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Matrix representation of an operator in a new basis

  • #1
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Homework Statement


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.

Homework Equations




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>.
 

Answers and Replies

  • #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.
 
  • #3
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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?
 
  • #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>?
 
  • #5
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Row and column vectors?
Aij are the elements of matrix A, I think.
 
  • #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?
 
  • #7
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I am not sure I understand what you mean by "inserting that into what I wish to compute". Please clarify.
 
  • #8
Orodruin
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The definition of the matrix elements in the new basis is: Aij = <ψi| A |ψj>
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>.
If you insert the definition of the |ψj>, this is not what you get. The definition is:
I know that Unj|φn>=|ψj>.
 
  • #9
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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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