Parity Operator

Operator

For a spinless wavefunction, parity acts by spatial inversion:

$$(\mathsf P\psi)(\mathbf r) = \psi(-\mathbf r).$$

It satisfies

$$\mathsf P^2=I, \qquad \mathsf P^\dagger=\mathsf P,$$

so its eigenvalues are

$$\pi=\pm1.$$

On position and momentum operators,

$$\mathsf P\mathbf r\mathsf P^{-1} = -\mathbf r, \qquad \mathsf P\mathbf p\mathsf P^{-1} = -\mathbf p.$$

Assumptions

• The displayed action is for spinless scalar wavefunctions.
• Internal degrees of freedom may carry additional parity action.
• Parity is conserved only when $[\mathsf P,H]=0$.
• The spatial inversion operation is meaningful for the system’s configuration space.

Common Mistakes

• Assuming every Hamiltonian has parity symmetry.
• Confusing parity eigenvalue with energy sign.
• Forgetting internal parity factors for spinor, molecular, or field-like systems.
• Using parity language in a coordinate system where inversion has not been defined.

Canonical Links

• Parity as Spatial Inversion
• Parity
• Parity and Nodes
• Time Reversal
• Symmetry Constraints on Hamiltonians

References

• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.