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\heading{E1470: Current in a three site system} 
\auname{Daniel Fleischer, Jaron de Leeuw}


{\bf The problem:}

\Dn

A particle in a three site system is described using the standard basis: $|x=1 \rangle , |x=2 \rangle , |x=3 \rangle $, with the Hamiltonian   
\[
\mathcal{H} = \pppmatrix{ 0 & c & 0 \cr c & 0 & c \cr 0 & c & 0}
\]

\Dn

(1) Find the eigenvalues of the Hamiltonian: $ E_0,E_+, E_-$.

(2) Find the eigenstates of the Hamiltonian: $|A \rangle , |S_+ \rangle , |S_- \rangle $.

\Dn

The particle has a charge e.
\Dn

(3) Find the matrix representation of the current operator ${I}$  for current from site 1 to 2.

(4) Find the observable values of the current operator you defined.

\Dn

The particle has been prepared in the state $|x=1 \rangle $ at $t=0$.
\Dn

(5) Find the probability of measuring $I=0$ at time $t$.



\Dn\Dn


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{\bf The solution:}

\Dn

(1) Using diagonalization, we solve the equation,
\[ det(\mathcal{H}-\lambda \mathbb{I} )=0 \]
thus finding the eigenvalues:
\[E_0=0 \qquad 	E_\pm =\pm \sqrt{2}c  \]

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\Dn

(2)
The eigenstates of the Hamiltonian are: 
\[ |A\rangle  = \frac{1}{\sqrt{2}} (|1 \rangle - |3\rangle )                   \]
\[ |S_+ \rangle =  \frac{1}{2} (|1 \rangle + \sqrt{2} |2 \rangle + |3\rangle ) \]
\[ |S_- \rangle=   \frac{1}{2} (|1 \rangle - \sqrt{2} |2 \rangle + |3\rangle ) \]


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\Dn

(3)
We find the current operator in two ways:
\begin{enumerate}
\item Using the projection operator - the current from site 1 to site 2 being the rate of change of the projection operator $ \hat{P}^1$.

This can be inferred from the probabilty continuity equation: $\nabla \cdot J + \pd{\rho}{t} =0$.
\[
\hat{I}_{1\rightarrow 2}=-e \pd{ \hat{P}^1}{t}=-i e \left[\hat{\mathcal{H}}, \hat{P}^1 \right] = e \pppmatrix { 0 & i c & 0 \cr -i c & 0 & 0 \cr 0 & 0 & 0}
\]
\item Using spatial phase. We write the hamiltonian:
\[
\mathcal{H} = \pppmatrix { 0 & c e^{i \phi} & 0 \cr c e^{-i \phi} & 0 & c \cr 0 & c & 0}
\]
The current is defined as
$ \hat{I}= -e \pd{\hat{\mathcal{H}} }{ \phi}|_{\phi = 0} $,
\[
\hat{I}= -i e \pppmatrix { 0 & c e^{i \phi} & 0 \cr -c e^{-i \phi} & 0 & 0 \cr 0 & 0 & 0}\vert_{\phi = 0} = -e\pppmatrix { 0 &i c  & 0 \cr -i c  & 0 & 0 \cr 0 & 0 & 0}
\]

\end{enumerate}


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\Dn

(4)
Finding the observable values using diagonalization for the $2\times 2$ block:
\[I_0=0 \qquad 	I_\pm =\pm ce  \]


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\Dn

(5)
Using the energy basis, one can describe the wave function at t=0, and the current eigenstate corresponding to zero current.
\[|\psi(t=0)\rangle= |1\rangle = \frac{1}{\sqrt{2}}|A\rangle + \frac{1}{2}|S_+\rangle + \frac{1}{2}|S_-\rangle\]
\[|I=0\rangle = -\frac{1}{\sqrt{2}}   |A\rangle   +\frac{1}{2}  |S_+\rangle  +  \frac{1}{2}|S_-\rangle \]
\[|\langle\psi(t) | I=0\rangle|^2 = \left|\pppmatrix{\frac{1}{\sqrt{2}} & \frac{1}{2} \eexp{-i\sqrt{2}ct} & \frac{1}{2}\eexp{i\sqrt{2}ct} }\cdot \left(\begin{array}{c}\frac{-1}{\sqrt{2}} \\ \frac{1}{2} \\ \frac{1}{2}\end{array}\right)  \right|^2 = \sin^2\left(\frac{ct}{\sqrt{2}}\right)\]

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