Physlib.QuantumMechanics.Operators.Momentum

For a given dimension $d$ and an index $j \in \{0, 1, \dots, d-1\}$, this defines the $j$-th component of the momentum operator $\hat{p}_j$ as a continuous linear map on the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$. The operator acts on a Schwartz function $\psi$ by taking its partial derivative with respect to the $j$-th coordinate of the standard basis, multiplied by $-i\hbar$, where $\hbar$ is the reduced Planck constant: $$\hat{p}_j \psi = -i\hbar \frac{\partial \psi}{\partial x_j}$$

For any dimension $d \in \mathbb{N}$ and coordinate index $i \in \{0, \dots, d-1\}$, let $\psi: \text{Space } d \to \mathbb{C}$ be a Schwartz function. The $i$-th component of the momentum operator $\mathbf{p}_i$ applied to $\psi$ is given by: $$(\mathbf{p}_i \psi) = -i\hbar \partial_i \psi$$ where $i$ is the imaginary unit, $\hbar$ is the reduced Planck's constant, and $\partial_i$ denotes the partial derivative with respect to the $i$-th coordinate.

For any dimension $d$, coordinate index $i \in \{0, \dots, d-1\}$, Schwartz function $\psi: \text{Space } d \to \mathbb{C}$, and point $x \in \text{Space } d$, the $i$-th component of the momentum operator $\mathbf{p}_i$ applied to $\psi$ and evaluated at $x$ is given by: $$(\mathbf{p}_i \psi)(x) = -i \hbar \frac{\partial \psi}{\partial x_i}(x)$$ where $i$ is the imaginary unit, $\hbar$ is the reduced Planck's constant, and $\frac{\partial \psi}{\partial x_i}$ denotes the partial derivative of $\psi$ in the $i$-th spatial direction.

This definition introduces the mathematical notation $\mathbf{p}$ used to represent the momentum operator in quantum mechanics. The operator $\mathbf{p}$ acts on the space of Schwartz functions $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ such that its $i$-th component is given by $-i\hbar\frac{\partial}{\partial x_i}$.

Let $\text{Space } d$ be a $d$-dimensional real inner product space. Let $\mathbf{p}_i: \mathcal{S}(\text{Space } d, \mathbb{C}) \to \mathcal{S}(\text{Space } d, \mathbb{C})$ be the $i$-th component of the momentum operator (defined as $-i\hbar\partial_i$ on Schwartz functions). Let $\mathbf{P}_i$ be the corresponding $i$-th component of the unbounded momentum operator acting on the Schwartz submodule of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$. For any $\psi$ in this Schwartz submodule, the application of $\mathbf{P}_i$ to $\psi$ is equal almost everywhere to the application of $\mathbf{p}_i$ to the corresponding Schwartz function $\psi_S = \text{schwartzEquiv}^{-1}(\psi)$. That is, $$\mathbf{P}_i \psi = \mathbf{p}_i \psi_S$$ holds almost everywhere with respect to the volume measure on $\text{Space } d$.

For any dimension $d$ and any index $i \in \{0, \dots, d-1\}$, if $\psi$ is a function in the Schwartz submodule $\mathcal{S}(\text{Space } d, \mathbb{C})$ of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$, then the result of applying the $i$-th component of the momentum operator $p_i = -i\hbar\partial_i$ to $\psi$ is also an element of the Schwartz submodule.

For any dimension $d \in \mathbb{N}$ and coordinate index $i$, the $i$-th component of the momentum operator $p_i = -i \hbar \partial_i$, which maps between Schwartz functions $\mathcal{S}(\text{Space } d, \mathbb{C})$, has a dense domain in the Hilbert space $L^2(\text{Space } d, \mathbb{C})$. Here, $\hbar$ is the reduced Planck constant and $\partial_i$ is the partial derivative with respect to the $i$-th basis vector.

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