Homework Help: Electron position at time t

1. Sep 8, 2014

bowlbase

1. The problem statement, all variables and given/known data
Consider the electron on a six site chain. The Hamiltonian is:

$H = \begin{pmatrix} 0 & -1 & 0 & 0 & 0& 0\\ -1 & 0 & -1 & 0 & 0& 0\\ 0 & -1 & 0 & -1 & 0& 0\\ 0 & 0 & -1 & 0 & -1& 0\\ 0 & 0 & 0 & -1 & 0& -1\\ 0 & 0 & 0 & 0 & -1& 0\\ \end{pmatrix}$

This is, I think, the computed value of H:
$H_{diagonal} = \begin{pmatrix} e^{-it} & 0 & 0 & 0 & 0& 0\\ 0 & e^{-i2t} & 0 & 0 & 0& 0\\ 0 & 0 & e^{-i2t} & 0 & 0& 0\\ 0 & 0 & 0 & e^{-i2t} & 0& 0\\ 0 & 0 & 0 & 0 & e^{-i2t}& 0\\ 0 & 0 & 0 & 0 & 0& e^{-it}\\ \end{pmatrix}$

If the electron starts in $|x=0\rangle$, find where the electron is at time t.

2. Relevant equations

3. The attempt at a solution

I'm just not sure how to go about this. What exactly does $|x=0\rangle$ mean. Do I need to use the Dirac notation and $H|0\rangle$? Or is it that I need to multiply the diagonal matrix by

$\begin{pmatrix} 1\\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix}$

2. Sep 9, 2014

Simon Bridge

The electron starts at position x=0 in the 6 site chain.
Other possible positions are, presumably, x=1, x=2,...x=5.
You need to time-evolve that state.

3. Sep 10, 2014

vela

Staff Emeritus
You gave a matrix representation for the Hamiltonian. What basis is this representation with respect to? Answering that might provide you the answer to your question about what $|x=0\rangle$ means.