12.1 Symmetries and groups in classical and quantum mechanics

Symmetry in mechanics is all about transformations that keep physical properties unchanged. From translations to rotations, these symmetries connect to conservation laws and simplify equations. They're crucial in both classical and quantum realms.

Group theory provides a powerful toolkit for describing symmetries in physics. It helps us understand continuous and discrete symmetries, leading to conservation laws and influencing system dynamics. This mathematical approach is key in quantum mechanics and particle physics.

Symmetry Fundamentals

Symmetry transformations in mechanics

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  Noether’s Conservation Laws and Stability in Nonlinear Conservative Interactions View original

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  Symmetries and solutions – foreXiv View original

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  Noether’s Conservation Laws and Stability in Nonlinear Conservative Interactions View original

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  Noether's Theorems [The Physics Travel Guide] View original

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• Symmetry transformations preserve physical properties of systems unchanged include translations, rotations, reflections, and time reversal
• Role in classical mechanics connects to conservation laws through Noether's theorem and simplifies equations of motion by reducing degrees of freedom
• Role in quantum mechanics produces degeneracies in energy levels, establishes selection rules for transitions (dipole transitions), and determines conservation of quantum numbers (angular momentum)

Continuous vs discrete symmetries

• Continuous symmetries applied by any amount lead to conserved quantities like linear momentum from translations and angular momentum from rotations
• Discrete symmetries occur in distinct steps include reflections, inversions, and permutations associated with parity and charge conjugation in particle physics

Group Theory in Physics

Group theory for symmetry description

• Group theory basics require closure, associativity, identity, and inverse elements applied to symmetry operations
• Lie groups and algebras describe continuous symmetry transformations using generators and commutation relations (angular momentum operators)
• Point groups characterize discrete symmetries of molecules ($C_{2v}$ for water) and crystals (cubic, hexagonal)
• Symmetry groups in quantum mechanics encompass SU(2) for spin, SO(3) for rotations, and U(1) for phase transformations

Symmetry effects on system dynamics

• Conservation laws emerge from symmetries:
  1. Energy conservation from time translation invariance
  2. Momentum conservation from space translation invariance
  3. Angular momentum conservation from rotational invariance
• Symmetry breaking occurs spontaneously in phase transitions (ferromagnetic ordering) or explicitly by external fields (Zeeman effect)
• Symmetry in quantum systems creates degeneracies in energy spectra, establishes selection rules for transitions (Laporte rule), and forms symmetry-adapted wavefunctions (molecular orbitals)
• Consequences in particle physics enable classification of elementary particles (hadrons, leptons) and conservation of quantum numbers (baryon number, lepton number)