Dear Mohamed: Thank you, what i meant with this question is how to explain QM using classical physics/trajectories, and how to reconcile both in the context of Uncertainty Principle....Anyway thank you for your answer

this is a long,  if things match then one is lucky...

in schr\"odinger's second paper on the subject he said that the eikonal approximation of the wave equation is the same as the classical path

Dear Prof. El Naschie: thank you, yes i am aware of the argument suggesting haussdorf dimension =2 for QM, but it seems Mark's comment is appropriate too: eikonal approximation may be useful to see if classical trajectories correspond in some ways to QM path. see: http://arxiv.org/pdf/1504.04285.pdf

Dear Mark and Prof. El Naschie: i just found an interesting paper by Orefice, Giovanelli and Ditto, suggesting that correct interpretation of classical trajectories lead to non-probabilistic version of wave mechanics. What do you think?

the answer to the latest is unlikely.   I have the schr\"odinger ref Quantisierung cels Eigenwert problem ann physik 79(1926)489-527.   - the eikonal approximation of a wave equation,  which defines the geometric optics approximation where wave packets follow definite trajectories,  can be obtain by the opposite procedure ....

Dear Victor,

Bohr introduced the Planck constant h, as quantum of action to quantify different levels of energy. Quickly after the Bohr hydrogen atom theory, Sommerfeld extended the model to include the elliptic orbits. "The importance of this extension can be attributed to the accuracy with which these equations, with relativistic corrections, give the fine structure of energy levels and lines of the hydrogen spectrum."

This classical approach just considers the plane of gravitation of the electron. This is surprising because the interactions take place in the space and the object as proton and electron have a volume. As a result the perpendicular direction has to be quantified. The hypothesis is to share into two equal parts, the quantum of action between the plane of rotation and the axis of rotation for the first state 1s of the hydrogen atom. The periodicity impose to the additional action on each degree of freedom to be equal to an integer number of quantum of action h. As a result we get the half quantum number well verified in magnetism and the two sub-shells.

In this approach the differential equations of the quantum mechanics describe the variation of the action along the trajectory.

You will find more development in my paper “Quantum state and Periodicity, Ann. Fondation Louis de Brolgie, 36, 137-157, (2011)” or in “Atom and Crystal, Memories on sixties years of research.  

Very interesting observation Victor: "Uncertainty Principle predicts that there is no trajectory of any particle' but experiments show otherwise".

Indeed, to my knowledge, quantum mechanics is not being used in high energy accelerators because qm is unable to account for the very real and precise trajectories followed by beams of moving electrons or protons, whose accelerations, velocities and very localised paths are controled by precisely tuned magnetic fields.

If I understood wrong, I would appreciate being clearly explained how qm is involved in high energy accelerator operation while accounting for the verified inertia of the particles and how this can be reconciled with the moving non local qm wave packet and how magnetic fields can interact with the moving wave packet?.

Please discuss the trajectories shown in arXiv:1701.01168v4

A.Orefice

Could you be more specific? What about the simple and exact results of

arXiv:1701.01168v4     ?

A.Orefice

Dear Christian,

sorry I interrupted your dialogue with André. I apologize.

In any case: my work has NOTHING to do with Bohm. Bohmian quantum trajectories are simple probability flow lines. My trajectories (as you may easily control) are exact dynamical solutions of Schroedinger's equations.

Dear Christian,

in a nutshell: it's easily seen that any Helmholtz-like equation (including Schroedinger's time-independent eq.),  once supplied with a reasonable set of launching conditions, provides a set of stationary trajectories (or rails), together with the time table of the (point) particle motion along them.

to me, the article is on classical trajectories compatible with quantum mechanics.

That does not mean that there are quantum trajectories. And in a nutshell, there are no quantum trajectories, as is well known. So, the author's statement "Uncertainty Principle predicts that there is notrajectory of any particle' but experiments show otherwise" is not true as experiments have not shown this.

Dear Ken,

"to me, the article is on classical trajectories compatible with quantum mechanics.That does not mean that there are quantum trajectories. And in a nutshell, there are no quantum trajectories, as is well known."

I believe that the difference between "compatibility" and "existence" is vanishingly small.

"Compatible" descriptions constitute the charm of classical-looking physics, and provide a powerful tool for intuition. "Existence" in itself is a metaphysical extrapolation.

To me, Adriano Orefice's answer is spot on.  Yes, without classical mechanics working, we would not have: Newtonian physics, Maxwellian physics, possibly not the thermodynamics we now have, and much of Einstein's two Relativity physics. By the way on Einstein is often criticized because he did not see how the particles would 'know' wheer to go; many others had similar problems. He actually helped quantum physics out with his  work related to Planc's black body radiation formula:, he was the first to consider a photon quantized, which led to a better understanding of Planc's formula.I think quite likely more too (because much of his work was and is world of Physics shaking..

To my humble opinion: quantum dynamics is UNABLE to describe the trajectories we find in the macroscopic world. The quantum mechanical theory relies on the Schrodinger equation, and according to this differential equation it is impossible to describe any trajectory.

An electron in a cathode ray tube follows a clear path, as soon as it escapes from the vicinity of the electrically charged plates it is more or less a free particle. The Schrodinger equation would dictate that from then on it is dispersed, what we see however is something different: it follows a clear trajectory. Quantum mechanics may work well for a selected number of problems on the atomic scale but fails to describe simple Newtonian phenomena. It is therefore an incomplete theory.

Should you look at my last paper in researchGate, "The Dynamics of Wave-Particle Duality" (February 2018), you would see that the contrary is true. Exact classical looking trajectories are given by the mono-energetic Schroedinger equation, as well as by any other Helmholtz-like equation. The striking difference from the classical trajectories is due to the presence of a "Wave Potential function", encoded in this family of equations.

If one neglects the Wave Potential, one obtains, of course, the eikonal approximation, i.e. Classical Dynamics.

The main classical request in order to determine the trajectory of a body is its Energy. Now, we have, in principle, two Schroedinger equations: an energy dependent and an energy independent one. The second one, of course, is only able to provide statistical motions. But the first one - just as it occurs in the classical case - is perfectly able to give the correct set of exact trajectories, together with the time-table of the motion along them. This is what we say and show in our papers.