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Diagonization of matrix

  1. Dec 3, 2013 #1
    1. The problem statement, all variables and given/known data
    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


    2. Relevant 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


    3. 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?
     
  2. jcsd
  3. Dec 4, 2013 #2
    is there someone could make it?
     
  4. Dec 4, 2013 #3

    vela

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    You could try expressing H as ##H_{ij} = \delta_{i (j-1)}+\delta_{i (j+1)}##.
     
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