Physlib.QuantumMechanics.RectangularBarrier.Basic

For a rectangular potential barrier system $Q$, the mass $m$ of the particle is strictly positive, satisfying $m > 0$.

In the context of a rectangular potential barrier system, let $m$ be the mass of the particle. This theorem states that the mass $m$ is non-negative, satisfying $0 \le m$.

For a particle of mass $m$ in a rectangular potential barrier system, the mass is non-zero, denoted as $m \neq 0$.

For a one-dimensional rectangular potential barrier $Q$, the potential function $V: \mathbb{R} \to \mathbb{R}$ maps a position $x$ to the barrier height $V_0$ if $x$ lies within the closed interval $[a, b]$ (defined by the lower bound $a$ and upper bound $b$), and to $0$ otherwise. Mathematically, it is defined as $V(x) = V_0 \cdot \mathbb{1}_{[a, b]}(x)$.

For a one-dimensional rectangular potential barrier $Q$ with barrier height $V_0$, lower bound $a$, and upper bound $b$, the potential function $V: \mathbb{R} \to \mathbb{R}$ is defined by the indicator function of the closed interval $[a, b]$. Specifically, for a position $x$, $V(x) = V_0$ if $x \in [a, b]$ and $V(x) = 0$ otherwise. Mathematically, this is expressed as: $$V(x) = V_0 \cdot \mathbb{1}_{[a, b]}(x)$$ where $\mathbb{1}_{[a, b]}$ denotes the indicator function on the interval $[a, b]$.

For a quantum mechanical rectangular barrier, the potential function $V: \mathbb{R}^1 \to \mathbb{R}$, which is defined as $V_0$ on the closed interval $[x_{\text{lower}}, x_{\text{upper}}]$ and zero elsewhere, is almost everywhere (a.e.) strongly measurable.

The Hilbert space for a one-dimensional rectangular potential barrier is defined as the space $L^2(\mathbb{R}, \mathbb{C})$ of square-integrable functions mapping from the 1-dimensional real space to the complex numbers $\mathbb{C}$.

For a one-dimensional rectangular potential barrier $Q$ with particle mass $m$, the kinetic energy operator is the partial linear map on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ defined as $\frac{1}{2m} \mathbf{p}^2$, where $\mathbf{p}^2$ is the momentum-squared operator. In terms of the wavefunction $\psi$ in the domain, this operator acts as $-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \psi$.

For a one-dimensional rectangular potential barrier $Q$, the potential operator $\hat{V}$ is the partially defined linear operator on the Hilbert space $L^2(\mathbb{R}, \mathbb{C})$ defined by multiplication by the potential function $V(x)$. For a wave function $\psi(x)$ in its domain, the operator maps $\psi(x)$ to the product $V(x)\psi(x)$, where $V(x)$ is the real-valued potential function of the barrier $Q$ (mapped to the complex numbers).

In the quantum mechanical model of a one-dimensional rectangular potential barrier, the potential operator $\hat{V}$ acting on the system's Hilbert space $\mathcal{H}$ is self-adjoint.

In the quantum mechanical model of a one-dimensional rectangular potential barrier, the Hamiltonian operator $\hat{H}$ is defined as the sum of the kinetic energy operator $\hat{T}$ and the potential energy operator $\hat{V}$, acting on the Hilbert space $L^2(\mathbb{R})$.

The Hamiltonian operator $H$ associated with a one-dimensional rectangular potential barrier is essentially self-adjoint.

The definition characterizes the one-dimensional quantum mechanical system of a particle of mass $m$ subject to a rectangular potential barrier of height $V_0$. This system is defined by the Hilbert space $L^2(\mathbb{R})$ and its associated self-adjoint Hamiltonian operator $\hat{H} = \hat{T} + \hat{V}$, where $\hat{T}$ is the kinetic energy operator and $\hat{V}$ is the potential energy operator corresponding to a potential that is $V_0$ on a closed interval and zero elsewhere.