Introduction: Mirror Symmetry
The parity operator \(\hat{P}\) reflects space through the origin: \(\vec{r} \to -\vec{r}\). Unlike translations and rotations, parity is a discrete symmetry. Its eigenvalues are ±1, labeling states as even or odd.
Definition and Properties
\[\hat{P}\psi(\vec{r}) = \psi(-\vec{r})\]Properties:
- \(\hat{P}^2 = I\) (applying twice returns original state)
- Eigenvalues: \(\hat{P}|\pm\rangle = \pm|\pm\rangle\)
- \(\hat{P}\) is both Hermitian and unitary
Parity of Operators
- \(\hat{P}\hat{x}\hat{P}^{-1} = -\hat{x}\) (position: odd)
- \(\hat{P}\hat{p}\hat{P}^{-1} = -\hat{p}\) (momentum: odd)
- \(\hat{P}\hat{L}\hat{P}^{-1} = +\hat{L}\) (angular momentum: even)
Worked Examples
Example 1: Symmetric Potentials
If \(V(x) = V(-x)\), then \([\hat{H}, \hat{P}] = 0\) and energy eigenstates can be chosen with definite parity.
Example 2: Infinite Square Well
Ground state \(\psi_1(x) = \cos(\pi x/L)\): even parity (\(\hat{P}\psi_1 = +\psi_1\))
First excited \(\psi_2(x) = \sin(2\pi x/L)\): odd parity (\(\hat{P}\psi_2 = -\psi_2\))
Example 3: Selection Rules
\(\langle\psi_+|\hat{x}|\psi_+\rangle = 0\) (can't connect same-parity states with odd operator)
The Quantum Connection
Parity conservation implies selection rules for transitions. In atomic physics, electric dipole transitions require \(\Delta l = \pm 1\) (parity change). The weak nuclear force violates parity—a shocking discovery that won Lee and Yang the Nobel Prize in 1957.