Parity, Time Reversal, and Discrete Symmetries

The continuous symmetries of the previous lesson were all reached by exponentiating a Hermitian generator. Two of the most important symmetries cannot be. Parity, the reflection of space through the origin, and time reversal, the running of the film backward, are discrete: there is no infinitesimal version to exponentiate. One is unitary and supplies a two-valued quantum number and a set of selection rules; the other is antiunitary and, for half-integer spin, guarantees a degeneracy nothing weaker can remove.¹

The parity operator

The parity operator $\hat{\Pi }$ inverts every position coordinate,

Applying it twice restores the original, so ${\hat{\Pi }}^2=1$. It is both Hermitian and unitary, and its eigenvalues, squaring to one, are

with $+1$ for even states and $-1$ for odd states. Parity acts on the canonical operators by flipping the polar vectors and leaving the axial ones fixed:

Position and momentum reverse because they are ordinary (polar) vectors; angular momentum $\hat{\vec{L}}=\hat{\vec{r}}\times \hat{\vec{p}}$ is a cross product of two polar vectors and so does not reverse — it is a pseudovector. Spin $\hat{\vec{S}}$, sharing the transformation properties of angular momentum, is also even under parity.

Parity as a good quantum number

When the potential is symmetric, $V(-\vec{r})=V(\vec{r})$, the Hamiltonian commutes with parity, $[\hat{\Pi },\hat{H}]=0$. Then energy and parity are compatible observables, and a nondegenerate energy eigenstate is automatically an eigenstate of $\hat{\Pi }$ — it has definite parity. The proof mirrors the oscillator argument: if $\hat{H}\psi =E\psi$ with $E$ nondegenerate and $V$ even, then $\psi (-\vec{r})$ solves the same equation at the same energy, so it can differ from $\psi (\vec{r})$ only by a constant, and that constant squares to one.

The bound states of every symmetric potential organize this way. The infinite and finite square wells centered at the origin, the harmonic oscillator, and any even $V$ produce eigenstates alternating even, odd, even, ... as the number of nodes grows. Parity is exact here in a way it is only approximate for the weak interaction, where it is famously violated.

Parity selection rules

Parity constrains which transitions an atom can make. The rate of an electric-dipole transition between states $|i⟩$ and $|f⟩$ is set by the matrix element $⟨f|\hat{\vec{r}}|i⟩$. The position operator is parity-odd, so under $\hat{\Pi }$ the integrand picks up the product of the two states' parities times $-1$. If $|i⟩$ and $|f⟩$ have the same parity, the integrand is odd and the integral over all space vanishes:

A dipole transition connects only states of opposite parity. Written in terms of the orbital quantum number, where parity is $(-1{)}^{\ell }$, this is the Laporte rule: $\ell$ must change by an odd amount, and combined with the angular-momentum content of $\hat{\vec{r}}$ it sharpens to $\Delta \ell =\pm 1$. The oscillator selection rule$\Delta n=\pm 1$ is the one-dimensional case of the same parity argument.

The time-reversal operator

Time reversal reverses the direction of motion: positions are untouched but velocities and momenta flip. The Schrödinger equation $i\hslash {\partial }_t\psi =\hat{H}\psi$ is not invariant under $t\to -t$ alone, because the single time derivative changes sign while $\hat{H}$ does not. It becomes invariant if the transformation also conjugates the wavefunction, $\psi (\vec{r},t)\to {\psi }^{\ast }(\vec{r},-t)$. Complex conjugation is not a linear operation on the Hilbert space, and this is the essential feature.

The defining relation is $\hat{\Theta }i{\hat{\Theta }}^{-1}=-i$: time reversal conjugates the imaginary unit. From this and the requirement that it flip momentum, its action on the observables is

Angular momentum and spin reverse because they are built from, or transform like, $\vec{r}\times \vec{p}$ with one factor of $\vec{p}$ flipped. For a spinless particle the operator is simply complex conjugation, $\hat{\Theta }=\hat{K}$, and ${\hat{\Theta }}^2=+1$. A consequence is that the energy eigenfunctions of a time-reversal-invariant, spinless Hamiltonian can always be chosen real.

Kramers degeneracy

The spinless case squares to $+1$, but spin changes the sign. Building time reversal for a spin-$\frac{1}{2}$ particle requires a rotation in spin space alongside the conjugation, $\hat{\Theta }=-i{\hat{\sigma }}_y\hat{K}$, and this operator squares to ${\hat{\Theta }}^2=-1$. More generally ${\hat{\Theta }}^2=(-1{)}^{2j}$: integer spin gives $+1$, half-integer spin gives $-1$. The negative sign has a sharp physical consequence.

The pair $|\psi ⟩$ and $\hat{\Theta }|\psi ⟩$ is a Kramers doublet. An electron bound in any electrostatic environment, however irregular, keeps this twofold degeneracy: no purely electric field can split it, because electric fields are time-reversal invariant. Only a magnetic field, which is time-reversal odd and breaks $[\hat{\Theta },\hat{H}]=0$, lifts the doublet — the Zeeman splitting.

Parity and time reversal together

The two discrete symmetries jointly forbid one striking possibility: a permanent electric dipole moment aligned with a particle's spin. Such a moment $\vec{d}=d\hat{\vec{S}}/|\vec{S}|$ would need $d\hat{\vec{S}}$ to transform like the polar, time-even vector $\hat{\vec{r}}$. But $\hat{\vec{S}}$ is parity-even and time-odd, the exact opposite of $\hat{\vec{r}}$ on both counts. A nonzero $d$ therefore requires violating both parity and time reversal. Searches for a neutron or electron electric dipole moment are precision tests of combined $P$ and $T$ violation, and their null results to date bound physics beyond the standard model.²

The four symmetries of these two lessons complete the basic catalog:

Parity and time reversal round out the symmetry structure that angular momentum and spin build on directly.