Physlib.QuantumMechanics.Operators.Position

For a given dimension $d \in \mathbb{N}$ and an index $\mu \in \{0, 1, \dots, d-1\}$, the operator $\text{positionCLM}(\mu)$ is a continuous linear map from the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ to itself. It acts on a Schwartz function $\psi$ by multiplying it by the $\mu$-th coordinate function $x_\mu$: $$(\text{positionCLM}(\mu) \psi)(x) = x_\mu \psi(x)$$ where $x \in \text{Space } d$ and $x_\mu$ is the $\mu$-th component of the vector $x$.

For any dimension $d \in \mathbb{N}$, coordinate index $i$, and Schwartz function $\psi : \text{Space } d \to \mathbb{C}$, the action of the $i$-th component of the position operator $\mathbf{x}_i$ on $\psi$ is given by the pointwise multiplication of $\psi$ by the $i$-th coordinate function. That is, for all $x \in \text{Space } d$, $(\mathbf{x}_i \psi)(x) = x_i \psi(x)$, where $x_i$ denotes the $i$-th coordinate of $x$.

For any dimension $d$ and index $i \in \{0, \dots, d-1\}$, let $\psi \in \mathcal{S}(\text{Space } d, \mathbb{C})$ be a Schwartz function and $x \in \text{Space } d$ be a point in the $d$-dimensional Euclidean space. The $i$-th component of the position operator $\mathbf{x}_i$ acts on $\psi$ such that its evaluation at $x$ is the product of the $i$-th coordinate of $x$ and the value of the function at that point: $$(\mathbf{x}_i \psi)(x) = x_i \psi(x)$$ where $x_i$ denotes the $i$-th component of the vector $x$.

For a given dimension $d$, a non-zero real regularization parameter $\epsilon \in \mathbb{R}^\times$, and an exponent $s \in \mathbb{R}$, this definition defines a continuous complex-linear map from the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ to itself. The operator acts on a Schwartz function $\psi$ by pointwise multiplication with the regularized norm power function: $$\psi(x) \mapsto (\|x\|^2 + \epsilon^2)^{s/2} \psi(x)$$ where $\|x\|$ denotes the Euclidean norm of $x \in \text{Space } d$.

For any dimension $d$, non-zero real number $\epsilon$, and real power $s$, the regularized radius power operator $\mathbf{r}_0(\epsilon, s)$ applied to a Schwartz function $\psi: \text{Space } d \to \mathbb{C}$ at a point $x \in \text{Space } d$ is given by: $$((\mathbf{r}_0(\epsilon, s) \psi)(x) = (\|x\|^2 + \epsilon^2)^{s/2} \psi(x)$$ where $\|x\|$ is the Euclidean norm on $\text{Space } d$.

For any dimension $d \in \mathbb{N}$ and any non-zero real number $\epsilon$, let $\mathbf{x}_i$ denote the $i$-th component of the position operator acting on the space of Schwartz maps $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ by multiplication by $x_i$. Let $\mathbf{r}_0(\epsilon, 2)$ be the regularized radius power operator with exponent $2$, which acts on Schwartz maps by multiplication by $\|x\|^2 + \epsilon^2$. The sum of the squares (compositions) of the position operators satisfies the identity: $$\sum_{i=1}^d \mathbf{x}_i \circ \mathbf{x}_i = \mathbf{r}_0(\epsilon, 2) - \epsilon^2 I$$ where $I$ is the identity operator on $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$.

For a given dimension $d$ and a real exponent $s \in \mathbb{R}$, this is a complex-linear map from the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ to the space of complex-valued functions on $\text{Space } d$. It maps a Schwartz function $f$ to the function $x \mapsto \|x\|^s f(x)$, where $\|x\|$ denotes the Euclidean norm of the position vector $x \in \text{Space } d$.

This definition introduces the notation $\mathbf{r}$ to represent the radius power operator. This operator acts on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ by multiplication by $\|x\|^s$ for a given power $s$, where $\|x\|$ denotes the Euclidean norm of the position vector.

For any dimension $d \in \mathbb{N}$, real exponent $s \in \mathbb{R}$, Schwartz function $\psi \in \mathcal{S}(\text{Space } d, \mathbb{C})$, and point $x \in \text{Space } d$, the radius power operator $\mathbf{r}^s$ acts on $\psi$ such that its value at $x$ is given by the multiplication: $$(\mathbf{r}^s \psi)(x) = \|x\|^s \psi(x)$$ where $\|x\|$ denotes the Euclidean norm of the vector $x$ in the $d$-dimensional real inner product space $\text{Space } d$.

For any dimension $d$, any real number $s$, and any $n \in \mathbb{N} \cup \{\infty\}$, if $\psi$ is a Schwartz function in $\mathcal{S}(\text{Space } d, \mathbb{C})$, then the function resulting from the application of the radius power operator, defined by $(r^s \psi)(x) = \|x\|^s \psi(x)$, is $n$-times continuously differentiable ($C^n$) at any point $x \in \text{Space } d$ such that $x \neq 0$. Here, $\text{Space } d$ is the $d$-dimensional Euclidean space $\mathbb{R}^d$ equipped with the standard inner product and norm.

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. For any real number $s \in \mathbb{R}$ and any Schwartz map $\psi \in \mathcal{S}(\text{Space } d, \mathbb{C})$, the function $(\mathbf{r} s) \psi$—resulting from the action of the radius power operator—is strongly measurable with respect to the Borel $\sigma$-algebra.

For any dimension $d \in \mathbb{N}$, let $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ be a Schwartz map. Suppose $\psi$ belongs to the submodule of maps $\text{PolyBddSchwartzMap}(d, a)$, meaning that for all $k \leq a$, there exists a constant $C$ such that $\|\psi(x)\| \leq C \|x\|^k$. For any real power $s \in \mathbb{R}$, if the condition $d + 2(a + s) > 0$ is satisfied, then the function $(\mathbf{r}^s \psi)(x) = \|x\|^s \psi(x)$ is square-integrable, i.e., $(\mathbf{r}^s \psi) \in L^2(\mathbb{R}^d)$.

The notation $\mathbf{x}$ denotes the position vector operator in quantum mechanics. It represents the operator that acts on functions in the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ by multiplication by the position vector $x$. Specifically, for a wavefunction $\psi$, the action is given by $(\mathbf{x} \psi)(x) = x \psi(x)$.

For any dimension $d \in \mathbb{N}$ and any coordinate index $i \in \{0, \dots, d-1\}$, the $i$-th component of the position operator $\mathbf{x}_i$, acting on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ as a subspace of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$, is a symmetric operator. This means that for all functions $\psi, \phi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the inner product satisfies $\langle \mathbf{x}_i \psi, \phi \rangle = \langle \psi, \mathbf{x}_i \phi \rangle$.

For any dimension $d \in \mathbb{N}$ and any coordinate index $i \in \{0, \dots, d-1\}$, the $i$-th component of the position operator $\mathbf{x}_i$ is self-adjoint. The operator $\mathbf{x}_i$ acts on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ by multiplying a Schwartz function $\psi$ by the $i$-th coordinate function $x_i$, i.e., $(\mathbf{x}_i \psi)(x) = x_i \psi(x)$.

For any natural number $d$ and any coordinate index $i \in \{0, \dots, d-1\}$, the $i$-th component of the position operator $\mathbf{x}_i$, which acts on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ by $(\mathbf{x}_i \psi)(x) = x_i \psi(x)$, is an unbounded operator.