Niels Bohr and the other quantum theorists of the 1920'and1930's developed the notion that there is an entirely different physics that prevails at the microscopic quantum level than prevails at the macroscopic level and is described as Classical Physics.

Correspondence Principles

Bohr articulated the Correspondence Principle, that for quantum analysis to be valid its results must approach those of classical analysis as the energy level of the physical system increases without bound. Bohr left out that it is the spatial averages of the quantum theoretic results that must approach the classical results. In Schrödinger it is shown that the spatial averages of Schrödinger's time-independent equation do indeed asymptotically approach the results of classical analysis as the energy of the system increases without bound.

The Correspondence Principle was extended to include the limits of quantum analysis as Planck's Constant h→0 and as the mass of the increases without bound. In Planck's Constant and Mass it is shown that these cases also asymptotically approach the results of classical analysis

A Basic Model

Consider a particle of mass m moving in space subject to a potential function V(z), such that V(0)=0 and V(−z)=V(z) where z is the location coordinates of a point. The time-independent Schrödinger equation for the wave function φ(z) for this physical system is

(−h²/2m)∇²φ(z) + V(z)φ(z) = Eφ(z)

where h is Planck's Constant divided by 2π and E is the energy of the system. It can be reduced to

∇²φ = −μK(z)φ(z)

where μ= (2m/h²) and K(z)=E−V(z), the kinetic energy of the system as a function of particle location. For m→∞, μ→∞. Thus the analysis for m→∞ applies as well to h→0;.

The Helmholtz Equation

For a single dimension Schrödinger's equation is the generalized Helmholtz equation

d²φ/dx² = −H(x)φ(x)

If the coefficient function H(x) is a constant H₀ the solution is sinusoidal with a frequency of

ω = (H₀)^½

and a wavelength of

L = 2π/ω = 2π/(H₀)^½

The probability density is squared magnitude of the wave function φ of the form

φ²(x) = A·cos²(ωx)

Averaging the probability density over a half wave length of π/ω=π/H^½ eliminates the oscillations of cos²(ωx).

Here is the quantum analytic probability density function for a harmonic oscillator in which V(x)=½kx².

Thus, as the coefficient increases the oscillations become denser and denser, and consequently any degree of spatial or temporal averaging results in the appearance of the smooth pattern fitting classical analysis.

There is no transition from the dynamics of the quantum level to the classical level; it is the quantum dynamics all of the way out. It is just that the quantum fluctuation getter denser and denser and their time averaged observation give the appearance of satisfying classical dynamics.

Conclusions

Thus the fundamental physics is the quantum physics all the way out from the microscopic level through the macroscopic level but at some scale the time averaging inherent in observations produces results that are not significantly different from the results of classical analysis. There is no scale level at which quantum physics ceases to be valid and classical physics begins to prevail.

What Schrödinger's equation determines is a probability density function. The probrobability density function on the macro scale that it is associated with is the "time-spent" probrobability density function based upon the proportion of the time the physical system spends in various proportions of its trajectory.

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

where v(s) is the velocity at point s of the trajectory and T is the total time spent on the trajectory. But just as particle velocity can be translated into probability density, probability density can be translated into particle velocity. Thus if P_S is the probability density from quantum analysis using Schrödinger's equation the corresponding velocity |v_S(s)| at point s of the trajectory is proportional to 1/P_S(s). Since the quantum analytic probability density functions have oscillations like that of a harmonic oscillator the quantum analytic motion consists of sequences of slower-faster intervals. That sort of motion Schrödinger labeled zitterbewegung (trembling motion). So all motion involves zittering but at the macro level the zittering is so rapid it is not perceptible.