Feynman clock's Hamiltonian matrix reduction

[shantan]

Homework Statement

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|

Homework Equations

How can I write the correct hamiltonian for this system?

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" 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

 

[mark726]

Thank you for sharing your interesting research on the evolution of spin systems. From your description, it seems that you have already made good progress in creating a basis for your system and identifying the possible configurations of spin states.

In order to write the correct Hamiltonian for your system, you will need to incorporate the information about the allowed operations in your algebra. It is important to note that the Hamiltonian represents the total energy of the system and should take into account all possible interactions between spins.

One approach could be to start with the Pauli matrices and use the Kronecker product to combine them with the allowed operations in your algebra. This will help you create a more complete Hamiltonian that takes into account the allowed operations and their effects on the spin states.

Additionally, you may also want to consider any external factors or interactions that may affect the evolution of your spin system. For example, if there is an external magnetic field, this should be included in your Hamiltonian.

Once you have a complete Hamiltonian, you can use it to calculate the evolution of your system over time. This can be done by using the time-dependent Schrödinger equation or by using the time-evolution operator, as you mentioned in your post.

I hope this helps and I wish you success in your research. Please feel free to reach out if you have any further questions or would like to discuss your findings.