Physlib.QuantumMechanics.DDimensions.Basic

For a single-particle quantum system $Q$ with spatial dimension $d$, the associated Hilbert space is defined as $L^2(\text{Space } d, \mathbb{C})$, the space of square-integrable complex-valued functions on the $d$-dimensional space $\text{Space } d$.

In a single-particle quantum system $Q$ as defined in `SpaceDQuantumSystem`, let $m$ denote the mass of the particle. The mass $m$ is strictly positive, satisfying $0 < m$.

In a single-particle quantum system $Q$, the mass $m$ of the particle is non-negative, satisfying $0 \le m$.

For a single-particle quantum system $Q$, let $m$ be the mass of the particle. Then the mass is non-zero, i.e., $m \neq 0$.

For a single-particle quantum system $Q$ with mass $m$ and spatial dimension $d$, the kinetic energy operator is defined as the continuous linear map $T: \mathcal{S}(\mathbb{R}^d, \mathbb{C}) \to \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ given by: $$T = \frac{1}{2m} \mathbf{p} \cdot \mathbf{p}$$ where $\mathbf{p}$ is the momentum operator and $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ is the Schwartz space of rapidly decreasing functions on the $d$-dimensional space.

For a single-particle quantum system $Q$ with mass $m$ and spatial dimension $d$, the kinetic energy operator $T$, defined as a continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$, is equal to the momentum operator vector $\mathbf{p}$ squared divided by twice the mass: $$T = \frac{1}{2m} \mathbf{p} \cdot \mathbf{p}$$ where $\mathbf{p} \cdot \mathbf{p}$ (or $\mathbf{p}^2$) denotes the dot product of the momentum operator with itself.

For a single-particle quantum system $Q$ with mass $m$ and spatial dimension $d$, the kinetic energy operator $T$ is an unbounded linear operator on the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ defined by: $$T = \frac{1}{2m} \mathbf{p}^2$$ where $\mathbf{p}^2$ is the momentum-squared operator. The domain of the operator is the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$.

For a single-particle quantum system $Q$ with mass $m$, the kinetic energy operator $T$ is equal to the momentum-squared operator $\mathbf{p}^2$ scaled by the factor $\frac{1}{2m}$: $$T = \frac{1}{2m} \mathbf{p}^2$$

For a quantum system $Q$ in $d$ dimensions with potential function $V: \mathbb{R}^d \to \mathbb{R}$, the potential operator is the continuous linear map from the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ to itself defined by pointwise multiplication. Specifically, for any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the operator maps $\psi$ to the function $x \mapsto V(x)\psi(x)$.

For a $d$-dimensional single-particle quantum system $Q$ with potential function $V: \mathbb{R}^d \to \mathbb{R}$, the potential operator $Q.\text{potentialCLM}$ (defined as a continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$) is equal to the operator of pointwise multiplication by $V$. That is, $Q.\text{potentialCLM} = \text{smulLeftCLM}_{\mathbb{C}}(\text{ofReal} \circ V)$, where $\text{ofReal} \circ V$ denotes the potential function cast into complex values and $\text{smulLeftCLM}$ denotes the continuous linear map corresponding to left scalar multiplication.

For a quantum system $Q$ in $d$ dimensions with potential function $V: \mathbb{R}^d \to \mathbb{R}$, if $V$ has temperate growth, then the potential operator $\text{potentialCLM}$ applied to a Schwartz function $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ is the function given by pointwise multiplication: $$(\text{potentialCLM } \psi)(x) = V(x) \psi(x)$$ for all $x \in \mathbb{R}^d$.

Let $Q$ be a single-particle quantum system in $d$ dimensions with a potential function $V: \mathbb{R}^d \to \mathbb{R}$. If $V$ has temperate growth, then for any wave function $\psi$ in the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ and any point $x \in \mathbb{R}^d$, the potential operator $V_{\text{op}}$ applied to $\psi$ and evaluated at $x$ is given by the pointwise product: $$(V_{\text{op}} \psi)(x) = V(x) \psi(x)$$ where $V(x) \psi(x)$ denotes the scalar multiplication of the potential value at $x$ and the value of the wave function at $x$.

For a single-particle quantum system $Q$ in $d$ dimensions with a real-valued potential function $V: \text{Space } d \to \mathbb{R}$, the potential operator $\hat{V}$ (denoted by `potentialOperator`) is the partially defined (unbounded) linear operator on the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ that maps a wave function $\psi$ to the pointwise product $V \cdot \psi$. The domain of this operator is the set of all $\psi \in L^2(\text{Space } d, \mathbb{C})$ such that the resulting product $V \cdot \psi$ is also square-integrable.

For a single-particle quantum system $Q$ in $d$ dimensions with a real-valued potential function $V: \text{Space } d \to \mathbb{R}$, the potential operator $\hat{V}$ is defined as the multiplication operator $\mathcal{M}_V$ associated with the complex-valued function obtained by embedding $V$ into the complex numbers.

For a single-particle quantum system $Q$ in $d$ dimensions with a real-valued potential function $V: \text{Space } d \to \mathbb{R}$, if $V$ is almost everywhere strongly measurable, then the potential operator $\hat{V}$ (the multiplication operator $\mathcal{M}_V$ acting on the Hilbert space $L^2(\text{Space } d, \mathbb{C})$) is self-adjoint.

For a single-particle quantum system $Q$ in $d$ dimensions with potential function $V: \text{Space } d \to \mathbb{R}$, if $V$ has temperate growth, then the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ is contained within the domain of the potential operator $\hat{V}$.

For a single-particle quantum system $Q$ in $d$ dimensions, the Hamiltonian operator $H$ is the continuous linear map from the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ to itself defined as the sum of the kinetic energy operator $T$ and the potential energy operator $V$: $$H = T + V$$ where $T$ is the kinetic energy operator `kineticCLM` and $V$ is the potential energy operator `potentialCLM`. This definition assumes the potential function has temperate growth to ensure the operator is well-defined on the Schwartz space.

For a single-particle quantum system $Q$ in $d$ dimensions, the Hamiltonian operator $H$ is equal to the sum of the kinetic energy operator $T$ and the potential energy operator $V$, where each is defined as a continuous linear map on the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$: $$H = T + V$$ Here, $T$ represents the operator $\frac{1}{2m} \mathbf{p}^2$ and $V$ represents the operator of pointwise multiplication by the potential function.

For a single-particle quantum system $Q$ with spatial dimension $d$, the Hamiltonian operator $H$ is the unbounded linear operator on the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ defined as the sum of the kinetic energy operator $T$ and the potential energy operator $V$: $$H = T + V$$ This operator is a symmetric unbounded operator on the system's Hilbert space.

For a single-particle quantum system $Q$ in $d$ dimensions, the Hamiltonian operator $H$ is equal to the sum of the kinetic energy operator $T$ and the potential energy operator $V$: $$H = T + V$$ These operators are defined as unbounded linear operators on the system's Hilbert space $L^2(\text{Space } d, \mathbb{C})$.