Physlib.QuantumMechanics.FreeParticle.Basic

For a free quantum particle $Q$, the mass $m$ is strictly positive, i.e., $0 < m$.

For a free quantum particle $Q$, its mass $m$ is non-negative, satisfying the inequality $0 \le m$.

For a free quantum particle $Q$ in $d$ dimensions, its mass $m$ is non-zero ($m \neq 0$).

For a free particle in $d$ dimensions, the associated Hilbert space is defined as $L^2(\text{Space } d, \mathbb{C})$, which is the space of square-integrable complex-valued functions on the $d$-dimensional space $\text{Space } d$.

For a free particle $Q$ of mass $m$ in $d$ dimensions, the Hamiltonian $H$ is defined as a partial linear map on the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ given by $$H = \frac{1}{2m} \mathbf{p}^2$$ where $\mathbf{p}^2$ is the momentum-squared operator (proportional to the Laplacian). This operator represents the kinetic energy of the particle in the absence of a potential.

The Hamiltonian operator $H$ for a free particle in $d$ dimensions is essentially self-adjoint on its domain in the Hilbert space.

The free particle in $d$ dimensions is represented as a quantum system. This structure packages the Hilbert space $\mathcal{H}$ (typically $L^2(\mathbb{R}^d)$) and the self-adjoint Hamiltonian operator $H = \frac{\mathbf{p}^2}{2m}$ into a single quantum system object, where $\mathbf{p}$ is the momentum operator and $m$ is the mass of the particle.