(θ/2)∣+z⟩+eiϕsin⁡(θ/2)∣−z⟩|\psi\rangle=\cos(\theta/2)|+z\rangle+e^{i\phi}\sin(\theta/2)|-z\rangle. Its mean Pauli vector is (sin⁡θcos⁡ϕ,sin⁡θsin⁡ϕ,cos⁡θ)(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta). A magnetic field rotates this direction by Larmor precession.

P(+n)=1+r⋅n2,P(−n)=1−P(+n)P(+\mathbf n)=\frac{1+\mathbf r\cdot\mathbf n}{2},\qquad P(-\mathbf n)=1-P(+\mathbf n)

Spin measurements and statistics

An ideal Stern–Gerlach analyzer uses a magnetic-field gradient so the force on the magnetic moment depends on spin along the analysis axis. Each particle reaches one branch; an ensemble divides according to the probabilities. An analyzer perpendicular to the preparation axis gives equal probabilities.

Example: Example: measurement along a tilted axis

Prepare ∣+z⟩|+z\rangle, then measure along an axis at 60∘60^\circ to zz. Then P(+)=cos⁡2(θ/2)=3/4P(+)=\cos^2(\theta/2)=3/4 and P(−)=1/4P(-)=1/4. These are frequencies expected over repetitions, not fractional outcomes for one particle.

Solution

Born's rule gives P(+)=∣⟨+n∣+z⟩∣2=(1+cos⁡θ)/2=3/4P(+)=|\langle +\mathbf n|+z\rangle|^2=(1+\cos\theta)/2=3/4.

A general spin state is ∣χ⟩=α∣+z⟩+β∣−z⟩|\chi\rangle=\alpha|+z\rangle+\beta|-z\rangle, with ∣α∣2+∣β∣2=1|\alpha|^2+|\beta|^2=1. Measuring along zz gives ±ℏ/2\pm\hbar/2 with probabilities ∣α∣2,∣β∣2|\alpha|^2,|\beta|^2. The relative phase does not affect this measurement but matters for interference along another axis. For many electrons, Pauli exclusion applies to the complete one-particle state, including position and spin; antisymmetrization explains atomic shell structure.

Quick check

For spin-1/2 along z, what are the two S_z outcomes?

If the measurement axis matches the polarization, what is the positive-result probability?

References

  1. David J. Griffiths, Darrell F. Schroeter (2018). Introduction to Quantum Mechanics