Symmetry and Selection-Rule Checking

A symmetry supplies predictions that can be checked before trusting a spectrum or transition amplitude. Useful diagnostics include commutator residuals, forbidden off-block entries, conserved sector weights, and selection-rule zeros. A symmetry test must use the symmetry of the actual Hamiltonian and boundaries; imposing a symmetry that the physical problem breaks is an error.

Required background. Symmetry constraints on Hamiltonians and selection rules derive the theoretical statements. Array representations fixes matrix conventions.

Commutators and sector leakage

For a unitary symmetry $U$ satisfying $U^\dagger HU=H$, compute a dimensionless defect such as

$$\epsilon_U= \frac{\|U^\dagger HU-H\|}{\max(\|H\|,E_*)},$$

where $E_*>0$ is a declared energy scale. For an additive Hermitian conserved quantity $Q$, normalize $\|[H,Q]\|$ by a compatible product of energy and $Q$ scales. An absolute commutator norm changes when units change.

If $P_q$ projects onto a symmetry sector, $[H,P_q]=0$ implies $P_{q'}HP_q=0$ for distinct sectors. Start a propagation in one sector and monitor

$$\ell_q(t)=1-\langle\psi(t)|P_q|\psi(t)\rangle$$

for a normalized state. Nonzero leakage can reveal operator assembly, time integration, or projector mistakes. A calculation restricted to the sector by construction cannot use zero leakage as an independent validation of the omitted couplings: inspect the full small-system operator too.

Parity on a symmetric grid

On a grid symmetric about the origin, parity reverses the ordering of samples: $(P\psi)_j=\psi_{N+1-j}$ for one-based interior indices. A symmetric potential and symmetric endpoint conditions should give $PHP=H$ and $P^2=I$. In a weighted representation, reflection must also preserve the quadrature weights.

For parity eigenstates $P|n\rangle=p_n|n\rangle$ and an operator with $POP=s_OO$, where $p_n,s_O=\pm1$,

$$\langle m|O|n\rangle =p_m p_n s_O\langle m|O|n\rangle.$$

Thus the matrix element vanishes unless $p_mp_ns_O=1$. The position operator is odd, so it connects opposite parity. An even operator such as $x^2$ connects equal parity. These are independent checks of eigenvectors and observable assembly.

A small asymmetric grid or boundary error can create forbidden elements. Near a degeneracy, a numerical eigensolver may return mixed parity vectors even when $H$ is correct. Diagonalize parity within the nearly degenerate eigenspace before labeling individual states, and check the subspace’s Hamiltonian residual afterward.

A controlled symmetry-breaking experiment

Use oscillator units and $H=p^2/2+x^2/2+\lambda x$. Completing the square gives

$$H=\frac{p^2}{2}+\frac{(x+\lambda)^2}{2}-\frac{\lambda^2}{2}.$$

At $\lambda=0$, $\langle x\rangle$ vanishes in every stationary state. At nonzero $\lambda$, parity about the original origin is broken and the exact ground-state expectation is $\langle x\rangle=-\lambda$. The ground energy is $1/2-\lambda^2/2$.

This tests both the symmetric limit and a predictable departure from it. A program that silently projects onto even states may produce perfect parity diagnostics but fail to reproduce the linear displacement. Symmetry should reduce computation only when it belongs to the model being solved.

Angular momentum and antiunitary checks

For a spherical tensor $T_q^{(k)}$, angular selection rules constrain allowed $J,M$ values, but radial or reduced matrix elements may produce further zeros. An allowed transition need not have nonzero strength. Use Clebsch–Gordan computation to check angular factors and their phases separately from reduced matrix elements.

Time reversal is antiunitary. If represented by $\Theta=UK$ with complex conjugation $K$, invariance is $UH^*U^\dagger=H$, not generally $UHU^\dagger=H$. Testing the latter may falsely reject a valid spin Hamiltonian or accept the wrong representation. For a spin-one-half degree of freedom, the matrix part and conjugation together implement the sign reversal of all spin components.

Tolerances and interpretation

Compare forbidden matrix elements with a meaningful allowed amplitude or operator norm, and inspect their trend under refinement. A single absolute threshold can hide a symmetry violation in a weak transition. Exact arithmetic or structural construction can enforce some zeros, while quadrature and finite precision yield small residuals in others.

A symmetry-consistent implementation can still have the wrong coupling strengths. Combine these checks with analytic benchmarks and the target observable’s convergence study; no single conserved quantity certifies the entire calculation.

Exercises

For two same-parity normalized states, which matrix elements must vanish: $\langle m|x|n\rangle$ and $\langle m|x^2|n\rangle$?

Solution

The position matrix element must vanish because $p_mp_n=1$ and $s_x=-1$. The $x^2$ element is allowed, but may still vanish for additional reasons. Selection rules supply necessary conditions, not guaranteed transition strengths.

References

• Edmonds, A. R., Angular Momentum in Quantum Mechanics, Princeton University Press, 1957, reprinted 1996 — tensor-operator selection rules.
• Sakurai, J. J., and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2021 — parity, time reversal, and symmetry constraints.
• Trefethen, L. N., and D. Bau III, Numerical Linear Algebra, SIAM, 1997 — invariant subspaces and residual-based numerical checks.