Symmetry in quantum mechanics

A symmetry in quantum mechanics is a transformation that leaves a system’s physical predictions unchanged. Symmetry operators and groups constrain states, organize energy levels, and connect continuous invariances to conserved quantities.

Unitary operator: A linear operator that preserves inner products and therefore quantum probabilities. Quantum symmetries preserving transition probabilities are represented by unitary operators.
Group theory: The mathematical study of sets equipped with an associative operation, identity, and inverses. Symmetry transformations form groups whose representations classify quantum states.
Angular momentum: The conserved quantity associated with rotational motion, represented by operators in quantum mechanics. Rotational symmetry produces angular momentum as its conserved generator.
Gauge symmetry: A redundancy in the mathematical description of a physical system that leaves observables unchanged. It differs from ordinary global symmetry because gauge-related descriptions represent the same physical state.
Eugene Wigner: A physicist whose work established major applications of symmetry principles in quantum mechanics. His theorem formalized probability-preserving transformations of quantum states.
Wigner's theorem: A theorem stating that transformations preserving transition probabilities are represented by unitary or antiunitary operators. It establishes the operator forms available for quantum symmetries.
Hilbert space: A complete inner-product space used to represent quantum states and their superpositions. Quantum symmetry operators act on the Hilbert space of states.
Parity (physics): The spatial inversion transformation that sends position coordinates to their negatives. Parity symmetry assigns states even or odd spatial-inversion eigenvalues.
Global symmetry: A transformation applied uniformly throughout a system that leaves its physical laws invariant. It contrasts with local gauge transformations, which can vary from point to point.
Hermann Weyl: A mathematician and physicist who developed influential applications of group theory in quantum mechanics. His work helped make group representations a tool for classifying quantum states.

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Degenerate energy levelsRelated: Symmetries can force different states to have identical energies.