Quantum Mechanics Course

Total: 450+ pages covering mathematical foundations through advanced quantum mechanics

← Part VII/Symmetrization Postulate

1. Symmetrization Postulate

Reading time: ~32 minutes | Pages: 8

Fundamental principle: identical particles are truly indistinguishable in quantum mechanics.

The Problem of Indistinguishability

Classical physics: Particles can be labeled and tracked

Quantum mechanics: Identical particles cannot be distinguished

  • Two electrons are absolutely identical - same mass, charge, spin
  • No way to "tag" or track which electron is which
  • Wave function must reflect this indistinguishability

Two-Particle States

For distinguishable particles (e.g., electron and proton):

$$|\psi\rangle = |\psi_a(1)\rangle \otimes |\psi_b(2)\rangle$$

Particle 1 in state $\psi_a$, particle 2 in state $\psi_b$

Wave function:

$$\psi(\vec{r}_1, \vec{r}_2) = \psi_a(\vec{r}_1)\psi_b(\vec{r}_2)$$

Exchange Operator

Operator that swaps particles 1 and 2:

$$\hat{P}_{12}\psi(\vec{r}_1, \vec{r}_2) = \psi(\vec{r}_2, \vec{r}_1)$$

Properties:

  • Hermitian: $\hat{P}_{12}^\dagger = \hat{P}_{12}$
  • Involution: $\hat{P}_{12}^2 = 1$ (two swaps return to original)
  • Eigenvalues: $\pm 1$

Symmetric and Antisymmetric States

Symmetric state: $\hat{P}_{12}|\psi_S\rangle = +|\psi_S\rangle$

$$\psi_S(\vec{r}_1, \vec{r}_2) = \psi_S(\vec{r}_2, \vec{r}_1)$$

Antisymmetric state: $\hat{P}_{12}|\psi_A\rangle = -|\psi_A\rangle$

$$\psi_A(\vec{r}_1, \vec{r}_2) = -\psi_A(\vec{r}_2, \vec{r}_1)$$

The Symmetrization Postulate

For a system of identical particles, the physical state must be either completely symmetric or completely antisymmetric under particle exchange.

Which symmetry? Determined by particle spin:

  • Integer spin (s = 0, 1, 2, ...): Symmetric (Bosons)
  • Half-integer spin (s = 1/2, 3/2, 5/2, ...): Antisymmetric (Fermions)

Constructing Symmetric States

Given single-particle states $|a\rangle, |b\rangle$:

$$|\psi_S\rangle = \frac{1}{\sqrt{2}}(|a\rangle_1|b\rangle_2 + |b\rangle_1|a\rangle_2)$$

If $a = b$ (both in same state):

$$|\psi_S\rangle = |a\rangle_1|a\rangle_2$$

Bosons can occupy same quantum state

Constructing Antisymmetric States

Given single-particle states $|a\rangle, |b\rangle$:

$$|\psi_A\rangle = \frac{1}{\sqrt{2}}(|a\rangle_1|b\rangle_2 - |b\rangle_1|a\rangle_2)$$

If $a = b$:

$$|\psi_A\rangle = \frac{1}{\sqrt{2}}(|a\rangle_1|a\rangle_2 - |a\rangle_1|a\rangle_2) = 0$$

Fermions cannot occupy same quantum state - Pauli Exclusion Principle!

N-Particle Generalization

Bosons: Symmetrize over all permutations

$$|\psi_S\rangle = \frac{1}{\sqrt{N!}}\sum_{P} |a\rangle_{P(1)}|b\rangle_{P(2)}\cdots|z\rangle_{P(N)}$$

Fermions: Antisymmetrize with sign of permutation

$$|\psi_A\rangle = \frac{1}{\sqrt{N!}}\sum_{P} (-1)^P |a\rangle_{P(1)}|b\rangle_{P(2)}\cdots|z\rangle_{P(N)}$$

Slater Determinant

Elegant notation for fermion states:

$$|\psi_A\rangle = \frac{1}{\sqrt{N!}}\begin{vmatrix}|a\rangle_1 & |b\rangle_1 & \cdots & |z\rangle_1\\|a\rangle_2 & |b\rangle_2 & \cdots & |z\rangle_2\\\vdots & \vdots & \ddots & \vdots\\|a\rangle_N & |b\rangle_N & \cdots & |z\rangle_N\end{vmatrix}$$

Determinant automatically ensures antisymmetry

Two identical rows (same state) → determinant = 0

Physical Consequences

  • Bosons: Tend to bunch together (Bose-Einstein condensation)
  • Fermions: Spread out (degeneracy pressure in white dwarfs)
  • Exchange forces: Pure quantum effect from symmetrization
  • Chemical bonding: Electrons (fermions) obey Pauli principle
  • Statistics: Different counting rules for bosons vs fermions

Spin-Statistics Theorem

Deep connection between spin and statistics:

  • Integer spin → Bosons → Symmetric wave functions
  • Half-integer spin → Fermions → Antisymmetric wave functions

Proven rigorously in relativistic quantum field theory

Follows from Lorentz invariance + causality + positive energy

No known exceptions in nature

Part VII OverviewFermions & Bosons →