Definite parity of nondegenerate eigenstates and the position-matrix-element selection rule

Adopt the postulates P1, P2, P4, P5 and P6 in the readings of The massive spin-s particle model of the framework: state space, operators, units and frame readings, Assigned sharp observables and the outcome law of the framework, Assigned energy observable, unitary flow and the two pictures of autonomous evolution and Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables for a declared particle of mass parameter $m>0$ and spin label $s$, with state space $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$, $r=2s+1$, componentwise operators $Q_j,P_k$ and the parity action $\mathrm{\Pi }$ (The spatial parity action on the component-valued wave-function space); assume the Axiom of Countable Choice, the Axiom of Choice and the Axiom of Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Let $V:{\mathbb{R}}^3\to \mathbb{R}$ be real, bounded and measurable with $V(-x)=V(x)\text{for almost every }x\in {\mathbb{R}}^3,$ let $H_V=\frac{1}{2m}{\sum }_{k=1}^3{P}_k^2+M_V,D(H_V)=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^r),$ and put $V_P(x):=V(-x)$. Then:

(a) if $\psi \in D(H_V)$ is an eigenvector of $H_V$ with eigenvalue $E\in \mathbb{R}$, $H_V\psi =E\psi$ and $\psi \mathrm{\ne }0$, then $\mathrm{\Pi }\psi$ is again an eigenvector of $H_V$ with the same eigenvalue $E$; consequently every eigenspace of $H_V$ is invariant under $\mathrm{\Pi }$;

(b) if the eigenspace of $E$ is one-dimensional, then $\mathrm{\Pi }\psi =\lambda \psi$ with $\lambda \in \{-1,1\}$: the state has definite parity;

(c) if $\psi ,{\psi }^{\mathrm{\prime }}\in H_s$ are unit vectors with $\mathrm{\Pi }\psi =\sigma \psi$ and $\mathrm{\Pi }{\psi }^{\mathrm{\prime }}=\tau {\psi }^{\mathrm{\prime }}$ for signs $\sigma ,\tau \in \{-1,1\}$ and $\psi \in D(Q_j)$, then $\sigma =\tau ⟹⟨Q_j\psi ,{\psi }^{\mathrm{\prime }}⟩=0;$ the position matrix element of two states of equal parity vanishes, while no nonvanishing statement is made for opposite parities;

(d) Scope. The example is conditional on the postulates and the cited mathematics; the matrix element of (c) is an identity about the model's assigned position observables. No transition rate, no spectral intensity, no time-dependent coupling, no measurement model and no empirical consequence is computed; the hypothesis that the relevant eigenvalue is nondegenerate is essential for (b).

• Why the same-parity case vanishes. The parity action conjugates the position operator to its negative, so a matrix element between two states of equal parity equals its own negative; the argument uses the involution structure and self-adjointness of $\mathrm{\Pi }$, not the details of the potential beyond its evenness.
• What a selection rule is not. The vanishing statement is an identity for the model's assigned position observables in the declared even-potential model. Turning it into a statement about observed transition intensities requires a time-dependent coupling model, the corresponding transition-rate analysis, a detection model and a sampling model, none of which is supplied here.