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Simple question concerning unitary transformation

  1. Apr 6, 2014 #1
    Is the transformation of an operator under INFINITESIMAL unitary transformation, U^-1AU or UAU^-1?? I saw that two books defined it differently?
     
  2. jcsd
  3. Apr 6, 2014 #2

    Matterwave

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    Remember that the unitary matrices form a group. So if U is a unitary matrix, then U^-1 is also a unitary matrix. Therefore, the distinction is superficial depending on which transformation you want to define as your U.
     
  4. Apr 6, 2014 #3
    So it doesn't matter?
     
  5. Apr 6, 2014 #4

    Matterwave

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    Well, it does matter a little bit. If you define U to be the unitary transformation that transforms kets |a>, then U^-1 will be the unitary transformation that makes the same transformation on the bras <b|. Using this, one should see that the matrix element <b|S|a> under a transformation goes to <b|U^-1SU|a> which means that the matrix S'=U^-1SU is the matrix that represents the operator in this new basis. If I defined U the opposite way, as U is the transformation on bras, then I get the other definition.
     
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