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QM: Infinitesimal Generator for Scale Transformation

  1. Nov 17, 2008 #1
    1. The problem statement, all variables and given/known data

    The scale transformation is a continuous transformation which acts on a function f(x) according to

    [tex]D_{s}[/tex]f(x) = f(sx)

    where s is a real number. There is a continuous family of such transformations, including the identity transformation corresponding to s = 1. Calculate the infinitesimal generator for scale transformation in terms of familiar quantum operators.

    2. Relevant equations

    3. The attempt at a solution

    This question was in a foreign language to me. I don't recall ever hearing of such a thing as an 'infinitesimal generator' in my quantum mechanics course, so I have absolutely NO clue what this question means or how to do it. Any guidance is much appreciated.
  2. jcsd
  3. Nov 18, 2008 #2
    To demonstrate how we found the generator, lets consider the case of the translation operator

    [tex] T(x)[f(x)] = f(x-a) [/tex] (translation by a)

    For an infinitesimal translation [tex]\delta[/tex]

    [tex] T(\delta)[f(x)] = f(x-\delta) = f(x) - \delta \frac{df(x)}{dx} = \left( 1 - \delta \frac{d}{dx} \right) f(x) = \left( 1 - i\delta\frac{p}{\hbar} \right) f(x) [/tex]

    In this case, [tex]\frac{p}{\hbar}[/tex] (or just p) is called the generator of the infinitesimal translation.

    Maybe you can proceed accordingly for the scale trasformation?

    For further information you may consider any graduate-level textbooks for QM. (e.g. Sakurai)
    Last edited: Nov 18, 2008
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