PhyslibAlpha.CondensedMatter.TightBindingChain.ElementalUncertainty

In an open tight-binding chain $T$, the energy and position fluctuations of the maximal current state $\omega_{\text{max}}$ are uncorrelated. That is, the covariance between the open Hamiltonian observable $H$ and the position observable $X$ in the state $\omega_{\text{max}}$ is zero: $$\text{Cov}_{\omega_{\text{max}}}(H, X) = 0$$ where $\text{Cov}_\omega(H, X) = \text{Re}(\omega((H - \omega\langle H \rangle)(X - \omega\langle X \rangle)))$ is the real part of the expectation of the product of the fluctuations.

For an open tight-binding chain with $N$ sites, lattice spacing $a$, and hopping parameter $t$, let $\omega_{\text{max}}$ be the maximal current vector state, $H_{\text{open}}$ be the open Hamiltonian observable, and $X$ be the position observable. The product of the variances of $H_{\text{open}}$ and $X$ in the state $\omega_{\text{max}}$ satisfies the identity: $$\text{Var}_{\omega_{\text{max}}}(H_{\text{open}}) \cdot \text{Var}_{\omega_{\text{max}}}(X) = \left( a t \cos \frac{\pi}{N+1} \right)^2 + \mathcal{D}_{\omega_{\text{max}}}(H_{\text{open}}, X)$$ where $\mathcal{D}_{\omega_{\text{max}}}(H_{\text{open}}, X)$ is the centered Gram defect of $H_{\text{open}}$ and $X$ in the state $\omega_{\text{max}}$, defined as $\text{Var}_{\omega}(H) \text{Var}_{\omega}(X) - |\omega(\Delta H \Delta X)|^2$.

For an open tight-binding chain $T$, the constant $C_{\text{Nava}}$ is defined as the ratio of the square root of the product of the variances of the Hamiltonian $H_{\text{open}}$ and the position operator $X$ in the maximal current state $\omega_{\text{max}}$, to the absolute value of the product of the lattice spacing $a$, the hopping parameter $t$, and $\cos(\pi / (N+1))$. Mathematically, $$C_{\text{Nava}} = \frac{\sqrt{\text{Var}_{\omega_{\text{max}}}(H_{\text{open}}) \cdot \text{Var}_{\omega_{\text{max}}}(X)}}{\left| a t \cos\left( \frac{\pi}{N+1} \right) \right|}$$ where $N$ is the number of sites in the chain.

For an open tight-binding chain with lattice spacing $a$, hopping parameter $t$, and number of sites $N$, if $t \neq 0$ and $N \geq 2$, then the expression $at \cos(\pi / (N + 1))$ is non-zero.

For an open tight-binding chain with a non-zero hopping parameter $t \neq 0$ and number of sites $N \geq 2$, the uncertainty constant $C_{\text{Nava}}$ is at least one: $$C_{\text{Nava}} \geq 1$$ where $C_{\text{Nava}}$ is the ratio of the square root of the product of the variances of the Hamiltonian and the position operator in the maximal current state to the absolute value of the product of the lattice spacing, hopping parameter, and $\cos(\pi / (N+1))$.

For an open tight-binding chain $T$ with number of sites $N \geq 2$ and a non-zero hopping parameter $t \neq 0$, the uncertainty constant $C_{\text{Nava}}$ is equal to $1$ if and only if $N = 2$ or $N = 3$. The constant $C_{\text{Nava}}$ is defined as the ratio: $$C_{\text{Nava}} = \frac{\sqrt{\text{Var}_{\omega_{\text{max}}}(H_{\text{open}}) \cdot \text{Var}_{\omega_{\text{max}}}(X)}}{\left| a t \cos\left( \frac{\pi}{N+1} \right) \right|}$$ where $H_{\text{open}}$ is the Hamiltonian of the chain, $X$ is the position operator, $a$ is the lattice spacing, and $\omega_{\text{max}}$ is the maximal current state.

For an open tight-binding chain with a non-zero hopping parameter $t \neq 0$, if the number of sites $N$ is at least 4, then the uncertainty constant $C_{\text{Nava}}$ is strictly greater than 1. The constant $C_{\text{Nava}}$ is defined as the ratio of the square root of the product of the variances of the Hamiltonian $H_{\text{open}}$ and the position operator $X$ in the maximal current state, to the absolute value of the product of the lattice spacing $a$, the hopping parameter $t$, and $\cos(\pi / (N+1))$: $$C_{\text{Nava}} = \frac{\sqrt{\text{Var}_{\omega_{\text{max}}}(H_{\text{open}}) \cdot \text{Var}_{\omega_{\text{max}}}(X)}}{\left| a t \cos\left( \frac{\pi}{N+1} \right) \right|}$$

For an open tight-binding chain $T$ with $N \ge 2$ sites, lattice spacing $a$, and hopping parameter $t \neq 0$, let $H_{\text{open}}$ be the open Hamiltonian observable, $X$ be the position observable, and $\omega_{\text{max}}$ be the maximal current vector state. The **Nava–Robertson–Schrödinger elemental dimensional uncertainty inequality** states that: 1. The product of the variances of the Hamiltonian and the position operator in the maximal current state satisfies: $$\text{Var}_{\omega_{\text{max}}}(H_{\text{open}}) \cdot \text{Var}_{\omega_{\text{max}}}(X) = \left( a t \cos\left( \frac{\pi}{N + 1} \right) \right)^2 + \text{defect}_{\omega_{\text{max}}}(H_{\text{open}}, X)$$ where $\text{defect}_{\omega_{\text{max}}}(H_{\text{open}}, X)$ is the centered Gram defect. 2. The uncertainty constant $C_{\text{Nava}}$ satisfies $C_{\text{Nava}} \geq 1$. 3. $C_{\text{Nava}} = 1$ if and only if $N = 2$ or $N = 3$.