After the great success of Niels Bohr's solar system model of the hydrogen atom in 1913 and its extension to hydrogen-like atoms there was not much progress in quantum physics until the 1920's. In Copenhagen there developed a school of quantum physics consisting mainly of Niels Bohr and Werner Heisenberg from Germany.

Heisenberg developed a quantum model to explain the spectra of elements with multiple valence electrons, but it involved quantum numbers with half-unit values. Bohr insisted that quantum numbers had be integral and Heisenberg stopped any further work on that model and went on to develop a model in In fact half-unit quantum numbers could be justified on the basis of an electron being partially shielded from the positive charge of the nucleus by other elecrons in the same shell.

And then seemingly out of nowhere came a set of articles by a physicist from Vienna, Austria named Erwin Schrödinger. Schrödinger's system was based on solutions to partial differential equations Initially Schrödinger called his system "Undulatory Mchanis" but ultimately it became known as "Wave Mechanic." The quantum numbers of a physical system are, according to Schrödinger, the eigenvalues of its partial differential equations. These may or may not be integral numbers. Heisenberg wrote disparagingly of Schrödinger's theory but Schrödinger wrote an article showing that Wave Mechanics is equivalent to Matrix Mechanics.

Max Born of Gottingen suggested to Bohr that the solution to a Schrödinger equation is a function, ultimately known as the wave function, whose squared value is the probability density for the physical system. Bohr replied from Copenhagen that they never considered it to be anything else. Schrödinger did not agree and neither did Albert Einstein. Born and the Copenhagen people were right but they went on to make a disasterous and silly error. They asserted that a material particle could not have a probability density distribution. In fact any material particle in motion has a probability density distribution based upon the proportions of the time it spends in the various parts of its trajectory. Schrödinger's equation for a system relates to its dynamic appearance rather than to its static nature.

Let s denote the path position of a particle and v(s) its velocity. Then the probability density P(s) at point s of the path is given by

P(s) = 1/(T|v(s))

where T is the total time spent in the trajectory.

This probability density distribution can appropriately be called its, time-spent probability density distribution. In effect it is

P(s) = dt/T
but v(s) = (ds/dt)
so dt=1/|v(s)|

So the Copenhagen School not only missed the Time-Spent probability density distributions which do exist but hypothesized some disembodied probability density distributions which do not exist.

Just as v(s) = (ds/dt) so does acceleration a(s) = (dv/dt) = (d²s/dt²) and hence

dt = 1/(|a(s)|)

So a material particle in motion automatically has a time-spent probability density distribution and it does not have any other.

It is shown mathematically Elsewhere that the solution of the time-independent Schrödinger equation is always asymptotically equal to the solution corresponding to the time-spent probability density distribution for the system.

The Separation of
Variables Assumption

The conventional theory makes use of the Separation of Variables assumption. If the coordinate system is (x, y, z) then the assumption is that the wave function ψ(x, y, z} is the product of three separate functions; i.e.,

ψ(x, y, z} = X(x)Y(y)Z(z)

This must be eliminated from the analysis because it is incompatible with particleness.

Correspondence Principle

Niels Bohr stated in the 1920's that the validity of classical physics is well established so a quantum theory can only be valid if it coincides with classical physics in the limit as the energy of the system increases without bound. Bohr neglected to state that it is the spatial average of the results of the quantum analysis which in the limit must coincide with classical analyis.

This illustrated by the case a harmonic oscillator. (A harmonic oscillator is a mass on a spring.)

The light line is the probability density from the quantum analysis and the heavy one is the probability density from classical analysis. As can be seen it is the spatial average of the quantum results that approximates the classical results. The principal quantum number, which corresonds to energy, for this case is 30. As this number increases the density of the oscillations increases rather than the general level of their spatial average. .

The Nature of Motion
at the Quantum Level

Just as the velocity of a particle can converted into a time-spent probability density distribution a probability density distribution can be converted into the velocity of the particle over its trajectory. The velocity in classical analysis is smooth; at the quantum level motion is a sequence of slower and then faster motion. It is what Schrödinger called zitterbewungen (trembling motion).

The Copenhagen School interpreted the regions of slower motion as the allowed states for the particle and the regions of faster motion as instantaneous jumps between allowed states.

Uncertainty Principle

According to the Uncertainty Principle the product of the standard deviations of location and momentum for a particle must equahe Uncertainty Principlel or exceed Planck's Constant divided by 4π. For simple models for which explicit mathematical solutions are known there is no problem in satisfying the Uncertainty Principle. In fact for the harmonic oscillator the product is four times the required level.

Conclusions

There are a number of corrections that should be made to the Copenhagen Interpretation of quantum physics.

1. There never was any basis for the Copenhagen School's assertion that generally particles do not have a material existence because they have a probability density distribution. All material particles in motion have probability density distribution based upon the proportion of the time they spend in the various parts of their trajectories.
2. Particles' time-spent probability density distributions can satisfy the Uncertainty Principle without losing their materiality.
3. Particles do not simultameous exist in all of their allowable states nor jump instantaneously between alloable states. Instead at the quantum level they move through a trajectory in a sequence of fast-slow-fast motions.