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Unitary operator/matrix

  1. Jul 9, 2009 #1
    I have a very basic question. I'm confused because I've read in a text that the matrix representation of a unitary operator is a unitary matrix if the basis is orthogonal, however I believe that the matrix is unitary whatever basis one uses. I'd appretiate any comments on this.
     
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  3. Jul 10, 2009 #2

    malawi_glenn

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    which text?
     
  4. Jul 10, 2009 #3
    Quantum Mechanics(third edition) E. Merzbacher-Chapter 17 Page 418,I quote :"Since the operators U_a were assumed to be unitary, the representation matrices are also unitary if the basis is orthonormal"
     
  5. Jul 10, 2009 #4

    George Jones

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    Try a simple example. Take the identity matrix on a 2-dimensional space, which is clearly unitary. Use linearity to compute the matrix elements with respect to the basis [itex]e_{1}' = e_1[/itex] and [itex]e_{2}' = e_1 + e_2[/itex], where [itex]e_1[/itex] and [itex]e_1[/itex] make up an orthonormal basis.

    Does this give a unitary matrix?
     
  6. Jul 10, 2009 #5

    Fredrik

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    A unitary matrix U satisfies [itex]\sum_j U^*_{ji}U_{jk}=\delta_{ik}[/itex]. Is this satisfied by the matrix representation of a unitary operator?

    [tex]\sum_j U^*_{ji}U_{jk}=\sum_j\langle j|U|i\rangle^*\langle j|U|k\rangle=\sum_j\langle i|U^\dagger|j\rangle\langle j|U|k\rangle=\langle i|U^\dagger\Big(\sum_j|j\rangle\langle j|\Big)U|k\rangle[/tex]

    This reduces to [itex]\delta_{ij}[/itex] if the parenthesis is the identity operator. I can prove that it is, if I use that the basis is orthonormal, but not without that assumption. So it looks like your book is right. What makes you think it's wrong?
     
  7. Jul 10, 2009 #6
    thank you people. The orgin of my mistake goes like this : Let T be a untiary operator and |a_i> (i=1,...n) a basis then the matrix elements satisfy,
    <a_i|T|a_k>=<a_k|T^{\dag}|a_i>*=<a_k|T^{-1}|a_i>*
    what a did not realize was that the matriz elementes in the basis |a_i> are <a_i|T|a_k> only if the basis is orthonomal.
     
  8. Jul 10, 2009 #7

    Fredrik

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    Oops, I didn't realize that myself. :redface:
     
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