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Operator which is written in k space

  1. Nov 20, 2013 #1
    I have an operator which is written in k space as something like:

    H = Ʃkckak
    where a_k and c_k are operators. So it is a sum of operators of different k but there are no crossterms as you can see. Being no crossterms does this mean that the operator is diagonalized in the language of linear algebra. That a matrix is diagonalized means that it is represented in a basis of eigenstates but this is an expansion of an operator so I am not sure.
    Edit; any operator is of course an expansion in operators related to the basis of eigenstates {e_k} by the equation:
    Ʃi,jle_i><e_jl a_ij
    Now to be diagonalized would mean the vanishing of all matrix elements where i≠j. Can this be said about the given Hamiltonian above. How do I know how the c_k and a_k are related to my eigenbasis.
    Last edited: Nov 20, 2013
  2. jcsd
  3. Nov 20, 2013 #2


    Staff: Mentor

    What's the actual question? Your sum looks like the product of two diagonal matrices C and A, where ci is an entry on the diagonal of matrix C and ai is an entry on the diagonal of matrix A. As it seems to me, C and A are the operators, but the entries in the matrices aren't.

    Without knowing what the entries in the matrices are, it's hard to say what H is. Otherwise, your sum looks like some inner product to me. More information would be helpful.
  4. Nov 20, 2013 #3
    The a_k and c_k are operators as stated? So its a sum of operators. The a_k is the creation operator for the momentum state lk> and c_k is the anihillation operator for the same state.
  5. Nov 20, 2013 #4
    Wait I know now. It is of course because the product a_k c_k is just the number operator.
  6. Nov 20, 2013 #5


    Staff: Mentor

    This thread is probably more appropriate in the Adv. Physics section, so I'm moving it.
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