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Hamiltonian Matrix?

  1. Apr 24, 2010 #1
    1. The problem statement, all variables and given/known data

    I need to find the 2x2 Hamiltonian matrix for the Hamiltonian, which is written in second-quantized form as below for a system consisting of the electrons and photons.

    H = h/ωb†b + E1a†1a1 + E2a†2a2 + Ca†1a2b† + Ca†2a1b,

    a's are creation and annihilation operator for electrons, and b's are for photons.

    2. Relevant equations

    Need to be written in the basis of the following states

    |φ1> = 1 √(n−1)! a2† (b†)^(n−1)|0>
    |φ2>= 1 √(n)! a1† (b†)^(n)|0>


    I'm not sure where to begin. I'm guessing I have to find the eigenvalues and vectors of the hamiltonian but not sure how. Could someone help me start this problem? Thanks.
     
  2. jcsd
  3. Apr 24, 2010 #2

    gabbagabbahey

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    No need to compute eigenvalues/vectors here...Just use the fact that any operator [itex]A[/itex] can be expressed in a basis [itex]\{|v_1\rangle,|v_2\rangle,\ldots|v_n\rangle\}[/itex] as a matrix with entries given by [itex]A_{ij}=\langle v_i|A|v_j\rangle[/itex]. So, for example [itex]H_{12}=\langle \psi_1|H|\psi_2\rangle[/itex]...just calculate the 4 inner products to get your 4 components.
     
  4. Apr 24, 2010 #3

    Cyosis

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    Recall [itex]\hat{H} |e_j \rangle= H_{ij} |e_i \rangle[/itex]. Summation over i implied. How can you find the matrix elements from this?
     
  5. Apr 24, 2010 #4
    Thanks to both of you. That helped a lot. I managed to do inner product for one of the terms of the Hamiltonian for just the first component of the matrix. Seems like this is going to be a long process. Is there a trick that I maybe missing that will reduce computations?
     
  6. Apr 24, 2010 #5

    gabbagabbahey

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    Just begin by calculating the effect of [itex]H[/itex] on each of your two basis states. After that, all 4 inner products should be straightforward.
     
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