Physlib.QuantumMechanics.OneDimension.HilbertSpace.PlaneWaves

For a given wave vector $k \in \mathbb{R}$, the plane wave functional is the continuous linear functional on the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$ defined by the composition of the Dirac delta distribution at $k$ (denoted $\delta_k$) and the Fourier transform operator $\mathcal{F}$. For any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$, this functional maps $\psi$ to its Fourier transform evaluated at $k$, i.e., $\hat{\psi}(k)$. This functional represents the plane wave $x \mapsto e^{2\pi i k x}$ acting as a tempered distribution.

For any wave vector $k \in \mathbb{R}$ and any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$, the value of the plane wave functional associated with $k$ applied to $\psi$ is equal to the Fourier transform of $\psi$ evaluated at $k$, denoted $\mathcal{F}(\psi)(k)$.