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Sakurai Chapter 1 Question 4c

  1. Feb 17, 2008 #1
    Evaluate [itex]\exp (i f(A))[/itex] in ket-bra form, where A is a Hermitian operator whose eigenvalues are known.

    [itex]\exp (i f(A)) = \exp(i f(\sum_i a_i \langle a_i |))[/itex]. I'm a little bit stuck on where to go from here. Is f supposed to be a matrix values function of a matrix variable or what?
     
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  3. Feb 17, 2008 #2

    George Jones

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    Assume [itex]f:\mathbb{R} \rightarrow \mathbb{R}[/itex] can be expressed using a power series expansion:

    [tex]f \left( x \right) = \sum_j c_j x^j.[/tex]

    In a standard abuse (should use a different symbol, maybe [itex]\hat{f}[/itex]) of notation, define

    [tex]f \left( A \right) = \sum_j c_j A^j.[/tex]

    Now use

    [tex]A = \sum_i a_i \left| a_i \right> \left< a_i \right|[/tex]

    with [itex]\left\{ \left| a_i \right> \right\}[/itex] chosen to be orthonormal.
     
  4. Feb 17, 2008 #3
    Interesting. I've seen this done for exp, but never thought about generalizing to arbitrary [itex]f: \mathbb{R} \to\mathbb{R}[/itex]. I wish physicists would be more careful with their notation sometimes. Thanks.
     
  5. Feb 17, 2008 #4

    George Jones

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    Sometimes it might be [itex]f: \mathbb{C} \rightarrow \mathbb{C}[/itex]. I think [itex]f: \mathbb{R} \rightarrow \mathbb{R}[/itex] is okay here, since the eigenvalues of [itex]A[/itex] are real.
     
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