Physlib.QuantumMechanics.DDimensions.Operators.Position

For a given index $i$, the $i$-th component of the position operator is a continuous linear map from the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ to itself. It maps a Schwartz function $\psi$ to the product $x_i \psi$, where $x_i$ is the $i$-th coordinate function on $\mathbb{R}^d$.

The notation $\mathbf{x}$ is defined to represent the position operator `positionOperator`, which acts as a continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$.

The notation $\mathbf{x}[d]$ denotes the position vector operator for a $d$-dimensional space. It acts on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ via pointwise multiplication by the coordinate vector $x = (x_1, \dots, x_d)$, mapping a function $f(x)$ to the vector of functions $(x_1 f(x), \dots, x_d f(x))$.

For any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the application of the $i$-th component of the position operator $\mathbf{x}_i$ to $\psi$ results in the function defined by the pointwise multiplication of the $i$-th coordinate function $x \mapsto x_i$ and $\psi$. That is, $\mathbf{x}_i \psi = (x \mapsto x_i) \cdot \psi$.

For any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ and any point $x \in \mathbb{R}^d$, the result of applying the $i$-th component of the position operator $\mathbf{x}_i$ to $\psi$ and evaluating it at $x$ is given by the product of the $i$-th coordinate of $x$ and the value of $\psi$ at $x$. That is, $(\mathbf{x}_i \psi)(x) = x_i \psi(x)$.

For a given dimension $d \in \mathbb{N}$ and real parameters $\epsilon, s \in \mathbb{R}$, the function maps a position vector $x \in \text{Space } d$ to the value $(\|x\|^2 + \epsilon^2)^{s/2}$. This function serves as a smooth regularization of the power of the norm $\|x\|^s$.

Let $d$ be a natural number and $\epsilon \in \mathbb{R}^\times$ be a non-zero real number. For any vector $x$ in the space $\text{Space } d$, the sum of the squared norm of $x$ and the square of $\epsilon$ is strictly positive, i.e., $$\|x\|^2 + \epsilon^2 > 0$$

For any dimension $d \in \mathbb{N}$, any non-zero real number $\epsilon \in \mathbb{R}^\times$, and any real number $s$, the regularized norm power function evaluated at any vector $x \in \text{Space } d$ is strictly positive, i.e., $$(\|x\|^2 + \epsilon^2)^{s/2} > 0$$

For any dimension $d \in \mathbb{N}$, non-zero real number $\epsilon \in \mathbb{R}^\times$, and real power $s \in \mathbb{R}$, the regularized norm power function on $\text{Space } d$, defined by $x \mapsto (\|x\|^2 + \epsilon^2)^{s/2}$, has temperate growth.

For a given dimension $d \in \mathbb{N}$, a non-zero real regularization parameter $\epsilon \in \mathbb{R}^\times$, and an exponent $s \in \mathbb{R}$, the regularized radius power operator is the continuous linear map from the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ to itself defined by the pointwise multiplication of a function $\psi$ by the smooth function $x \mapsto (\|x\|^2 + \epsilon^2)^{s/2}$. That is, $(\mathbf{r}_0 \psi)(x) = (\|x\|^2 + \epsilon^2)^{s/2} \psi(x)$.

The notation $\mathbf{r}_0$ denotes the regularized radius power operator `radiusRegPowOperator`. For a given dimension $d$, an invertible real regularization parameter $\epsilon$, and a power $s \in \mathbb{R}$, the operator $\mathbf{r}_0$ acts on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ by pointwise multiplication with the function $x \mapsto (\|x\|^2 + \epsilon^2)^{s/2}$.

The notation $\mathbf{r}_0[d]$ denotes the regularized radius power operator in dimension $d$. This operator acts on functions in the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ via multiplication by the smooth function $x \mapsto (\|x\|^2 + \epsilon^2)^{s/2}$, which serves as a regularization of the radius power $\|x\|^s$.

For any dimension $d \in \mathbb{N}$, a non-zero real regularization parameter $\epsilon \in \mathbb{R}^\times$, and a real exponent $s \in \mathbb{R}$, the regularized radius power operator $\mathbf{r}_0(\epsilon, s)$ applied to a Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ is equal to the function $x \mapsto (\|x\|^2 + \epsilon^2)^{s/2} \psi(x)$.

For any dimension $d \in \mathbb{N}$, a non-zero real regularization parameter $\epsilon \in \mathbb{R}^\times$, and a real exponent $s \in \mathbb{R}$, the value of the regularized radius power operator $\mathbf{r}_0(\epsilon, s)$ applied to a Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ and evaluated at a point $x \in \mathbb{R}^d$ is given by $(\mathbf{r}_0(\epsilon, s) \psi)(x) = (\|x\|^2 + \epsilon^2)^{s/2} \psi(x)$.

