The Copenhagen Interpretation of quantum theory maintains that particles exist only as probability distributions over their allowed quantum states. What will be shown below is that the solutions to the time-independent Schrödinger's equation correspond to the time-spent probability distributions. The time-spent probability density distribution is based the proportion of the time the particle spends at various locations. Such time-spent probability distributions are entirely consistent with the continued physical existence of particles.

Consider the following two situations in mathematical physics.

• A charge is distributed over some geometric region R and the charge density is known. The charge could be gravitational mass, electrical charge, magnetic charge or nucleonic charge (the charge associated with the nucleonic interaction force). The charge density is given by a function ρ(Z), where Z is a position vector. This is the static situation.
• A point particle of charge q traverses the same geometric region R. A probability density function is determined based on the proportion of the time the particle spends at various locations. This is its time-spent probability distribution. The expected value of the force at any location can be computed based upon that probability density function. This is the dynamic situation.

Suppose the function giving the force dF on a unit charge at position Z due to a charge of dq located at position ζ is given by

dF = F(s)dq

where s is the distance between point Z and point ζ. This distance is given by |Z−ζ|. The increment of charge is given by ρ(ζ)dR.

The intensity of the force vector G of the field at Z is then given by

G(Z) = ∫_R F(|Z−ζ|)ρ(ζ)dR

The intensity at Z is constant so the expected value of G, E{ }, is just G; i.e.,

E{G(Z)} = G(Z) = ∫_R F(|Z−ζ|)ρ(ζ)dR

On the other hand in the dynamic situation the field intensity at Z fluctuates. When the charge Q is at point ζ the force on a unit charge at point Z is given by

H(t) = F(|Z−ζ|)Q

Let P(ζ) be the probability density function representing the proportion of the time the charge spends at point ζ. For a particle traveling along a path at a velocity of v the probability density function is given by

P(ζ) = 1/(T|v(ζ|)

where T is the total time required to traverse the path.

The expected value of the vector of force field intensity H is then given by

E{H(Z)} = ∫_R F(|Z−ζ|)QP(ζ)dR

The Correspondence

Note that if the charge density ρ(ζ) is given by P(ζ)Q then the expected value for the dynamic case is mathematically identical to the static case problem.

This correspondence points up the significance of the time-spent probability density function to mechanics. It is as significant as the spatial distribution of any charge.

The expected value in this case is just the same as the time-averaged value.

Periodicity of Atomic
and Subatomic Motion

The rate of rotation of an electron in a hydrogen atom, according to the Bohr model, is easily determined. The electrostatic force between an electron and a proton has to be balanced by the centrifugal force on the electron; i.e.,

Γe²/r² = mrω²

where Γ is a constant, e is the electrical charge of the electron and the proton, m is the mass of the electron and r is the orbit radius for the electron. Therefore

ω² = Γe²(m/r³)

The frequency ν is equal to ω/(2π). This works out to be 6.6×10¹⁵ times per second; i.e., 6.6 quadrillion times per second. Any observation will involve a time-average and at the above frequency the observation will be equal to the expected value. Thus the observed world is the world of the time-averaged dynamic appearances of physical systems. In the case of electrons this would tubular elliptical rings.

At the nuclear level Aage Bohr and Ben Motellson found that rotations satisfied the equation

E_rot = (h²/(2J))I(I+1)

where E_rot is the energy of rotation, h is Planck's constant divided by 2π, J is the moment of inertia and I is an integer. This is equivalent to the angular of momentum L of the system being equal to

L = h(I(I+1))^½

Since L is equal to Jω

ω = h(I(I+1))^½/J

The smaller the scale of a system the smaller is its moment of inertia and thus the more rapid is its rate of rotation. The moment of inertia of a system is directly proportional to the mass of the system and to the square of its scale. The ratio of the radius of an atom to that of its nucleus is on the order of 10⁵ whereas the ratio of the mass of the outer band of electrons of an atom to that of the nucleus is on the order of 10⁻³. Thus the moment of inertia of the electrons in an atom is about 10⁷ times greater than the moment of inertia of the nucleus. Thus nuclear particles are involved in rotations at rates about ten million times higher than those of the electrons. Therefore the time-averaged observations for atomic and subatomic systems are essentially the same as their static equivalents. In other words, it is impossible to distinguish between an atomic or subatomic particle engaged in periodic motion and that the static equivalent. When quantum analysis relies upon the time-independent Schrödinger equation the solution corresponds to the static equivalent of the dynamic system. It gets the blurred disk of the rapidly rotating fan and not the rotating fan itself.

The Correspondence Principle

The Copenhagen Interpretation is largely due to Niels Bohr and Werner Heisenberg. But Bohr also articulated the Correspondence Principle. He said that the validity of classical physics was well established so for a piece of quantum theoretic analysis to be valid its limit when scaled up to the macro level had to be compatible with the classical analysis. It is very important to note that the observable world at the macro level involved averaging over time and space. Physical systems are not observed at instants because no energy can be transferred at an instant. Likewise there can be no observations made at a point in space. Therefore for a quantum analysis to be compatible with the classical analysis at the macro level it must not only be scaled up but also averaged over time or space.

For an example, consider a harmonic oscillator; i.e., a physical system in which the restoring force on a particle is proportional to its deviation from equilibrium. The graph below shows the probability density function for a harmonic oscillator with a principal quantum number of 60.

The heavy line is the probability density function for a classical harmonic oscillator. That probability density is proportional to the reciprocal of the speed of the particle. As can be seen that heavy line is roughly the spatial average of the probability density function derived from the solution of Schrödinger's equation for a harmonic oscillator.

As the energy of the quantum harmonic oscillator increases fluctuations in probability density become more dense and hence no matter how short the interval over which they are averaged there will be some energy level at which the average is equal to the classical time-spent probability density function.

There is no question but this analysis applies to one dimensional physical system. For two and three dimensional systems conventional quantum analysis relies on the Separation-of-Variables assumption. This assumption is incompatible with particleness and not consistent with the Correspondence Principle. Thus it is not valid analysis.

Extensions

The above derivation which was for a single particle applies equally as well to multiple particles. The multiple particles may be independent or interlinked. As a particular case it would apply to structures that rotate. It also applies to particles with a spatially distributed charge.

The analysis was carried out using a force vector. It equally applies to a scalar, other vector or a tensor. It applies to any quantity generated throughout space due the generic charge.

Conclusions

A particle with a generic charge of Q which traverses a periodic path with a spent-time probability density function of P(ζ) has the same expected value effect as a static charge distribution density equal to P(ζ)Q.

Thus when the time-independent Schrödinger equation is solved for a particle in motion what is obtained is the static charge distribution density equal to P(ζ)Q. The Copenhagen Interpretation interprets this as though it applied to the dynamic situation. It is in effect the blurred translucent disk of a rapidly rotating fan and it tells one nothing about the physical nature of the fan. The fan has a continuous physical existence despite the blurriness of its image while rotating.

This also applies to multiple particle systems whether they are independent particles or linked together in a structure. It also applies with a little modification to particles with spatially distributed charges.