Physlib.QuantumMechanics.HilbertSpaces.SpaceD.MomentumStates

For a given momentum vector $p \in \text{Space } d$ (where $\text{Space } d$ is a $d$-dimensional real inner product space), the momentum state is defined as a tempered distribution in the strong dual of the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$. Formally, it is the composition of the continuous linear Fourier transform $\mathcal{F}$ and the Dirac delta distribution $\delta$ centered at $(2\pi)^{-1}p$. For any test function $\psi \in \mathcal{S}(\text{Space } d, \mathbb{C})$, the momentum state maps $\psi$ to the value of its Fourier transform at the point $(2\pi)^{-1}p$, which corresponds to the integral against the non-normalizable plane wave $x \mapsto e^{i p \cdot x}$.

For any momentum vector $p \in \text{Space } d$ and any test function $\psi$ in the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$, the evaluation of the momentum state distribution $|p\rangle$ on $\psi$ is equal to the value of the Fourier transform of $\psi$ (denoted $\mathcal{F}\psi$) at the point $\frac{1}{2\pi}p$. Mathematically, this is expressed as: $$\langle p, \psi \rangle = \mathcal{F}\psi\left(\frac{1}{2\pi} p\right)$$