For a given dimension $d \in \mathbb{N}$, a non-zero real regularization parameter $\epsilon \in \mathbb{R}^\times$, and exponents $s, t \in \mathbb{R}$, the composition of the regularized radius power operators $\mathbf{r}_0(\epsilon, s)$ and $\mathbf{r}_0(\epsilon, t)$ acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ satisfies: $$\mathbf{r}_0(\epsilon, s) \circ \mathbf{r}_0(\epsilon, t) = \mathbf{r}_0(\epsilon, s + t)$$ where $\mathbf{r}_0(\epsilon, \cdot)$ is the operator defined by pointwise multiplication by the function $x \mapsto (\|x\|^2 + \epsilon^2)^{\cdot/2}$.

For any dimension $d \in \mathbb{N}$ and any non-zero real regularization parameter $\epsilon \in \mathbb{R}^\times$, the regularized radius power operator $\mathbf{r}_0(\epsilon, s)$ for the exponent $s = 0$ is equal to the identity continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. That is, $\mathbf{r}_0(\epsilon, 0) = \text{id}$.

For any dimension $d \in \mathbb{N}$ and any non-zero real regularization parameter $\epsilon \in \mathbb{R}^\times$, the sum of the squares of the $i$-th components of the position operator $\mathbf{x}_i$ on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ satisfies the identity: $$\sum_{i=1}^d \mathbf{x}_i^2 = \mathbf{r}_0(\epsilon, 2) - \epsilon^2 I$$ where $\mathbf{x}_i^2$ denotes the composition $\mathbf{x}_i \circ \mathbf{x}_i$, $\mathbf{r}_0(\epsilon, 2)$ is the regularized radius power operator for $s=2$ (which acts as multiplication by $\|x\|^2 + \epsilon^2$), and $I$ is the identity operator on the Schwartz space.

Given a real number $s \in \mathbb{R}$, the radius power operator $\mathbf{r}^s$ is a $\mathbb{C}$-linear map from the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ to the space of functions $\mathbb{R}^d \to \mathbb{C}$. For a Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the operator is defined by $(\mathbf{r}^s \psi)(x) = \|x\|^s \psi(x)$, where $\|x\|$ denotes the Euclidean norm of the position vector $x \in \mathbb{R}^d$.

The symbol $\mathbf{r}$ is the mathematical notation for the radius power operator. For a parameter $s \in \mathbb{R}$, the operator $\mathbf{r}^s$ (formally defined as `radiusPowOperator s`) acts on a Schwartz function $f \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ by pointwise multiplication by the $s$-th power of the Euclidean norm of the position vector, specifically $(\mathbf{r}^s f)(x) = \|x\|^s f(x)$.

For any dimension $d \in \mathbb{N}$, real power $s \in \mathbb{R}$, and Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the radius power operator $\mathbf{r}^s$ applied to $\psi$ is the function $x \mapsto \|x\|^s \psi(x)$, where $\|x\|$ is the Euclidean norm of $x \in \mathbb{R}^d$.

For any dimension $d \in \mathbb{N}$, real exponent $s \in \mathbb{R}$, Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, and position vector $x \in \mathbb{R}^d$, the radius power operator $\mathbf{r}^s$ applied to $\psi$ and evaluated at $x$ satisfies $(\mathbf{r}^s \psi)(x) = \|x\|^s \psi(x)$, where $\|x\|$ denotes the Euclidean norm of $x$.

For any dimension $d \in \mathbb{N}$, real exponent $s \in \mathbb{R}$, and Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the function $(\mathbf{r}^s \psi)(x) = \|x\|^s \psi(x)$ is of class $C^n$ at any point $x \in \mathbb{R}^d$ such that $x \neq 0$, for any $n \in \mathbb{N} \cup \{\infty\}$.

For any dimension $d \in \mathbb{N}$, real exponent $s \in \mathbb{R}$, and Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the function $(\mathbf{r}^s \psi)(x) = \|x\|^s \psi(x)$ is strongly measurable, where $\|x\|$ denotes the Euclidean norm on $\mathbb{R}^d$.

For any dimension $d \in \mathbb{N}$ and real exponent $s \in \mathbb{R}$ such that $d + 2s > 0$, the function defined by $(\mathbf{r}^s \psi)(x) = \|x\|^s \psi(x)$ is square-integrable for any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$. That is, $\mathbf{r}^s \psi$ belongs to the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$.

The filter on the set of non-zero real numbers $\mathbb{R}^\times$ (the units of $\mathbb{R}$) defined as the preimage of the neighborhood filter of $0$ in $\mathbb{R}$ under the inclusion map $\mathbb{R}^\times \hookrightarrow \mathbb{R}$. A set $S \subseteq \mathbb{R}^\times$ belongs to this filter if there exists an $\epsilon > 0$ such that the punctured neighborhood $(-\epsilon, 0) \cup (0, \epsilon)$ is contained in $S$.

Let $\mathbb{R}^\times$ denote the set of non-zero real numbers. The filter of punctured neighborhoods of $0$ in $\mathbb{R}$ (defined as the neighborhood filter of $0$ in the subspace $\mathbb{R}^\times$) is non-trivial, meaning it is not the bottom filter and thus every set in the filter is non-empty.

