Introduction to Quantum Mechanics · Problem 6.1

Symmetries and Conservation Laws — David J. Griffiths

Physics — Quantum Mechanics · parity_operator

Problem

Consider the parity operator in three dimensions. (a) Show that Π^ψ(r)=ψ′(r)=ψ(−r)\hat{\Pi} \psi(\mathbf{r}) = \psi'(\mathbf{r}) = \psi(-\mathbf{r}) is equivalent to a mirror reflection followed by a rotation. (b) Show that, for ψ\psi expressed in polar coordinates, the action of the parity operator is Π^ψ(r,θ,ϕ)=ψ(r,π−θ,ϕ+π).\hat{\Pi} \psi(r, \theta, \phi) = \psi(r, \pi - \theta, \phi + \pi). (c) Show that for the hydrogenic orbitals, Π^ψnℓm(r,θ,ϕ)=(−1)ℓψnℓm(r,θ,ϕ).\hat{\Pi} \psi_{n\ell m}(r, \theta, \phi) = (-1)^\ell \psi_{n\ell m}(r, \theta, \phi). That is, ψnℓm\psi_{n\ell m} is an eigenstate of the parity operator, with eigenvalue (−1)ℓ(-1)^\ell. Note: This result actually applies to the stationary states of any central potential V(r)=V(r)V(\mathbf{r}) = V(r). For a central potential, the eigenstates may be written in the separable form Rnℓ(r)Yℓm(θ,ϕ)R_{n\ell}(r) Y_\ell^m(\theta, \phi) where only the radial function RnℓR_{n\ell}—which plays no role in determining the parity of the state—depends on the specific functional form of V(r)V(r).

Browse the problem library