It is not really a good strategy to follow the historical development of classical theory to quantum theory by “quantizing” a classical theory. Quantum theory is not only the change of a government; it is even the change of the constitution.
For example, a classical Lagrangian with the separation in a free kinetic and an interaction term is not a good starting point to solve a quantal bound-state problem. The ontology and interpretation of a quantum description are completely different from those of a classical one. A classical–quantum relationship and distinction may be found in the different representation structures of time and position operations. In classical mechanics, the time orbits are valued in position, \( t \mapsto \vec x\left( t \right) \), mass points have a position, and the concept of a “point particle” makes sense. In quantum mechanics, the time orbits are valued in a Hilbert space with probability amplitudes. Now, in quantum mechanics, there are also orbits of position (“of,” not “in”): The Schrödinger wave functions (“information catalogues”) for bound-state vectors are matrix elements of infinite-dimensional Hilbert representations of noncompact position operations \( t \mapsto \psi \left( \vec x \right) \). The concept of a mass point is very restricted. For example, it does not make sense to call an electron, e.g., “in” a hydrogen atom, a point particle, as there are no orbits in position.
KeywordsPrincipal Quantum Number Time Translation Position Representation Quantum Algebra Time Orbit
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