- #1
qubitor
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Homework Statement
sorry for my english..
I was asked to find a base to diagonize a Hamiltonian, which could been written in the given base as below:
n×n matrix
0 1 0 0 0 ... 0 0
1 0 1 0 0 ... 0 0
0 1 0 1 0 ... 0 0
0 0 1 0 1 ... 0 0
... ... ... ... ...
0 0 0 0 0 ... 0 1
0 0 0 0 0 ... 1 0
Homework Equations
In order to diagonize this Hamiltonian, I think one could calculate its eigenvalues in this base to get
eigenfuntions, hence one can use matrix of eigenfunctions to tranforms this base to obtain a new base which diagonize Hamiltonian as
λ1 0 ... 0
0 λ2 ... 0
...
0 ... λn
The Attempt at a Solution
I tried to calculate the determinant to obtain eignenvalues of Hamiltonian by
det|λId - H|=0
But it is too complicated and I didn't find a way to calculate it in n dimensions. Is there some way that I didn't know to calculate the determinant?