Physlib.QuantumMechanics.DDimensions.Operators.Momentum

The $i$-th component of the momentum operator is a continuous linear map from the space of Schwartz functions $\mathcal{S}(\text{Space } d, \mathbb{C})$ to itself. It maps a function $\psi$ to $-i \hbar \partial_i \psi$, where $i$ (in the coefficient) is the imaginary unit, $\hbar$ is the reduced Planck constant, and $\partial_i$ denotes the partial derivative with respect to the $i$-th basis vector of the coordinate space.

The notation $\mathbf{p}$ represents the momentum operator acting on the space of Schwartz functions $\mathcal{S}(\text{Space } d, \mathbb{C})$. Mathematically, this operator corresponds to $-i\hbar \partial_i$ (where $\hbar$ is the reduced Planck constant and $\partial_i$ is the partial derivative with respect to the $i$-th coordinate).

The notation $\mathbf{p}[i]$ denotes the $i$-th component of the momentum operator vector. According to the definition of `momentumOperator`, this corresponds to the operator $\hat{p}_i = -i\hbar \frac{\partial}{\partial x_i}$ acting on the space of Schwartz functions $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, where $\hbar$ is the reduced Planck constant.

For any Schwartz function $\psi \in \mathcal{S}(\text{Space } d, \mathbb{C})$, the action of the $i$-th component of the momentum operator $\mathbf{p}_i$ on $\psi$ is given by the identity $\mathbf{p}_i \psi = -i \hbar \partial_i \psi$, where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, and $\partial_i$ denotes the partial derivative with respect to the $i$-th coordinate.

For any Schwartz function $\psi \in \mathcal{S}(\text{Space } d, \mathbb{C})$ and any 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 $(\mathbf{p}_i \psi)(x) = -i \hbar \partial_i \psi(x)$, where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, and $\partial_i$ is the partial derivative with respect to the $i$-th coordinate.

The momentum-squared operator $\mathbf{p}^2$ is a continuous linear map from the space of Schwartz functions $\mathcal{S}(\text{Space } d, \mathbb{C})$ to itself. It is defined as the sum over the coordinate indices $i$ of the composition of the $i$-th momentum operator $\mathbf{p}_i$ with itself, expressed as $\mathbf{p}^2 = \sum_i \mathbf{p}_i \circ \mathbf{p}_i$.

The notation $\mathbf{p}^2$ denotes the momentum-squared operator `momentumOperatorSqr`, which is a continuous linear operator acting on the space of Schwartz functions $\mathcal{S}(\text{Space } d, \mathbb{C})$. It represents the sum of the squares of the momentum operator components, $\sum_i \mathbf{p}_i^2$.

The momentum-squared operator $\mathbf{p}^2$ acting on the space of Schwartz functions $\mathcal{S}(\text{Space } d, \mathbb{C})$ is equal to the sum of the compositions of the $i$-th momentum operator components $\mathbf{p}_i$ with themselves: $$\mathbf{p}^2 = \sum_{i=1}^d \mathbf{p}_i \circ \mathbf{p}_i$$ where $\mathbf{p}_i$ is the continuous linear operator defined by $\mathbf{p}_i = -i \hbar \partial_i$.

For any Schwartz function $\psi \in \mathcal{S}(\text{Space } d, \mathbb{C})$ and any point $x \in \text{Space } d$, the action of the momentum-squared operator $\mathbf{p}^2$ at $x$ is given by the sum over the components of the momentum operator applied twice to $\psi$: $$\mathbf{p}^2 \psi(x) = \sum_{i} \mathbf{p}_i (\mathbf{p}_i \psi)(x)$$ where $\mathbf{p}_i = -i\hbar \partial_i$ is the $i$-th component of the momentum operator.

The $i$-th component of the momentum operator, acting as a linear map on the Schwartz submodule of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$. For a function $\psi$ in this submodule, the operator maps it to $-i\hbar\partial_i\psi$, where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, and $\partial_i$ denotes the partial derivative with respect to the $i$-th coordinate.

The $i$-th component of the momentum operator $p_i = -i\hbar \partial_i$, acting as a linear operator on the Schwartz submodule of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$, is symmetric. That is, for any two functions $\psi, \phi$ in the Schwartz submodule, the inner product satisfies: $$\langle p_i \psi, \phi \rangle = \langle \psi, p_i \phi \rangle$$ where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, and $\partial_i$ denotes the partial derivative with respect to the $i$-th coordinate.

The $i$-th component of the momentum operator $p_i$ is an unbounded linear operator on the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$. It is defined as the unique unbounded operator associated with the symmetric linear map acting on the dense Schwartz submodule $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. For a function $\psi$ in this domain, the operator acts as $p_i \psi = -i\hbar\partial_i\psi$, where $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, and $\partial_i$ denotes the partial derivative with respect to the $i$-th coordinate.