# Feynman clock's Hamiltonian matrix reduction

1. Jun 6, 2013

### shantan

1. The problem statement, all variables and given/known data

I have this 2^n*2^n matrix that represent the evolution of a system of $n$ spin.
I know that I can have only one excited spin in my configuration a time.
(eg: 0110 nor 0101 ar not permitted, but 0100 it is)

s_+ is defined with s_x+is_y and s_- is defined with s_x-is_y
(creation and annichilation operators)

I've created the hamiltonian composing pauli matrices with kronecker product.
I know that I have only few operation in my algebra: I, s_z, s_+[k].s_-[k-1] where k is the spin number where this operation is made.
This algebra sends valid state to valid states, preserving the constant of my motion.

So, for reducing my Hamiltonian, I've created new basis of my space, listing all the possibile (and valid) configuration of my system. With n spin, using the consideration that i can have only one excited spin, i have only n possibile configuration.
I've created my algebra with the following operator: I, s_3[k] (similar to I but with -1 in the k-th position on the diagonal, and |k\rangle\langle k-1| for moving my spin-up forward and it's complex conjugate: |k\rangle \langle k+1|

2. Relevant equations
How can I write the correct hamiltonian for this system?

3. The attempt at a solution
Summing up the "moving right" and "moving left" operator, as described in [URL="http://www.cs.princeton.edu/courses/archive/fall05/frs119/papers/feynman85_optics_letters.pdf" [Broken] Quantum mechanical computer[/URL] at page 5, I've obtained a matrix with the main diagonal filled with 0 and the upper diagonal and lower diagonal filled with 1.
Applaying this matrix to a n-bit state [0,0,1,0,0] i got [0,1,0,1,0] which is NOT a valid state (it has 2 excited bit).
Shoul i use this hamiltonian in e^iHt for the evolution of my system? How?
Thanks

Last edited by a moderator: May 6, 2017