In Franz Schwabl's QM book the idea of active transformation has been put in this way: "transformation of a state Z to Z' and view Z' from the same reference frame".The statement follows by the argument that the state which arises through the transformation \(\Lambda^{-1}\) is given as \(\psi\ '(\ x) =\psi\ (\Lambda^{-1}\ x)\) \(\psi\ (\ x)\) has been actively moved from \(\ x \) to \(\ x'\) where \(\ x'=\Lambda^{-1}\ x\). That is, here x' denotes a point in the same reference gotten from \(\ x'=\Lambda^{-1}\ x\) In the same reference frame \(\psi\rightarrow\psi\ '\); But how can that \(\psi\ '(\ x)\) be equal to \(\psi\ (\Lambda^{-1}\ x)\)? I know that active transformation is expressed in literature most commonly as \(\psi\ '(\ x) =\psi(\ x')\),[as opposed to passive transformation \(\psi\ '(\ x') =\psi\ (\ x)\)---here primed co-ordinate means a new primed co-ordinate system],however,I am struggling a bit with the definition...
The scenario is clearer if the following viewpoint is taken: You take the wavefunction \(\psi\).Give some spatial translation to \(\psi\),w.r.t. the co-ordinate system where you are.This retains the form (dependence on the variable) of the wave function the same.\(\psi\) remains \(\psi\),but the location of its peak etc. is shifted up to a constant distance.Thus, \( \psi(\ x) =\psi\ (\ x\ +\ a) \) It is a bit obscure to see if you first shift the point,and according to that,shift your wavefunction. From the book I saw,they depicted the transformation as rotation in that frame.Obviously,that is a bit difficult to see;because,it looks like \(\psi\rightarrow\psi'\) What do people think
You've really got nothing to worry about since you've essentially described things correctly: an active transformation is a physical transformation (translation, rotation etc.) of a system, while a passive transformation is the same system redefined in terms of a transformed coordinate system. There's no real mathematical difference between the two. You need the same mathematics to describe an active transformation as to describe the inverse passive transformation. Well technically the notation you use here specifies a periodicity condition. A translated function is a new function so you should really write something like: \(\psi^{\prime}(x) \,=\, \psi(x - a) \quad \forall x\)or as a function composition: \(\psi^{\prime} \,=\, \psi \,\circ\, \Lambda^{-1}\) This information isn't contained within the notation. If you're thinking of \(\psi^{\prime}\) as a physically translated wave function, you're performing an active transformation. If you're thinking of \(\psi^{\prime}\) as essentially the same thing as \(\psi\) just redefined in terms of new coordinates then you're performing a passive transformation. What the notation "means" is determined by you and the problem you're considering.
Yea...for an operator \(\ T_\ a\),which shifts by a constant distance a, \(\psi\ (\ x)\rightarrow\psi\ (\ x-\ a)\) Even though the two transformations are the same,the expressions are not identical: I found it confusing with many books which does not explicitly state these definitions...
