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Topic 7 of 8

SU(2) and Angular Momentum

$SU(2)$ is the group of $2\times2$ unitary matrices with determinant 1. It is the mathematical home of quantum angular momentum — spin. The quantization of spin, the ladder operators, the Clebsch-Gordan decomposition of tensor products — all follow inevitably from the structure of $SU(2)$ and its representations. It also double-covers $SO(3)$, explaining why spin-$\frac{1}{2}$ particles pick up a minus sign under $2\pi$ rotation.

Key Concepts

SU(2)

SU(2) = \{U \in M_{2\times2}(\mathbb{C}) : U^\dagger U = I,\, \det U = 1\}

. Diffeomorphic to

S^3

as a manifold. Dimension 3.

Pauli Matrices

\sigma^1,\sigma^2,\sigma^3

— Hermitian, traceless, each squaring to

I

. Generators:

T^i = \sigma^i/2

. They satisfy

\{\sigma^i,\sigma^j\}=2\delta^{ij}I

and

[\sigma^i,\sigma^j]=2i\varepsilon^{ijk}\sigma^k

Spin-

j

Representation

For each

j = 0, \frac{1}{2}, 1, \frac{3}{2}, \ldots

, an irrep of dimension

2j+1

with basis states

|j,m\rangle

m = -j, -j+1, \ldots, j

Ladder Operators

J_\pm = J_x \pm iJ_y

. Action:

J_\pm|j,m\rangle = \sqrt{(j\mp m)(j\pm m+1)}\,|j,m\pm1\rangle

. These raise or lower

m

by 1.

Double Cover of

SO(3)

The map

SU(2)\to SO(3)

sending

U\mapsto R_{ij}=\tfrac{1}{2}\text{Tr}(\sigma^i U \sigma^j U^\dagger)

is a 2-to-1 surjective homomorphism. Both

U

and

-U

map to the same rotation.

Key Equations

SU(2) Commutation Relations

[J_i,\, J_j] = i\,\varepsilon_{ijk}\,J_k

The angular momentum algebra — fundamental to all of quantum mechanics.

Casimir Operator

J^2 = J_x^2 + J_y^2 + J_z^2,\quad J^2|j,m\rangle = j(j+1)|j,m\rangle

The Casimir commutes with all

J_i

and labels irreps by

j

Pauli Matrix Algebra

\sigma^i \sigma^j = \delta^{ij}\,I + i\,\varepsilon^{ijk}\,\sigma^k

Encodes both the anticommutator and commutator in one formula.

Worked Example

Eigenvalues for Spin- $3/2$

Problem

For a particle with spin $j = 3/2$ , find: (a) all eigenvalues of $J_z$ , (b) the Casimir eigenvalue $j(j+1)$ , and (c) the dimension of the representation.

Solution

(a) Eigenvalues of $J_z$ are $m = -j, -j+1, \ldots, j$ :

m \in \left\{-\tfrac{3}{2},\,-\tfrac{1}{2},\,+\tfrac{1}{2},\,+\tfrac{3}{2}\right\}

(b) Casimir eigenvalue: $j(j+1) = \frac{3}{2}\cdot\frac{5}{2} = \frac{15}{4} = 3.75$ .

Answer

J_z

eigenvalues:

-3/2, -1/2, +1/2, +3/2

; Casimir:

15/4

; dimension:

4

Practice

Exercises

7 problems

1 of 7

For $j = 2$ , what is the Casimir eigenvalue $j(j+1)$ ?

Your answer

2 of 7

What is the dimension of the $j = 5/2$ representation of $SU(2)$ ?

Your answer

3 of 7

The Pauli matrix $\sigma^1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ . What is $\mathrm{Tr}(\sigma^1)$ ?

Unlock Exercise 3