5.2 Spin angular momentum and Pauli matrices

Spin angular momentum is a mind-bending quantum property that particles have, even when they're not spinning. It's like a built-in compass that can only point in certain directions. This quirky feature plays a huge role in how particles behave and interact.

Pauli matrices are the math tools we use to describe and work with spin. They're like a Swiss Army knife for quantum mechanics, helping us calculate how spin changes and how particles with spin interact with each other and magnetic fields.

Spin Angular Momentum of Particles

Fundamental Quantum Mechanical Property

• Spin angular momentum represents an intrinsic property of particles not associated with physical rotation
• Quantized nature of spin allows only specific discrete values characterized by spin quantum number s
• Magnitude of spin angular momentum calculated using $\sqrt{s(s+1)}\hbar$
• Intrinsic property of elementary particles remains unchanged regardless of motion or external fields
• Spin magnetic moment directly relates to spin angular momentum causing various magnetic phenomena in materials
• Crucial role in Pauli exclusion principle and behavior of multi-particle systems

Spin Quantization and Measurement

• Spin quantum number s determines allowed spin states (s = 1/2 for electrons)
• Spin projection along z-axis quantized with values $m_s = -s, -s+1, ..., s-1, s$
• Stern-Gerlach experiment demonstrates quantization of spin for silver atoms
• Measurement of spin along any axis yields only two possible outcomes for spin-1/2 particles (up or down)
• Uncertainty principle applies to simultaneous measurements of spin components along different axes

Spin States and Pauli Matrices

Spinor Representation

• Two-component complex vectors (spinors) describe quantum state of spin-1/2 particles
• General form of spinor for spin-1/2 particle $|\psi\rangle = \alpha|\uparrow\rangle + \beta|\downarrow\rangle$ with complex coefficients α and β
• Normalization condition requires $|\alpha|^2 + |\beta|^2 = 1$
• Bloch sphere provides geometric representation of spin-1/2 states
• Spinor transformations under rotations involve SU(2) group

Pauli Matrices and Their Properties

• 2x2 complex matrices ($\sigma_x, \sigma_y, \sigma_z$) serve as generators of rotations in spin space
• Explicit forms of Pauli matrices: $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
• Satisfy commutation relations $[\sigma_i, \sigma_j] = 2i\varepsilon_{ijk}\sigma_k$ ($\varepsilon_{ijk}$ Levi-Civita symbol)
• Eigenstates of $\sigma_z$ correspond to spin-up $|\uparrow\rangle$ and spin-down $|\downarrow\rangle$ states along z-axis
• Arbitrary spin state expressed as linear combination of eigenstates of chosen Pauli matrix
• Expectation value of spin along any direction calculated using appropriate Pauli matrix and spinor representation

Operations with Pauli Spin Operators

Pauli Spin Operators and Commutation Relations

• Pauli spin operators ($S_x, S_y, S_z$) related to Pauli matrices by $S_i = (\hbar/2)\sigma_i$ (i = x, y, z)
• Satisfy commutation relations $[S_i, S_j] = i\hbar\varepsilon_{ijk}S_k$ reflecting non-commutative nature of spin measurements
• Total spin operator $S^2 = S_x^2 + S_y^2 + S_z^2$ commutes with individual spin component operators
• Application of Pauli spin operator to spin state results in rotation about corresponding axis
• Raising ($S_+$) and lowering ($S_-$) operators constructed from Pauli spin operators for transitions between spin-up and spin-down states

Spin Measurements and Time Evolution

• Measurement of spin along arbitrary direction represented by linear combination of Pauli spin operators
• Expectation values of spin components calculated using $\langle S_i \rangle = \langle \psi | S_i | \psi \rangle$
• Time evolution of spin states under magnetic fields described using Pauli spin operators in Hamiltonian
• Larmor precession of spin in uniform magnetic field with frequency $\omega_L = -\gamma B$ ($\gamma$ gyromagnetic ratio)
• Rabi oscillations occur when spin-1/2 particle exposed to oscillating magnetic field

Bosons vs Fermions: Spin Properties

Spin-Statistics Connection

• Particles with integer spin (including 0) classified as bosons (photons, Higgs bosons)
• Particles with half-integer spin classified as fermions (electrons, quarks, neutrinos)
• Spin-statistics theorem connects spin of particles to exchange symmetry and statistical behavior
• Fermions obey Pauli exclusion principle prohibiting identical fermions from occupying same quantum state
• Bosons allow unlimited occupation of same quantum state
• Wave function of identical fermions antisymmetric under particle exchange, symmetric for bosons

Statistical Behavior and Physical Consequences

• Fermions follow Fermi-Dirac statistics governing behavior in many-particle systems
• Fermi-Dirac distribution describes occupation probabilities of energy states for fermions
• Electron degeneracy pressure in white dwarf stars results from Pauli exclusion principle
• Bosons follow Bose-Einstein statistics allowing for formation of Bose-Einstein condensates
• Bose-Einstein condensation occurs when large fraction of bosons occupy lowest energy state (liquid helium superfluidity)
• Particle spin influences behavior of quantum gases, superconductivity, and other collective phenomena