According to the Copenhagen Interpretation of Quantum Theory subatomic particles such as electrons generally do not have a physical presence but instead exist only as probability distributions over sets of allowable states. This webpage presents an argument that the Copenhagen Interpretation is a misinterpretation of the nature of the wave function in the Schroedinger's equation for a physical system and hence a misinterpretation of the nature of quantum reality.

When Erwin Schrödinger revolutionized quantum theory with the brilliant notion that quantization was a matter of discrete eigenvalues of differential equations rather than integral numbers per se he left the wave function variable of his equations unspecified. Max Born of the University of Götingen speculated that the wave function is such that its squared value is equal to a probability density function. When he communicated this speculation to Niels Bohr in Copenhagen the reply was that they had never considered it to be anything else. Erwin Schrödinger himself disagreed with this interpretation, as also did Albert Einstein. Nevertheless Niels Bohr and the Copenhagen Interpretation prevailed.

There is evidence for and evidence against the Copenhagen Interpretation. There was a recent convention of quantum theorists. They were queried as to which interpretation of quantum mechanics they favored. Only 42 percent said the Copenhagen Interpretation. This was more than any other interpretation so the Copenhagen Interpretation can be considered the dominant philosophy, but still it meant than 58 percent of the respondents thought the evidence against the Copenhagen Interpretation was more significant than the evidence for it. Yet the evidence for the Copenhagen Interpretation is presented by its true believers as settling the issue definitively.

One alternative to the Copenhagen Interpretation was proposed by Erwin Schrödinger. He suggested that particles are in constant motion but going more slowly through some intervals and faster through other intervals. He called this motion zitterbewegung (trembling motion). The motion is so fast that the particle appears to be in all of allowable states simultaneously.

Probability Density Functions

It is generally accepted that the squared value of the wave function solution to the Schroedinger equation for a system is a probability density function. The problem is the nature of this probability density function. The Copenhagen Interpretation takes this probability density function to be disembodied probabilities that shows the probability that the system will be in any one of the various allowable states when subjected to measurement. Thus subatomic particles such as electrons, protons and neutrons do not have a physical existence until they subjected to measurement. There are perplexing problems with what happens to the mass and charge when a particle exists only as a probability density function.

Here it is argued that the probability density function from Schroedinger's time-independent equation when averaged over time is just the proportion of the time spent in the allowable states given the dynamics of the system. There are no metaphysical perplexities involved with this probability density function. The time-averaged effect of a particle executing a periodic path is the same as if its generic charge (gravitational mass or electric charge) is spread over its path in proportion to its time spent probability density distribution. See Static and Dynamic.

The Static versus Dynamic
Appearances of a System

First consider a propeller standing still and rapidly rotating.

The still picture is the static appearance of the propeller and the blurred disk of the rapidly rotating propeller is its dynamic appearance.

The Copenhagen Interpretation asserts that a system is not in any of its allowable states or alternatively that it is in all of its allowable states simultaneously. Furthermore a particle does not have a trajectory involving a definite location and velocity as a function of time. This is because of the Uncertainty Principle.

Notice how well the dynamic appearance of the propeller fits the Copenhagen Interpretation. The propeller seems to be smeared over the disk and nowhere and everywhere at once. If one can only observe the dynamic appearance it seems to be unchanged over time. Therefore there is no trajectory for the blurred disk.

The analogy of the blurred disk of the propeller may be extended a bit further. Suppose a heavy object is plunged into the blurry disk of the rotating propeller. The result: CLANG! The propeller which had no perceived existence before is miraculously solidly there. The blurred disk of the rotating propeller has, a la the Copenhagen Interpretation, collapsed into a spiked probability density distribution.

Thus the Copenhagen Interpretation treats the dynamic appearance of the rotating propeller as though it is the static appearance of some object.

Time-Spent
Probability Distributions

The time-spent probability distribution for a particle executing periodic motion is simply 1/(T|v|) where |v| is the absolute value of the velocity of the particle and T is the time period of the motion. In notable cases, such as for a harmonic oscillator, the time-spent probability density distributions for the location and momentum of a particle satisfy the Uncertainty Principle. In the case of the rapidly rotating propeller they do not because it is presumed that the propeller can be rotating at constant (zero uncertainty) velocity. However in general velocity has a time-spent probability distribution given by

dt = dv/|dv/dt| = dv/|a|

where |a| is the absolute value of acceleration. The time-spent probability density distribution for velocity P_v(x) is then

P_v(x) = 1/(T|a|)

where T is the time period of periodic motion of the particle.

The analysis on this topic is given in Uncertainty. A limited version is given below for the simple case of a harmonic oscillator.

A Harmonic Oscillator

A mass m subject to a restoring force proportional to its deviation x from equilibrium executes harmonic motion. Its energy function is

E = ½mv² + ½kx²

The coefficient k is called the spring constant of the oscillator.

The Hamiltonian form of the total energy is

H = ½p²/m + ½kx²

The solution for the oscillator can be represented as

½mv² = K(x)
where
K(x) = E − ½kx²
is the kinetic energy
of the particle

Therefore

v(x) = [2K(x)/m]^½

The time-spent probability density function P_tsp(x) is then

P_tsp(x) = [K(x)]^−½/T

where T = ∫K(z)]^−½dz and the integration is over the path of the particle. The solution is independent of the mass m of the particle.

