The
#paperofTheDay is 99 years old: "Winkelvariablen und kanonische Transformationen in der Undulationsmechanik" by Fritz London 1927.
If classical
#physics is formulated in terms of Hamilton's canonical equations, the variables (p,q) initially are position and momentum, but they can be transformed in various ways by what is called "canonical transformations". Usually, one tries to find "action-angle variables", which are such that the transformed momentum becomes a constant (and the corresponding transformed position is then a linear, or periodic, function of time, hence an "angle"). Typical simple examples of this kind are rotation symmetric problems, where the action-angle variables are spherical coordinates, and the transformed (and conserved) momentum is angular momentum. In particular, the classical action S is the generator of a canonical transformation.
The present paper is one of the very early papers of
#quantumMechanics , and it deals with the question how canonical transformations can be realized in quantum mechanics. An important special case is the transformation generated by e^(i/h S). There are two conceptual challenges: Firstly, in Schrödinger's formulation, instead of a "particle position", which is a number, one has a "wave function", which is a function of the spacial coordinate. So, the canonical transformations operate on an infinite dimensional Hilbert space of functions, instead of an ordinary vector space. Secondly, the momentum q becomes the differential operator i d/dq, and the canonical transformation must transform this object. Since these concepts were very new and foreign at that time, the paper includes many clarifying comments.
https://link.springer.com/article/10.1007/BF01400361
Jack C.M
Sean Lynch
Mille e Una Avventura
Paul Balduf
Scientific Frontline
Knowledge Zone
