QM: Infinitesimal Generator for Scale Transformation

[brooke1525]

Homework Statement

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

D_{s}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.

Homework Equations

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.

 

[weejee]

To demonstrate how we found the generator, let's consider the case of the translation operator

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

For an infinitesimal translation \delta

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)

In this case, \frac{p}{\hbar} (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)