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: QM Series Representation of Bras, Dirac Brackets

  1. Nov 17, 2008 #1
    URGENT: QM Series Representation of Bras, Dirac Brackets

    1. The problem statement, all variables and given/known data

    Suppose the kets |n> form a complete orthonormal set. Let |s> and |s'> be two arbitrary kets, with representation

    |s> = [tex]\sum c_n|n>[/tex]

    |s'> = [tex]\sum c'_n|n>[/tex]

    Let A be the operator

    A = |s'><s|

    a) Give the series representation of the bras <s| and <s'|.
    b) Express [tex]c_n and c'_n[/tex] in terms of Dirac brackets.
    c) Calculate in terms of Dirac brackets the matrix elements of A[tex]^+[/tex], the Hermitian conjugate of A.

    2. Relevant equations



    3. The attempt at a solution

    a) I know that the bras corresponding to the kets are row vectors where we take the complex conjugate of each element. That's all I've got.

    b) c_n = <s|n> and c'_n = <s'|n>
    Of course to go any further, I need the answer to part a.

    c)To find the matrix elements of A (NOT [tex]A^+[/tex], I don't know how to do that exactly) we do the following for the element in row m, column n:

    [tex]A_{mn}[/tex] = <m|s'><s|n>

    What is the Hermitian conjugate of A so that I can use the same method? Once I know the matrix for A+, can I tell if A is Hermitian without finding the matrix for A?
     
  2. jcsd
  3. Nov 17, 2008 #2
    Re: URGENT: QM Series Representation of Bras, Dirac Brackets

    UPDATE:

    For c) Can I just find the matrix for A and take its transpose and the complex conjugate of each element to get the matrix of the Hermitian conjugate of A? If the transpose is the same as the original, then I know A is Hermitian.
     
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