What if Correspondence principle in QM happening in classical sense?

Question

I assume a simple set up (hope to generalize it later)... suppose there is a position pdf (need not be position but any) (probability density function) which is the magnitude of a $\Psi(x)$ predicted by QM (lets call it P_quantum) and let there be a classical prediction pdf (P_classic), according to CP, for large quantum numbers, P_quantum agree with P_classic in an average sense, but my question is for suppose, what if it agress to converge to P_classic(x) in pointwise convergence, while still explaining all the microscopic things like hydrogen atom..blah blah, what would that mean philosophically? for example see Particle in a 1-D box and the correspondence principle

For supose What if classical mechanics originate from QM in a classical sense ( it currently does in only statistical sense) but still explaining all the microscopic things like Hydrogen atoms, atomic spectra etc..., what would that mean?

Right now they converge only in statistical sense, hence they say you need not converge pointwise as anyway you lack measuring device at that microscopic level for macroscopic things. What if they did converge pointwise, would that mean the lack of measuring device argument disappear from the QM scene?

Answer 1

People were not sadomasochists to invent quantum mechanics and the riddles it carries with it in order to set difficult problems to students.

They tried this method of extending classical physics laws to the microcosm and it did not work.

It gave infinities for example:

a) in the black body radiation the classical solution gives infinities at the ultraviolet, not observed in nature:

look at the classical curve.

b) the classical atoms, electrons in classical orbits about the nucleus would radiate down and fall in, so no atomic structure could be built. The history that leads to the orbitals is long and classical extrapolations to the microcosm could not work.

As experiments progressed more and more experimental data from the microcosm showed that a new framework/model was necessary to describe the observations. Extrapolations from the classical did not work.

Anybody looking over the same data will come to the same conclusion : classical physics is limited to certain phase spaces and dimensions.

Right now they converge only in statistical sense, hence they say you need not converge pointwise as anyway you lack measuring device at that microscopic level for macroscopic things. What if they did converge pointwise, would that mean the lack of measuring device argument disappear from the QM scene?

Converging pointwise as you call it would mean that classical physics would extent to the microcosm and it is an experimental fact it does not. The argument about measuring devises is useful the other way around. "Why we do not see quantum mechanical effects macroscopically ( except in very special cases)". It is the logical tool to show the convergence of the underlying nature which is quantum mechanical, to the everyday observable world which is described classically.