Physlib.QuantumMechanics.OneDimension.Operators.Momentum

The one-dimensional momentum operator is a transformation that maps a function $\psi: \mathbb{R} \to \mathbb{C}$ to another function $\mathbb{R} \to \mathbb{C}$ defined by the expression $$x \mapsto -i \hbar \frac{d\psi}{dx}(x)$$ where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, and $\frac{d\psi}{dx}$ denotes the derivative of $\psi$.

For any function $\psi: \mathbb{R} \to \mathbb{C}$, the one-dimensional momentum operator applied to $\psi$ is the function mapping $x$ to the product of the constant $-i\hbar$ and the derivative $\psi'(x)$, where $i$ is the imaginary unit and $\hbar$ is the reduced Planck constant. That is, $$(\hat{p}\psi)(x) = -i\hbar \frac{d\psi}{dx}(x).$$

For any function $\psi: \mathbb{R} \to \mathbb{C}$ that is continuously differentiable (of class $C^1$), the function resulting from the application of the one-dimensional momentum operator $\hat{p}$, defined by $(\hat{p}\psi)(x) = -i \hbar \frac{d\psi}{dx}(x)$, is a continuous function. Here, $i$ is the imaginary unit and $\hbar$ is the reduced Planck constant.

For any differentiable function $\psi: \mathbb{R} \to \mathbb{C}$ and any complex scalar $c \in \mathbb{C}$, the one-dimensional momentum operator $\hat{p}$ satisfies the homogeneity property $\hat{p}(c \psi) = c \hat{p}\psi$, where $\hat{p}$ is defined as $-i\hbar \frac{d}{dx}$.

For any differentiable functions $\psi_1, \psi_2: \mathbb{R} \to \mathbb{C}$, the one-dimensional momentum operator $\hat{p}$ satisfies the additivity property: $$\hat{p}(\psi_1 + \psi_2) = \hat{p}\psi_1 + \hat{p}\psi_2$$ where $\hat{p}$ is defined by $(\hat{p}\psi)(x) = -i\hbar \frac{d\psi}{dx}(x)$, with $i$ being the imaginary unit and $\hbar$ being the reduced Planck constant.

The momentum operator on the space of Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$ is defined as the continuous linear map $\hat{p}: \mathcal{S}(\mathbb{R}, \mathbb{C}) \to \mathcal{S}(\mathbb{R}, \mathbb{C})$ that maps a function $\psi$ to $-i \hbar \frac{d\psi}{dx}$, where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, and $\frac{d\psi}{dx}$ denotes the derivative of $\psi$.

For any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$ and any real number $x$, the momentum operator on Schwartz maps $\hat{p}$ satisfies: $$(\hat{p}\psi)(x) = -i\hbar \frac{d\psi}{dx}(x)$$ where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, and $\frac{d\psi}{dx}(x)$ is the derivative of $\psi$ evaluated at $x$.

The momentum operator $\hat{p}$ is defined as an unbounded operator on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ with its domain consisting of the space of Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$. It is constructed from the continuous linear map on the Schwartz space that sends a function $\psi$ to $-i \hbar \frac{d\psi}{dx}$, where $i$ is the imaginary unit and $\hbar$ is the reduced Planck constant.

For any real number $k \in \mathbb{R}$, the plane wave functional associated with $k$ is a generalized eigenvector of the unbounded momentum operator $\hat{p}$ with corresponding eigenvalue $2\pi\hbar k$. Here, $\hat{p}$ is defined as the unbounded operator on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ whose domain is the space of Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$ and which acts as $-i\hbar \frac{d}{dx}$, where $i$ is the imaginary unit and $\hbar$ is the reduced Planck constant.

The momentum operator $\hat{p}$, defined as an unbounded operator on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ with the domain of Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$, is self-adjoint. The operator is given by $(\hat{p}\psi)(x) = -i\hbar \frac{d\psi}{dx}(x)$, where $i$ is the imaginary unit and $\hbar$ is the reduced Planck constant.