Quantum Mechanics Course

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6. Group Theory Basics

Reading time: ~40 minutes | Pages: 8

Symmetries are at the heart of quantum mechanics. Group theory is the mathematical language for describing symmetries, and it provides profound insights into conservation laws, selection rules, and the structure of quantum states.

What is a Group?

Definition: Group

A group $G$ is a set equipped with a binary operation $\cdot: G \times G \to G$that satisfies four axioms:

  1. Closure: For all $g, h \in G$, $g \cdot h \in G$
  2. Associativity: $(g \cdot h) \cdot k = g \cdot (h \cdot k)$
  3. Identity: There exists $e \in G$ such that $e \cdot g = g \cdot e = g$ for all $g \in G$
  4. Inverses: For each $g \in G$, there exists $g^{-1} \in G$ such that $g \cdot g^{-1} = g^{-1} \cdot g = e$

Physical Interpretation

In quantum mechanics, group elements typically represent symmetry operations:

  • Rotations in 3D space
  • Translations in space and time
  • Reflections and inversions
  • Permutations of identical particles
  • Gauge transformations

Examples of Groups

Example 1: The Rotation Group SO(3)

The group of all rotations in 3D space. Elements are rotations $R(\theta, \hat{n})$ by angle$\theta$ about axis $\hat{n}$.

$$SO(3) = \{R \in \mathbb{R}^{3\times 3} : R^T R = I, \det R = 1\}$$

"SO" stands for "Special Orthogonal" — special means determinant +1 (proper rotations).

Example 2: The Translation Group

Translations in space form a group under composition: $T(\vec{a}) \cdot T(\vec{b}) = T(\vec{a}+\vec{b})$

$$T(\vec{a})\psi(\vec{r}) = \psi(\vec{r} - \vec{a})$$

Example 3: The Permutation Group $S_N$

The group of all permutations of $N$ objects. For identical particles, wave functions must transform under $S_N$ representations.

Order: $|S_N| = N!$

Example 4: The Unitary Group U(1)

Phase transformations $|\psi\rangle \to e^{i\theta}|\psi\rangle$. This is the gauge group of electromagnetism!

$$U(1) = \{e^{i\theta} : \theta \in [0, 2\pi)\}$$

Continuous vs. Discrete Groups

Continuous (Lie Groups)

  • Infinitely many elements
  • Parametrized by continuous variables
  • Examples: SO(3), SU(2), U(1)
  • Described by Lie algebras

Discrete Groups

  • Finite or countable elements
  • Examples: $\mathbb{Z}_n$, $S_N$, point groups
  • Important for crystal symmetries
  • Character tables for representations

Representations

A representation of a group $G$ is a homomorphism from $G$ to linear operators on a vector space:

$$\rho: G \to GL(V)$$

such that $\rho(g_1 g_2) = \rho(g_1)\rho(g_2)$ (preserves group multiplication).

Physical Meaning:

Representations tell us how quantum states transform under symmetries. For example, the spin-1/2 representation of SU(2) acts on the 2D spinor space $\mathbb{C}^2$.

Irreducible Representations (Irreps)

A representation is irreducible if it has no proper invariant subspaces. Irreps are the "building blocks" — any representation can be decomposed into irreps.

$$\rho = \bigoplus_i n_i \rho^{(i)}$$

where $\rho^{(i)}$ are irreducible and $n_i$ are multiplicities.

Generators and Lie Algebras

For continuous groups (Lie groups), we can study infinitesimal transformations near the identity. These are described by generators.

Example: SO(3) Generators

Infinitesimal rotations are generated by angular momentum operators:

$$R(\delta\theta, \hat{n}) = I - \frac{i}{\hbar}\delta\theta\, \hat{n} \cdot \hat{\vec{J}} + O(\delta\theta^2)$$

The generators $\hat{J}_x, \hat{J}_y, \hat{J}_z$ satisfy the Lie algebra:

$$[\hat{J}_i, \hat{J}_j] = i\hbar\epsilon_{ijk}\hat{J}_k$$

Schur's Lemma

Schur's Lemma:

If $\hat{A}$ commutes with all operators in an irreducible representation, then$\hat{A} = \lambda I$ (i.e., $\hat{A}$ is a multiple of the identity).

Consequence: In an irreducible representation, observables that commute with the symmetry group must be constant on each irrep subspace. This explains degeneracy and quantum numbers!

Applications in QM

1. Conservation Laws (Noether's Theorem)

Continuous symmetries lead to conserved quantities:

  • Time translation → Energy conservation
  • Space translation → Momentum conservation
  • Rotation → Angular momentum conservation

2. Quantum Numbers

Quantum states are labeled by irreps of the symmetry group. For example, hydrogen eigenstates are labeled by $(n, \ell, m)$ where $\ell, m$ come from SO(3) irreps.

3. Selection Rules

Group theory determines which matrix elements $\langle f|\hat{O}|i\rangle$ are zero by symmetry, giving selection rules for transitions (e.g., $\Delta \ell = \pm 1$ for electric dipole).

4. Identical Particles

Wavefunctions must transform according to 1D irreps of $S_N$: symmetric (bosons) or antisymmetric (fermions).

5. Particle Physics

Particles are classified by irreps of Poincaré group (spacetime symmetries) and internal symmetries like SU(3) (color), SU(2) (weak isospin), U(1) (hypercharge).

Important Groups in Quantum Mechanics

GroupPhysical Meaning
$SO(3)$3D rotations (orbital angular momentum)
$SU(2)$Spin, weak isospin (double cover of SO(3))
$U(1)$Phase symmetry, electric charge
$S_N$Permutation of N identical particles
PoincaréSpacetime symmetries (relativity)
$SU(3)$Color symmetry (QCD), flavor symmetry

Summary

  • ✓ Groups formalize the concept of symmetry mathematically
  • ✓ Representations show how quantum states transform under symmetries
  • ✓ Irreducible representations are building blocks, related to quantum numbers
  • ✓ Generators of Lie groups form Lie algebras (commutation relations)
  • ✓ Schur's lemma explains degeneracy and selection rules
  • ✓ Conservation laws arise from continuous symmetries (Noether)
  • ✓ Essential for atomic physics, particle physics, and quantum field theory
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