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Matrix representation of an operator in a new basis

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  1. Dec 26, 2014 #1
    1. The problem statement, all variables and given/known data
    Let Amn be a matrix representation of some operator A in the basis |φn> and let Unj be a unitary operator that changes the basis |φn> to a new basis |ψj>. I am asked to write down the matrix representation of A in the new basis.

    2. Relevant equations


    3. The attempt at a solution
    Should I try to evaluate <m|UnjA|n>? Is that how this should be approached?
    I know that Unjn>=|ψj>.
     
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  3. Dec 26, 2014 #2

    Orodruin

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    No. The definition of the matrix elements in the new basis is: Aij = <ψi| A |ψj>. This is what you have to evaluate.
     
  4. Dec 27, 2014 #3
    I realise that, which is precisely why I asked whether I should evaluate the effect Unj has on the basis elements, namely through evaluating <m|UnjA|n>. Or thus I presumed. Am I totally off?
     
  5. Dec 27, 2014 #4

    Orodruin

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    Not totally, but a bit. What is |ψj> and what is <ψi|? What happens when you insert these expressions into Aij = <ψi| A |ψj>?
     
  6. Dec 27, 2014 #5
    Row and column vectors?
    Aij are the elements of matrix A, I think.
     
  7. Dec 27, 2014 #6

    Orodruin

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    I mean in terms of the original basis vectors. What happens when you insert this into what you want to compute?
     
  8. Dec 27, 2014 #7
    I am not sure I understand what you mean by "inserting that into what I wish to compute". Please clarify.
     
  9. Dec 27, 2014 #8

    Orodruin

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    If you insert the definition of the |ψj>, this is not what you get. The definition is:
     
  10. Dec 27, 2014 #9
    Let's try this. I know that the i-th column is obtained through U|i>=|Ai> and <Ai|=<i|U. Is this somewhat closer to what you intended?
     
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