Physlib.QuantumMechanics.InfiniteSquareWell.Basic

For an infinite square well $Q$, the mass $m$ of the particle is strictly positive, such that $0 < m$.

For a particle in an infinite square well $Q$, the mass $m$ of the particle is non-negative, satisfying $0 \le m$.

For an infinite square well system $Q$, the mass $m$ of the particle is non-zero, i.e., $m \neq 0$.

The domain of an infinite square well $Q$ in $d$ dimensions is the set of points in the space that fall within the axis-aligned cuboid defined by the $d$-dimensional closed interval $[Q.\text{lower}, Q.\text{upper}]$. Mathematically, it is the preimage of the closed interval $[Q.\text{lower}, Q.\text{upper}]$ under the coordinate valuation map of the space.

The measure $\mu$ on the $d$-dimensional space $\text{Space } d$ associated with an infinite square well $Q$ is defined as the standard Lebesgue measure (denoted as `volume`) restricted to the well's domain $Q.\text{well}$. Mathematically, for any measurable set $A \subseteq \text{Space } d$, the measure is given by $\mu(A) = \lambda(A \cap Q.\text{well})$, where $\lambda$ is the Lebesgue measure and $Q.\text{well}$ is the axis-aligned cuboid defined by the interval $[Q.\text{lower}, Q.\text{upper}]$.

For an infinite square well $Q$ in a $d$-dimensional space, the Hilbert space is defined as the space $L^2(\text{Space } d, \mathbb{C})$ of complex-valued square-integrable functions, where the underlying measure is the Lebesgue measure restricted to the well's domain $Q.\text{well} = [Q.\text{lower}, Q.\text{upper}]$.

The Hamiltonian operator $H$ for a particle of mass $m$ in an infinite square well is defined as $H = \frac{1}{2m} \hat{p}^2$, where $\hat{p}^2$ is the momentum squared operator acting on the Hilbert space defined by the measure of the well.

The Hamiltonian operator $H$ for a particle in an infinite square well is essentially self-adjoint on its domain in the Hilbert space.

This definition constructs the quantum mechanical system for a particle in an infinite square well. The system is defined by a Hilbert space $\mathcal{H} = L^2(\Omega)$ over an axis-aligned cuboid $\Omega \subset \mathbb{R}^n$ and a self-adjoint Hamiltonian operator $H$ acting on this space, representing the particle's energy under Dirichlet boundary conditions.