Paritätstransformationsoperator

The Parity Transformation Operator is a fundamental concept in quantum mechanics that describes the spatial inversion of a system. Mathematically, it is represented by the operator P. When P acts on a quantum state |ψ⟩, it transforms the spatial coordinates of the system by inverting them: P|ψ(r)⟩ = |ψ(-r)⟩. This means that the wavefunction at a position r is mapped to the wavefunction at the position -r. The parity operator has two eigenvalues, +1 and -1, corresponding to even and odd parity states, respectively. A state with even parity remains unchanged under spatial inversion, while a state with odd parity changes its sign.

The parity operator is Hermitian, meaning that P† = P, which ensures that its eigenvalues are real. Furthermore, P² = I, where I is the identity operator. This implies that an object undergoing two parity transformations returns to its original state. The parity operator commutes with the Hamiltonian of a system, [H, P] = 0, if and only if the potential energy of the system is an even function of position, i.e., V(r) = V(-r). In such cases, parity is a conserved quantity, and the energy eigenstates can be chosen to be simultaneous eigenstates of both the Hamiltonian and the parity operator. This has important implications for the symmetry properties of quantum systems and simplifies the analysis of their spectra. For systems with parity-conserving Hamiltonians, states can be classified as having definite even or odd parity, which can lead to selection rules in spectroscopic transitions.