From the equation for v(x) the acceleration of the particle is found to be

a(x) = dv/dt = (dv/dx)(dx/dt) = (dv/dx)v
= [(1/2m)^½K(x)^−½](dK/dx)v
= [(1/2m)^½K(x)^−½][2K(x)/m]^½](dK/dx)
= (1/m)(dK/dx) = (dK/dx)/m

and thus the probability density for velocity at x is inversely proportional to (dK(x)/dx). In the vicinity of K(x) and U(x) being an extreme their derivatives with respect to x are zero so the dynamics of particle motion are determined by their second derivatives.

For a harmonic oscillator the second derivative of the energy function divided by its mass is equal to the square of its oscillation frequency.

The Quantum Mechanical Solutions

The Hamiltonian operator for a harmonic oscillator is

H = −h²/(2m)(∂²/∂x²) + ½kx²

The time-independent Schroedinger equation for the oscillator is

Hφ = Eφ
which is the same as
−h²/(2m)(∂²φ/∂x²) + ½kx²φ = Eφ
which reduces to
the Helmholtz equation
(∂²φ/∂x²) = −(2m/h²)(E − ½kx²)φ

The solution to this equation has rapidly oscillations of φ², which is the probability density, between maxima and minima of zero.

Here is an example of such a solution.

The heavy line is the time-spent probability density function for the harmonic oscillator of the same energy. As can be seen a spatial average of the probability density from Schroedinger's equation for this case is equal to the time-spent probability density.

The proof of this proposition for a harmonic oscillator with a more general potential energy function is given in Particle Averaging.

The higher the level of energy E, the more dense are the oscillations. So for any spatial averaging there is a level energy for a quantum mechanical harmonic oscillator such that the spatial average is equal to the time-spent probability density distribution for a classical harmonic oscillator.

The Necessity of an Observation
Occurring Over an Interval of Time

At an instant of time zero energy is transferred and thus no observation can be achieved. Therefore any observation represents a time average over some non-zero interval of time. A time average of repetitive motion is equivalent to a spatial average.

P = ∫Pdt = ∫(P/(|dx/dt|))dx = ∫(P/|v|)dx

The only reality for a physical system that can be observed is a time average. With rotations at the nuclear level occurring at rates of billions of billions of times per second time averages are all that can be discerned.

If the probability density function from the time independent Schroedinger equation corresponds to a time spent probability distribution the Schroedinger probability density can be converted into a an absolute value of the velocity of a particle at the quantum level. What emerges then is a picture of quantum motion as a sequence of slowly-fast-slowly…. This is in contrast to the Copenhagen picture of a particle resting in an allowable state and then randomly jumping instantaneously to another allowed state. The particle does not rest motionless in an allowed state; an allowed state is just the vicinity through which the particle travels relatively slowly. There are not instantaneous jumps but instead vicinities through which the particle travels relatively quickly. When Bohr insisted on the notion of quantum jumps, Schroedinger replied,

" If we have to go on with these damned quantum jumps,
then I'm sorry that I ever got involved."

Schroedinger later formulated the concept of Zitterbewegung (trembling motion) in the movement of particles at the quantum level.

The Correspondence Principle

Niels Bohr articulated the Correspondence Principle. He noted that the classical analysis of macroscopic physical systems had been thoroughly validated. Therefore, for the microscopic quantum analysis to be valid it has to such that the limit of its results as parameters such energy increase to macroscopic levels is consistent with classical analysis. But the solution to the time-independent Schroedinger's equation is a probability density function and its limit at the macroscopic level is also a probability density function. The relevant probability density function is a time averaged one. The only probability density function that a classical macroscopic system has is its time-spent probability density function. Therefore the time averaged version of the limit of the solution to the time independent Schroedinger equation for a system is its time spent probability density function. Thus the probability density function derived from the solution to the time independent Schroedinger equation is a time spent probability density function at the quantum level. The time-spent probability density distribution corresponds to the dynamic appearance of the system.

The estimated rates of rotation and revolution of particles and sub-atomic systems are fantastically high, on the order of billions of billions of times per second. In ordinary observation all that can be seen is the dynamic appearances of the systems. For more on this see Nuclear Rotation.

The Nature of the Probability Density
Distributions in Quantum Theory

The fact that the probability density distribution that derives from the Schroedinger equation reduces to the time-spent probability density distribution of classical analysis strongly suggest that it is itself in the nature of a time-spent probability density distribution. This means that if P(x) is the probability density from the Schroedinger equation then the velocity in the quantum level is given by :

|v(x)| = 1/(TP(x))

where T is the time period of the periodic motion of the system. This gives quantum motion as involving sequences of slow-fast-slow-fast …. The regions of slow motion are what are designated as allowable states. There are no quantum jumps as asserted in the Copenhagen Interpretation, only periods of relatively rapid motion.

The point that the Copenhagen Interpretation is a misinterpretation of quantum reality can be reduced a simple proposition. See Quantum Reality.

Conclusions

When the probability density distribution which comes from the solution to the time-independent Schroedinger equation for a physical system is averaged over time or space the rapid oscillations are eliminated and what is left is just the time-spent probability distribution for the system. Quantum mechanics based on Schroedinger's equation is not wrong; it just only applies to the dynamic appearance of a system and has nothing to say about the static nature of the system. Thus there is no reason to doubt the continuous material existence of subatomic particles.

The nature of physical reality with respect to quantum theory and classical mechanics can be summed up as in the following diagram.

The Correspondence Principle between quantum theory and classical mechanics works as follows. The probability density distribution from Schroedinger's equation involves more and more rapid fluctuations the greater the energy level. When those fluctuations are averaged over time and/or space the result is the time-spent probability density distribution arising from the classical motion.

For more on this topic see Helmholtz.