Diagonalizable matrix tight-binding model

Question

Given the following model for a particle on a lattice: $$ \hat{H} = -t \sum_j[\left|\ j-2\right\rangle \left\langle j \right|+ \left|j+2\right\rangle \left\langle j \right|]$$

We introduce vectors $\vec{e_j} = (0,0,\dots,1, \dots, 0) \equiv \left|\vec{e_j}\right\rangle$ where 1 is the $j$-th column. Similarly introduce the vectors $\vec{k_n} = (0,0,\dots,1,\dots,0)$ which is equivalent to: $$\left|k_n\right\rangle = \frac{1}{\sqrt{N}} \sum_j \exp(i k_nja)\left|e_j\right\rangle$$ with $k_n = \frac{2\pi n }{aN}$

Now, as part of an assignment, I would like to construct a matrix $Q$ such that $H_e = Q^{-1}H_kQ$ which has the property that $Q^{\dagger} = Q^{-1}$ where $H_e$ is the $N \times N $ matrix with elements $[H_e]_{jj'} = \left\langle j|\hat{H}|j'\right\rangle$ and $H_k$ is the $N \times N $ matrix with elements $[H_k]_{kk'} = \left\langle k|\hat{H}|k'\right\rangle$

I suppose that the method is to first find matrices $H_e$ and $H_k$ but I'm not quite sure how to do this, since I don't fully understand the way the different elements are calculated?

Answer 1

There are several approaches to this, I am going to give you a few hints so you can follow through with the method you suggested in your question:

Representing your bras and kets as unit vectors in $\mathbb{C}^n$, you can express your outer products $| j \rangle \langle i | = e_j \otimes e_i = e_j^T e_i$, yielding a $N \times N$ matrix (because $(N \times 1 ) \times (1 \times N) = N \times N$)). This gives you your $[H_e]$-Matrix.

On the other hand, you can define the inverted fourier transform $|j \rangle = \frac{1}{\sqrt{N}} \sum_k e^{-i k j a} |k \rangle$ and plug this into your hamiltonian, using the fact that $\sum_k e^{i k (i - j)} \propto \delta_{i j}$ (please be careful with the normalizations here, since you are not looking at unit lattice spacing). This will yield you the diagonal hamiltonian in $k$-space, including the band structure $E(k)$. It should be easy for you to figure out the matrix representation of this (hint: its diagonal :-) ). Hence you have both matrix reps and can figure out how $Q$ needs to look like.