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A symmetry of a quantum system is a transformation, represented by an operator, that leaves the Hamiltonian unchanged. For a continuous symmetry the operator can be written exp(−iεG/ħ) with a Hermitian generator G, and the symmetry holds exactly when [G, H] = 0, which makes G a conserved quantity. Translations are generated by momentum and rotations by angular momentum, so these are conserved when space is uniform and isotropic. Discrete symmetries such as parity give conserved quantum numbers and selection rules, and time reversal is represented by an antiunitary operator.
Begin with the translation operator T(a) = exp(−ipa/ħ) and check that it shifts a wavefunction, T(a)ψ(x) = ψ(x − a). Then show that [T, H] = 0 for a free particle and connect this to conservation of momentum through the Heisenberg equation. Next treat parity: find the parity of oscillator and hydrogen eigenstates and use it to derive the dipole selection rule. Close with time reversal and Kramers degeneracy as a case where a symmetry forces a degeneracy.
In classical mechanics, Noether's theorem ties each continuous symmetry to a conserved quantity. Quantum mechanics expresses the same idea with operators. A transformation of a system is represented by an operator U acting on states. It is a symmetry if it leaves the dynamics unchanged, which for a unitary U means U†HU = H, or equivalently [U, H] = 0. A theorem of Eugene Wigner states that any transformation preserving transition probabilities can be represented by an operator that is either unitary or antiunitary.
Continuous symmetries are built from small steps. An infinitesimal transformation has the form U = 1 − iεG/ħ, where G is a Hermitian operator called the generator. A finite transformation is U = exp(−iaG/ħ). If U commutes with H for every a, then G commutes with H. By the Heisenberg equation, dG/dt = (i/ħ)[H, G] = 0, so the expectation value of G never changes, and states can be labelled by eigenvalues of both H and G.
Three cases form the core of the subject. Translations are generated by momentum: exp(−ipa/ħ) shifts ψ(x) to ψ(x − a). A Hamiltonian that looks the same everywhere conserves momentum. Rotations are generated by angular momentum, so a rotationally symmetric Hamiltonian, such as a central potential, conserves L. Time translations are generated by H itself, which conserves energy when H does not depend on time. Symmetries also explain degeneracies. Rotational symmetry is why each level of a central potential holds 2l + 1 states with different m.
Parity Π reflects every coordinate, Πψ(r) = ψ(−r). Applying it twice gives the identity, so its eigenvalues are +1 and −1. When Π commutes with H and a level is not degenerate, the eigenstate has definite parity. Harmonic oscillator states have parity (−1)ⁿ and hydrogen orbitals have parity (−1)l. Because the position operator is odd, electric dipole matrix elements ⟨f|r|i⟩ vanish unless the two states have opposite parity. In hydrogen this forbids Δl = 0 and every even Δl. Rotational symmetry limits a vector operator to |Δl| ≤ 1, and together the two give the rule Δl = ±1.
Time reversal Θ reverses momenta and spins while leaving positions alone. To be consistent with the Schrödinger equation it must include complex conjugation, so it is antiunitary. One consequence is Kramers' theorem: in a time-reversal-invariant system with an odd number of spin-½ particles, every energy level is at least doubly degenerate. Parity and time reversal were long assumed to be exact symmetries of nature. Experiments beginning in 1957 showed that the weak interaction violates parity, and later experiments showed that it also violates time-reversal symmetry, though only slightly.