Physlib.QuantumMechanics.OneDimension.HilbertSpace.PositionStates

For a given position $x \in \mathbb{R}$, the position state is defined as the continuous linear functional on the Schwartz space $\mathcal{S}(\mathbb{R}, \mathbb{C})$ that maps a function $\psi$ to its value at $x$, i.e., $\psi(x)$. This functional is the Dirac delta distribution $\delta_x$, viewed as an element of the space of tempered distributions $\mathcal{S}'(\mathbb{R}, \mathbb{C})$.

For any position $x \in \mathbb{R}$ and any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$, the application of the position state functional at $x$ to $\psi$ is equal to the value of the function at $x$, i.e., $\text{positionState}(x)(\psi) = \psi(x)$. This identifies the position state as the Dirac delta distribution $\delta_x$.