For any dimension $d$, real exponent $s$, and Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, let $x \in \mathbb{R}^d$ be a position vector such that $x \neq 0$. Then the value of the regularized radius power operator $(\mathbf{r}_0(\epsilon, s) \psi)(x)$ converges to the value of the radius power operator $(\mathbf{r}^s \psi)(x)$ as the regularization parameter $\epsilon$ approaches $0$ through non-zero real values.

For any dimension $d \in \mathbb{N}$, real exponent $s \in \mathbb{R}$, and Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, if either $s \ge 0$ or the function vanishes at the origin ($\psi(0) = 0$), then the regularized radius power function $x \mapsto (\|x\|^2 + \epsilon^2)^{s/2} \psi(x)$ converges pointwise to the radius power function $x \mapsto \|x\|^s \psi(x)$ as the regularization parameter $\epsilon$ approaches $0$ through non-zero real values.

For a positive dimension $d \in \mathbb{N}$, a real exponent $s \in \mathbb{R}$, and a Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the value of the regularized radius power operator $(\mathbf{r}_0(\epsilon, s) \psi)(x)$ converges to the value of the radius power operator $(\mathbf{r}^s \psi)(x)$ as the regularization parameter $\epsilon$ approaches $0$ (through non-zero real values) for almost every $x \in \mathbb{R}^d$. Here, $(\mathbf{r}_0 \psi)(x) = (\|x\|^2 + \epsilon^2)^{s/2} \psi(x)$ and $(\mathbf{r}^s \psi)(x) = \|x\|^s \psi(x)$.

For a positive dimension $d > 0$, real exponent $s \in \mathbb{R}$, Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, and function $\phi: \mathbb{R}^d \to \mathbb{C}$, the regularized radius power function $(\mathbf{r}_0(\epsilon, s) \psi)(x) = (\|x\|^2 + \epsilon^2)^{s/2} \psi(x)$ converges pointwise to $\phi(x)$ for almost every $x \in \mathbb{R}^d$ as the regularization parameter $\epsilon$ approaches $0$ through non-zero real values if and only if $\phi$ is equal to the radius power function $(\mathbf{r}^s \psi)(x) = \|x\|^s \psi(x)$ almost everywhere with respect to the Lebesgue measure.

For a given index $i$, the $i$-th component of the position operator on the Schwartz submodule is a linear map from the Schwartz submodule of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ to itself. It maps a Schwartz function $\psi$ to the product $x_i \psi$, where $x_i$ is the $i$-th coordinate function on $\mathbb{R}^d$. This definition specifically realizes the position operator $\mathbf{x}_i$ as an operator acting within the context of the Schwartz space as a dense subspace of the system's Hilbert space.

For a given index $i$, the $i$-th component of the position operator $\mathbf{x}_i$, acting as a linear operator on the Schwartz submodule of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$, is symmetric. That is, for any two functions $\psi$ and $\phi$ in the Schwartz submodule, the inner product satisfies $\langle \psi, \mathbf{x}_i \phi \rangle = \langle \mathbf{x}_i \psi, \phi \rangle$.

For a given dimension $d$ and an index $i$, the $i$-th component of the position operator $\mathbf{x}_i$ is an unbounded operator on the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ (represented by `SpaceDHilbertSpace d`). It is defined with the Schwartz space $\mathcal{S}(\mathbb{R}^d)$ as its dense domain. The operator acts by mapping a Schwartz function $\psi(x)$ to the product $x_i \psi(x)$, where $x_i$ is the $i$-th coordinate function. This definition utilizes the symmetry of the position operator on the Schwartz submodule to establish it as a symmetric unbounded operator on the entire Hilbert space.

For a given dimension $d \in \mathbb{N}$, a non-zero real regularization parameter $\epsilon \in \mathbb{R}^\times$, and an exponent $s \in \mathbb{R}$, the regularized radius power operator on the Schwartz submodule is a $\mathbb{C}$-linear map from the Schwartz submodule of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ to itself. This operator acts on a function $\psi$ by pointwise multiplication with the smooth regularization of the radius power function, $x \mapsto (\|x\|^2 + \epsilon^2)^{s/2}$.

For any dimension $d \in \mathbb{N}$, any non-zero real regularization parameter $\epsilon \in \mathbb{R}^\times$, and any real exponent $s \in \mathbb{R}$, the regularized radius power operator on the Schwartz submodule of $L^2(\mathbb{R}^d, \mathbb{C})$—defined by the pointwise multiplication of a function $\psi$ by the function $x \mapsto (\|x\|^2 + \epsilon^2)^{s/2}$—is a symmetric operator.

For a given dimension $d \in \mathbb{N}$, a non-zero real regularization parameter $\epsilon \in \mathbb{R}^\times$, and a real exponent $s \in \mathbb{R}$, the regularized radius power unbounded operator is an unbounded operator on the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$. It is defined with the dense submodule of Schwartz functions $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ as its domain, acting on a function $\psi$ by pointwise multiplication: $\psi(x) \mapsto (\|x\|^2 + \epsilon^2)^{s/2} \psi(x)$. The operator is constructed as a symmetric operator using the fact that this multiplication map is symmetric on the Schwartz space.