= Midpoint parity operator
{title2=$R_a\psi(x)=\psi(a-x)$}
= Reflection about the midpoint of an interval
{synonym}
On $L^2(0,a)$, the <midpoint parity operator> is $(R_a\psi)(x)=\psi(a-x)$. Substitution $y=a-x$ gives $\langle f,R_ag\rangle=\langle R_af,g\rangle$ and $\|R_af\|=\|f\|$, so it is a bounded <self-adjoint operator> and a <unitary operator>, with $R_a^2=I$. It is the <parity operator> in the centered coordinate $x-a/2$.
For an <infinite square well>, $\psi_n=\sqrt{2/a}\sin(n\pi x/a)$ obeys $R_a\psi_n=(-1)^{n+1}\psi_n$. Reflection preserves the <Dirichlet boundary conditions> and commutes with the second derivative, so $R_a$ commutes with the <Hamiltonian operator> on its <operator domain>. By contrast, differentiating once gives $PR_a=-R_aP$ for the <momentum operator>. This distinction permits conserved midpoint <parity> with changing mean momentum.
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