Chapter 2: Angular Momentum and Spin

Question 1

Symmetries

Answer 1

• Symmetries are represented by unitary operators
  
  • conserve scalar product and normalization
  • conserve energy → commute with Hamiltonian [H,U] = 0
  • NOTE: except, time-revresal symmetry is anti-unitary
• Each symmetry corresponds to some conserved physical quanitity

Question 2

Types of Symmetries and
Corresponding Conserved Quantity

Answer 2

1. Translational ⇐⇒ linear Linear momentum
2. Rotational ⇐⇒ Angular momentum

Question 3

Translational Symmetry

Answer 3

• U_<b>a</b> translates by a

Question 4

Rotational Symmetry

Answer 4

• U_{<strong>ω </strong>}rotates about axis ω through angle |ω|
• R_{<strong>ω </strong>}is representation of rotation in real-space
• R_{<em><strong>ω</strong></em>} ∈ SO(3) = {3 × 3 matrix : A⁻¹ = A^T , detA = 1}
• Rotations do not commute

Question 5

Generator of Translation

Answer 5

momentum operator

Question 6

Translation Commutation

Answer 6

Question 7

Generator of Rotation

Answer 7

angular momentum operator

Question 8

Rotation (and Angular Momentum) Commutators

Answer 8

Question 9

Angular Momentum

(Eigenstates and Eigenvalues)

Answer 9

NOTE: l is integer or half-integer

Question 10

Angular Momentum

(Lowering and Raising Operators)

Answer 10

• In analogy to harmonic oscillator, introduce lowering and raising operators
• λ_± gives m_min = −l
• For fixed l, H_L = {| l, m> : m = −l, . . . , l} is irreducible subspace with representation D_L on H_L

Question 11

Spin

(Motivation)

Answer 11

• Up to now, only 1D wavefunction
• Consider how rotations act on vector wavefunctions

Question 12

Generator of Infinitesimal Rotation of Vector Wavefunction

Answer 12

• For |ω| → 0
• Spin is internal degree of freedom that has special consideration under rotation

Question 13

Spin

(Eigenstates and Eigenvalues)

Answer 13

Question 14

Hydrogen Atom

(Hilbert Space)

Answer 14

Question 15

Hydrogen Atom

(Natural Basis)

Answer 15

Question 16

Spin

(Lowering and Rasiing Operators)

Answer 16

Question 17

Addition of Angular Momentum Generalization

(Overview)

Answer 17

• Consider J = J₁ + J₂
• Basis & Dimension
• New basis |j₁j₂jm> characters by eigens states of J², J_z for J = J₁ + J₂

Question 18

Addition of Angular Momentum Generalization

(Allowable j values)

Answer 18

Question 19

Addition of Angular Momentum Generalization

(General Basis Transformation)

Answer 19

• given by Glesh-Gordon Coefficients