OddParity

Odd parity is a property of functions or quantum states under spatial inversion, the transformation r -> -r. A function ψ is said to have odd parity if it changes sign under inversion: ψ(-r) = -ψ(r). If it remains the same, ψ has even parity: ψ(-r) = ψ(r). In quantum mechanics, states that have definite parity are eigenstates of the parity operator P, with eigenvalues ±1. The eigenvalue is called the parity, and values of -1 and +1 indicate odd and even parity, respectively.

Mathematically, the parity operator P acts on a wavefunction by Pψ(r) = ψ(-r). When a state is an eigenstate of P, it satisfies Pψ = ηψ, where η ∈ {+1, -1}. In many physical systems, parity is a useful quantum number because P commutes with the Hamiltonian in the absence of parity-violating interactions, allowing states to be labeled by their parity.

In atomic and nuclear physics, parity is closely related to orbital angular momentum. For a single-particle orbital with orbital angular momentum l, the intrinsic parity is (-1)^l. For a two-body system, the total parity is the product of the intrinsic parities times (-1)^L, where L is the orbital angular momentum of their relative motion. Consequently, p-waves (l = 1) have odd parity, while s-waves (l = 0) have even parity.

Parity is a conserved quantity in strong and electromagnetic interactions but is violated by the weak interaction. This violation was famously demonstrated in beta decay experiments in the 1950s. Understanding parity and its breaking has guided fundamental insights into the structure of symmetries in physics.

See also: parity operator, parity conservation, parity violation, spherical harmonics.