9.3 Spin angular momentum and Pauli matrices

Spin angular momentum is a mind-bending quantum property with no real-world equivalent. It's key to understanding particle behavior and explains phenomena like magnetism. Measuring spin led to groundbreaking discoveries about the quantum nature of reality.

Pauli matrices are mathematical tools that describe spin states and how they change. They're crucial for calculating probabilities in quantum mechanics and help us predict how particles will behave in different situations.

Spin and the Stern-Gerlach Experiment

Fundamental Concepts of Spin

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• Spin represents intrinsic angular momentum of particles
• Quantum mechanical property with no classical analogue
• Measured in units of ℏ (reduced Planck's constant)
• Spin-1/2 particles include electrons, protons, and neutrons
• Spin quantum number s determines allowed spin states
• Spin magnetic moment arises from particle's intrinsic spin

The Stern-Gerlach Experiment

• Conducted in 1922 by Otto Stern and Walther Gerlach
• Demonstrated quantization of angular momentum in atoms
• Experimental setup consists of beam of silver atoms passing through inhomogeneous magnetic field
• Observed discrete splitting of beam into two components
• Results contradicted classical physics predictions
• Provided evidence for spatial quantization of magnetic moments

Implications and Applications of Spin

• Spin explains Zeeman effect in atomic spectra
• Fundamental role in understanding atomic structure
• Crucial for explaining ferromagnetism and other magnetic phenomena
• Forms basis for nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI)
• Essential concept in particle physics and quantum field theory
• Spin statistics theorem relates particle spin to its quantum statistics (fermions vs bosons)

Pauli Matrices and Spin Operators

Mathematical Foundations of Spin

• Pauli matrices represent 2x2 complex Hermitian matrices
• Three Pauli matrices: σx, σy, and σz
• Pauli matrices satisfy specific algebraic properties (anticommutation relations)
• Spin operators derived from Pauli matrices: Sx = (ℏ/2)σx, Sy = (ℏ/2)σy, Sz = (ℏ/2)σz
• Commutation relations of spin operators: [Sx, Sy] = iℏSz (cyclic permutations)
• Total spin operator S² = Sx² + Sy² + Sz² commutes with individual spin components

Spin Eigenstates and Quantum Measurements

• Spin eigenstates represent definite spin states along specific axes
• Sz eigenstates commonly denoted as |↑⟩ (spin-up) and |↓⟩ (spin-down)
• Measurement of spin along arbitrary axis projects state onto corresponding eigenstates
• Probability of measuring spin-up or spin-down determined by quantum state
• Spin precession occurs when magnetic field applied perpendicular to spin axis
• Larmor frequency describes precession rate of spin in magnetic field

Spinors and Quantum State Representation

• Spinors describe quantum states of spin-1/2 particles
• Two-component complex vectors in Hilbert space
• General spinor state written as |ψ⟩ = α|↑⟩ + β|↓⟩, where |α|² + |β|² = 1
• Coefficients α and β determine probability amplitudes for spin measurements
• Spinors transform under rotations according to SU(2) group
• Pauli exclusion principle arises from antisymmetric nature of multi-electron spinor wavefunctions