Physlib.QuantumMechanics.HilbertSpaces.SpaceD.Fourier

For a natural number $d$, this defines the $L^2$ Fourier transform $\mathcal{F}$ as a unitary operator (specifically, a complex-linear isometric isomorphism) from the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ to itself. This operator maps functions in the space of square-integrable complex-valued functions over $d$-dimensional space to their corresponding frequency-domain representations while preserving the $L^2$ norm.

Let $L^2(\text{Space } d, \mathbb{C})$ be the Hilbert space of square-integrable complex-valued functions over a $d$-dimensional real inner product space. For any state $\psi \in L^2(\text{Space } d, \mathbb{C})$, the action of the Fourier unitary operator $\mathcal{F}$ (defined as `fourierUnitary d`) on $\psi$ is equal to the $L^2$ Fourier transform of $\psi$, denoted by $\mathcal{F} \psi$.

For any function $\psi$ in the Hilbert space $L^2(\text{Space } d, \mathbb{C})$, the application of the inverse of the Fourier unitary operator $\mathcal{F}$ to $\psi$ is equal to the inverse $L^2$ Fourier transform $\mathcal{F}^{-1} \psi$.

Let $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ be the Schwartz space of rapidly decreasing smooth functions and $L^2(\mathbb{R}^d, \mathbb{C})$ be the Hilbert space of square-integrable functions. Let $\iota: \mathcal{S}(\mathbb{R}^d, \mathbb{C}) \to L^2(\mathbb{R}^d, \mathbb{C})$ denote the canonical inclusion map that sends a Schwartz function to its equivalence class in $L^2$. For any Schwartz function $f \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, applying the $L^2$ Fourier transform $\mathcal{F}$ to the image of $f$ under $\iota$ is equivalent to taking the Schwartz Fourier transform of $f$ and then applying the inclusion $\iota$. That is, $$\mathcal{F}(\iota(f)) = \iota(\mathcal{F} f)$$

Let $\mathcal{S}(\text{Space } d, \mathbb{C})$ be the Schwartz space of rapidly decreasing functions and $L^2(\text{Space } d, \mathbb{C})$ be the Hilbert space of square-integrable functions. For any Schwartz function $f \in \mathcal{S}(\text{Space } d, \mathbb{C})$, applying the inverse $L^2$ Fourier transform $\mathcal{F}^{-1}$ to the equivalence class of $f$ in $L^2$ is equal to the equivalence class of the inverse Schwartz Fourier transform $\mathcal{F}^{-1} f$. That is, $$\mathcal{F}^{-1}(\iota(f)) = \iota(\mathcal{F}^{-1} f)$$ where $\iota: \mathcal{S}(\text{Space } d, \mathbb{C}) \to L^2(\text{Space } d, \mathbb{C})$ is the continuous linear inclusion map.

For any function $f$ in the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$, let $\iota: \mathcal{S}(\text{Space } d, \mathbb{C}) \to L^2(\text{Space } d, \mathbb{C})$ be the continuous linear inclusion into the Hilbert space of square-integrable functions. Then the inverse Fourier transform $\mathcal{F}^{-1}$ applied to the inclusion of the Fourier transform of $f$ equals the inclusion of $f$ itself: $$\mathcal{F}^{-1}(\iota(\mathcal{F} f)) = \iota(f)$$ where $\mathcal{F}$ denotes the Fourier transform.

For any dimension $d \in \mathbb{N}$, let $L^2(\text{Space } d, \mathbb{C})$ be the Hilbert space of square-integrable complex-valued functions. Let $\mathcal{S} \subseteq L^2(\text{Space } d, \mathbb{C})$ denote the Schwartz submodule, which is the image of the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ under the natural inclusion map. If $\mathcal{F}$ is the Fourier unitary operator on $L^2(\text{Space } d, \mathbb{C})$, then the image of the Schwartz submodule under $\mathcal{F}$ is equal to the Schwartz submodule itself, i.e., $\mathcal{F}(\mathcal{S}) = \mathcal{S}$.