# Hamiltonian reduction having constant of the motion

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 now 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.

How can i reduce the dimension of my matrix deleting row and columns to remain in a proper subspace? How can I have a $n*n$ matrix??

Thanks!

• Well, since you can list all the states $|i\rangle,|j\rangle$ you are interested you can just explicitly evaluate the matrix elements between only those states: $\langle i|H|j\rangle$. – Michael Brown May 29 '13 at 22:38

I've created new basis of my space, listing all the possibile (and valid) configuration of my system. So 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 $|k\rangle \langle k+1|$