8. Spin, identical particles and entanglement
Position in the course: Lesson 8 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Angular momentum and tensor products
1. Spin is intrinsic angular momentum
An electron has spin \(s=1/2\). This is not a charged sphere mechanically spinning; it is an intrinsic representation of rotations. In the \(S_z\) basis write \(|\alpha\rangle=(1,0)^T\) and \(|\beta\rangle=(0,1)^T\). The Pauli matrices give \(S_k=\hbar\sigma_k/2\), satisfying the same angular momentum commutators. The squared spin is \(S^2=3\hbar^2I/4\), while \(S_z\) outcomes are \(\pm\hbar/2\).
For \(|\alpha\rangle\), an \(x\) measurement gives both signs with probability \(1/2\): the \(x\) eigenvectors are \((|\alpha\rangle\pm|\beta\rangle)/\sqrt2\). A sharp spin component therefore does not imply a classical orientation with sharp projections on every axis.
2. Add two spins explicitly
The two-spin space has four product states. Total spin is \(S=S_1+S_2\). The three symmetric triplet states and one antisymmetric singlet are
Expand \(S^2=S_1^2+S_2^2+2S_1\cdot S_2\). The dot product expectation is \(\hbar^2/4\) in the triplet and \(-3\hbar^2/4\) in the singlet. These values are useful for exchange models, although a model coupling constant must still be derived or fitted rather than inferred from spin counting alone.
3. Antisymmetry concerns full coordinates
For two identical electrons exchanging both position and spin changes the total wavefunction sign. A symmetric spatial factor must accompany the antisymmetric singlet spin factor; an antisymmetric spatial factor can accompany a triplet. For orthonormal spatial orbitals \(a,b\), form \((a(1)b(2)\pm b(1)a(2))/\sqrt2\). If \(a=b\), the minus combination vanishes. Two opposite-spin electrons can share one spatial orbital because their complete spin orbitals differ.
4. Worked singlet measurement and reduced state
In the singlet, measuring electron 1 along \(z\) yields either sign with probability \(1/2\). Conditional on the plus result, electron 2 has the minus result along the same axis. Tracing over spin 2 gives
Thus each subsystem is mixed even though the combined pair is pure. No product of two single-spin vectors reproduces the singlet amplitudes. The anticorrelation does not permit controlled instantaneous messaging: the unconditioned marginal remains \(I/2\) regardless of the other observer's chosen measurement axis.
5. Exercises with solutions
Evaluate \(\langle S_z\rangle\) and \(\langle S_z^2\rangle\) for one spin in the singlet. Equal probabilities give zero mean and \(\hbar^2/4\) second moment. Zero average is not zero uncertainty.
Can a triplet occupy one common spatial orbital? No. Its symmetric spin part requires an antisymmetric spatial part, which vanishes when both orbitals are identical. This is the Pauli principle applied to a complete state.
6. Limits and bridges
Spin-free molecular Hamiltonians preserve total spin, but spin-orbit coupling can mix these labels. A single unrestricted determinant with unequal alpha and beta orbitals may be an eigenstate of \(S_z\) without being an eigenstate of \(S^2\); this is spin contamination. The molecular methods pathway develops determinants and explains why spin symmetry affects reference selection and correlation.
Further conceptual check
For a rotationally invariant singlet, the same-axis anticorrelation holds along any axis. A triplet with zero \(z\) projection is not rotationally invariant and has different correlations in other directions. Having zero mean spin along one axis therefore does not identify a singlet. Compute \(S^2\) or the full correlation structure to distinguish them.
7. References and study connections
The derivations and toy arithmetic are original teaching constructions. No molecular simulation is reported here.
8. Related theory and practice
Molecular methods · Quantum Monte Carlo · Gaussian · VASP