binding Hamiltonian defined above is just a function of the translation operator and can thus be written as $H=-\sum_n t_{n} T^n$. Since now $[H,T]=0$, we can diagonalise the Hamiltonian in the the eigen-basis of the translation operator which we know are the plane wave states defined as $$\begin{aligned}
\ket{k} &= &\frac{1}{\sqrt{N}}\sum_j e^{ikj}\ket{j}\end{aligned}$$ Action of $T^m$ on $\ket{k}$ gives $T^m\ket{k}=e^{-ikma}\ket{k}$. Therefore the eigenvalue spectrum for our tight binding Hamiltonian is given by $$\begin{aligned}
& H \ket{k} & = ~ \epsilon(k)\ket{k} \nonumber \\ 
& \epsilon (k) & = ~ -\sum_n t_n e^{-ikna}\end{aligned}$$ For a case when we only allow the nearest neighbour hopping i.e. $t_1=t\ne0$ and all others are zero. Then, we obtain $\epsilon(k)=-2t\cos{ka}$.

![Energy dispersion obtained from a 1D nearest neighbour tight binding Hamiltonian. The red portion represent the filled states.[]{data-label="tightbindingdispersion"}](metal.pdf)

This is the dispersion that is followed by a single electron hopping on a lattice made from hydrogen-like atoms. It turns out that this method, when applied to the many-electron problem by neglecting the $e^--e^-$ interaction, is not a bad approximation to describe many physical properties like metallic character. All the allowed $k$ falling in the line from $-\pi/a$ to $+\pi/a$ constitute a Brillouin zone. The edges at $k = 