Physlib.QuantumMechanics.OneDimension.Operators.Parity

The parity operator is a $\mathbb{C}$-linear map from the space of complex-valued functions on the real line, $\mathbb{R} \to \mathbb{C}$, to itself. It maps a function $\psi$ to the function defined by $x \mapsto \psi(-x)$.

The parity operator is defined as the continuous linear map from the space of Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$ to itself that maps a function $\psi$ to the function defined by $x \mapsto \psi(-x)$.

The unbounded parity operator $P$ is an unbounded operator defined on the domain of Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$. It is constructed as the composition $P = \iota \circ \mathcal{P}$, where $\mathcal{P}: \mathcal{S}(\mathbb{R}, \mathbb{C}) \to \mathcal{S}(\mathbb{R}, \mathbb{C})$ is the continuous linear map that sends a function $\psi$ to $\psi(-x)$, and $\iota: \mathcal{S}(\mathbb{R}, \mathbb{C}) \to L^2(\mathbb{R}, \mathbb{C})$ is the injective continuous linear inclusion of Schwartz functions into the Hilbert space of square-integrable functions.

For any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$, applying the parity operator twice results in the original function $\psi$. That is, if $P$ denotes the parity operator defined by $(P\psi)(x) = \psi(-x)$, then $P(P\psi) = \psi$.

The unbounded parity operator $P$ defined on the space of Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$ is self-adjoint. That is, for any two Schwartz functions $\psi_1, \psi_2 \in \mathcal{S}(\mathbb{R}, \mathbb{C})$, the inner product in the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ satisfies: $$\langle P \psi_1, \iota \psi_2 \rangle_{\mathbb{C}} = \langle \iota \psi_1, P \psi_2 \rangle_{\mathbb{C}}$$ where $(P\psi)(x) = \psi(-x)$ and $\iota : \mathcal{S}(\mathbb{R}, \mathbb{C}) \to L^2(\mathbb{R}, \mathbb{C})$ is the continuous linear inclusion.