Physlib.QuantumMechanics.QuantumSystem.Basic

For any quantum system $Q$, the underlying Hilbert space $\mathcal{H}_Q$ is equipped with the structure of a normed additive commutative group.

For any quantum system $Q$, the underlying Hilbert space $\mathcal{H}_Q$ is equipped with the structure of an inner product space over the complex numbers $\mathbb{C}$.

For any quantum system $Q$, the underlying Hilbert space $\mathcal{H}_Q$ is a complete space.

Let $HS$ be a complex Hilbert space and $\mathcal{H}$ be a partially defined linear operator on $HS$ (of type `LinearPMap`) that is essentially self-adjoint. This definition constructs a `QuantumSystem` by taking $HS$ as the underlying Hilbert space and the closure $\overline{\mathcal{H}}$ as the self-adjoint Hamiltonian operator.

The zero quantum system is defined as the system consisting of the zero-dimensional complex Hilbert space $\mathbb{C}^0$ and the zero operator $0$ as its Hamiltonian.

Two quantum systems $Q_1$ and $Q_2$ satisfy the unitary equivalence relation if there exists a unitary operator (expressed as a complex linear isometry equivalence) $e: \mathcal{H}_1 \to \mathcal{H}_2$ between their respective Hilbert spaces such that the domain of the Hamiltonian $H_1$ is mapped by $e$ onto the domain of the Hamiltonian $H_2$, and for every vector $\psi$ in the domain of $H_1$, the condition $e(H_1 \psi) = H_2(e \psi)$ holds.

For any quantum system $Q$, the relation of unitary equivalence is reflexive; that is, $Q$ is unitarily equivalent to itself.

For any two quantum systems $Q_1$ and $Q_2$, if $Q_1$ is unitarily equivalent to $Q_2$, then $Q_2$ is unitarily equivalent to $Q_1$. Here, unitary equivalence signifies that there exists a unitary operator between the respective Hilbert spaces that preserves the domain and action of the Hamiltonian operators.

Let $Q_1, Q_2,$ and $Q_3$ be quantum systems. If $Q_1$ is unitarily equivalent to $Q_2$ and $Q_2$ is unitarily equivalent to $Q_3$, then $Q_1$ and $Q_3$ are also unitarily equivalent.

The unitary equivalence relation between quantum systems is an equivalence relation. This means the relation is reflexive, symmetric, and transitive.

This definition establishes a setoid structure on the collection of all quantum systems. A setoid is a type equipped with an equivalence relation; here, the equivalence relation is defined as the unitary equivalence between quantum systems $Q_1$ and $Q_2$. Two quantum systems are equivalent if there exists a unitary transformation between their respective Hilbert spaces that intertwines their Hamiltonian operators.