PhyslibAlpha.CondensedMatter.TightBindingChain.Uncertainty

Given a tight binding chain $T$ with $N$ sites and its associated Hilbert space $\mathcal{H}_T \cong \mathbb{C}^N$, this definition converts a symmetric linear operator $A: \mathcal{H}_T \to \mathcal{H}_T$ into an **observable**. The observable is represented as a self-adjoint element in the algebra of continuous linear operators $\mathcal{L}(\mathbb{C}^N)$ by identifying the chain's Hilbert space with the standard complex Euclidean space $\mathbb{C}^N$ via a canonical isometric equivalence.

For a tight-binding chain $T$ with $N$ sites, the **open Hamiltonian observable** is the Hamiltonian operator $H_{\text{open}}$ with open boundary conditions, realized as a self-adjoint element of the algebra of continuous linear operators $\mathcal{L}(\mathbb{C}^N)$. It is constructed by applying the conversion function `toObservable` to the operator $H_{\text{open}}$ and the proof of its Hermiticity ($\langle H_{\text{open}} \psi, \phi \rangle = \langle \psi, H_{\text{open}} \phi \rangle$).

For a tight binding chain $T$ with $N$ sites and lattice spacing $a$, the position operator $X$ is defined as an observable on the Hilbert space $\mathcal{H} \cong \mathbb{C}^N$. This observable corresponds to the self-adjoint linear operator: $$X = \sum_{n=0}^{N-1} (a \cdot n) |n\rangle\langle n|$$ where $n \in \{0, \dots, N-1\}$ is the site index and $|n\rangle\langle n|$ is the projector onto the localized basis state at site $n$. It is constructed by identifying the Hermitian position operator with a self-adjoint element of the algebra of continuous linear operators on $\mathbb{C}^N$.

Let $T$ be a tight binding chain with $N$ sites and lattice spacing $a$. Let $H$ be the open boundary Hamiltonian observable and $X$ be the position observable. For any two site indices $m, n \in \{0, \dots, N-1\}$, the matrix element of the Lie bracket $\llbracket H, X \rrbracket = -\frac{i}{2}(HX - XH)$ between the localized basis states $|m\rangle$ and $|n\rangle$ is given by $$\langle m | \llbracket H, X \rrbracket | n \rangle = -\frac{i}{2} a (n - m) \langle m | H | n \rangle$$ where $i$ is the imaginary unit and $\langle m | H | n \rangle$ is the matrix element of the Hamiltonian.

For a tight-binding chain $T$ with $N$ sites, lattice spacing $a$, and hopping amplitude $t$, let $H$ be the open Hamiltonian observable and $X$ be the position observable. For any two site indices $m, n \in \{0, \dots, N-1\}$, the matrix element of the Lie bracket $\llbracket H, X \rrbracket = -\frac{i}{2}(HX - XH)$ between the localized basis states $|m\rangle$ and $|n\rangle$ is given by: $$\langle m | \llbracket H, X \rrbracket | n \rangle = \begin{cases} i \frac{at}{2} & \text{if } m + 1 = n \\ -i \frac{at}{2} & \text{if } n + 1 = m \\ 0 & \text{otherwise} \end{cases}$$ where $i$ is the imaginary unit.

For a tight-binding chain with $N$ sites and open boundary conditions, let $H$ be the open Hamiltonian observable and $X$ be the position observable. For any state $\omega$ on the C*-algebra of linear operators on $\mathbb{C}^N$, the Robertson–Schrödinger uncertainty relation holds: $$\text{Cov}_\omega(H, X)^2 + \omega\langle \llbracket H, X \rrbracket \rangle^2 \le \text{Var}_\omega(H) \cdot \text{Var}_\omega(X)$$ where $\text{Cov}_\omega(H, X)$ is the covariance of $H$ and $X$ in state $\omega$, $\text{Var}_\omega(H)$ and $\text{Var}_\omega(X)$ are the variances of the respective observables, and $\omega\langle \llbracket H, X \rrbracket \rangle$ is the expectation value of the Lie bracket $\llbracket H, X \rrbracket = -\frac{i}{2}(HX - XH)$.