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Question from Dirac's Principles of QM

  1. Apr 29, 2007 #1
    I have the fourth edition of Dirac's Principles of QM. I have a question concerning the equation in the middle of page 102 (between eqns 64 and 65 in section 25.)
    [tex]\lim_{\delta x \rightarrow 0}(De^{i\gamma} - 1)/\delta x = \lim_{\delta x \rightarrow 0}(D - 1 + i \gamma)/\delta x[/tex]
    If you make the substitution [itex]e^{i\gamma} \simeq 1 + i \gamma[/itex] then it seems to me you should get
    [tex]\lim_{\delta x \rightarrow 0}(De^{i\gamma} - 1)/\delta x = \lim_{\delta x \rightarrow 0}(D - 1 + i \gamma D)/\delta x[/tex]
    What am I missing?
  2. jcsd
  3. Apr 29, 2007 #2


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    You aren't missing anything. Is it a typo?
  4. Apr 30, 2007 #3
    I doubt it. The conclusion he correctly draws from this equation is that the displacement operator is indeterminate by an arbitrary additive pure imaginary number:
    [tex]ia_x = \lim_{\delta x \rightarrow 0}i \gamma/\delta x[/tex]
    If it were a typo, then the the right hand side would also need to be operated on by D and then would not be a number. Is there some reason that [itex]D(i\gamma) = i \gamma[/itex]?
  5. Apr 30, 2007 #4
    Never mind, I finally figured it out. Thanks for your help. The solution is that
    [tex]\lim_{\delta x \rightarrow 0}D = 1[/tex]
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