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A Reinterpretation of the Copenhagen
Interpretation of Quantum Theory

Historical Background

In the early 1920's Werner Heisenberg in Copenhagen under the guidance of the venerable Niels Bohr and Max Born and Pascual Jordan of Göttingen University were developing the New Quantum Theory of physics. Heisenberg, Born and Jordan were in their early 20's, the wunderkinder of physics. By 1925 Heisenberg had developed Matrix Mechanics, a marvelous intellectual achievement based upon infinite square matrices. Then in 1926 the Austrian physicist, Erwin Schrödinger, in six journal articles established Wave Mechanics based upon partial differential equations. The wunderkinder of quantum theory were not impressed by Schrödinger, an old man in his late thirties without any previous work in quantum theory and Heisenberg made some disparaging remarks about Wave Mechanics. But Schrödinger produced an article establishing that Wave Mechanics and Matrix Mechanics were equivalent. Wave Mechanics was easier to use and became the dominant approach to quantum theory.

Schrödinger's field had been optics and he had been prompted to start to work in quantum theory by the work of Louis de Broglie which asserted that particles have a wave aspect just as radiation phenomena have a particle aspect. Schrödinger's equations involved an unspecified variable which was called the wave function. He thought that it would have an interpretation similar to such variables involved in optics. However Niels Bohr and the wunderkinder had a different interpretation. Max Born at Göttingen University wrote to Bohr suggesting that the squared magnitude of the wave function in Schrödinger's equation was a probability density function. Bohr replied that he and the other physicists with him in Copenhagen had never considered any other interpretation of the wave function. This interpretation of the wave function became part of what was known as the Copenhagen Interpretation. Erwin Schrödinger did not agree with this interpretation. Bohr had a predelection to emphasize the puzzling aspects of quantum theory. He said something to the effect of:

If you are not shocked by the nature of quantum theory then you do not understand it.

The Copenhagen Interpretation came to mean, among other things, that

Some Simple Terminology

The static appearance of an object is its appearance when it is not moving. The dynamic appearance is that of an object moving so fast that it appears as a blur over its trajectory. This is because any observation involves a time-averaging. The homey example is that of a rapidly rotating fan that appears as a blurred disk.

There is a simple, yet profound, theorem that the expected value of the effect of a charged particle executing a periodic path is the same as that of an object in which the density of the charge is proportional to the time spent in various locations of the path. Charged here could be gravitational mass, electric charge, magnetic charge or charge with respect to the nuclear strong force.

It is very simple to compute the rate of revolution of subatomic particle. For an electron in a hydrogen atom it is about 7 quadrillion times per second. At this frequency any time-averaged observation is then equal to its expected value. Thus an electron revolving about the proton in a hydrogen atom dynamically appears to be a tubular ring. The Copenhagen Interpretation treats this tubular ring as a concatenation of nodules of probability density. But the probability density is the classical time-spent probability from the electron's motion. Equally well the tubular ring could be considered a static object with the properties of the electron smeared throughout its extent in proportion to the time spent by the electron in various parts of the path.

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.

A classical oscillator executing a closed path periodically is a deterministic system but there is still a legitimate probability density function for it which is the probability of finding the particle in some interval ds of its path at a randomly chosen time. The time interval dt spent in a path interval ds about the point s in the path is ds/v(s) where v(s) is the speed of the particle at point s of the path. The probability density function PTS is then given by

PTS(s) = 1/(Tv(s))

where T is the time period for the path; i.e., T=∫ds/v(s).

If the solution to the Schrödinger equation for a physical system gives a probability density function then the limit as the energy increases without bound is also a probability density function. The spatial averaged limit has to also be a probability density function. For compatibility according to the Correspondence Principle that spatially average limit of the quantum system has to be the time-spent probability density function. That indicates that the quantum probability density function from Schrödinger's equation also is in the nature of a time-spent probability density function. This means that the quantum probability density can be translated into the motion of quantum system. This involves sequences of relatively slow movement and then relatively fast movement. The positions of relatively slow movement correspond to what the Copenhagen Interpretation. designates as allowable states and the places of relatively fast movement are what the Copenhagen Interpretation designates as quantum jumps or leaps. When the periodic motion of quantum system is being executed at quadrillions of times per second it may seem like the particle exists simultaneously at multiple locations but that is not the physical reality. It is only the dynamic appearance. A rapidly rotating fan seems to have the fan smeared over a blurred disk.

For one dimensional systems there is no question but that the above is the proper interpretation of wave mechanics. For two and three dimensional systems the situation is murky. The Schrödinger equations for such systems cannot be solved analytically except through resort to the separation-of-variables assumption. But the separation-of-variables assumption is not compatible with a particle having a trajectory.

The Copenhagen Interpretation accepts such solutions and asserts that generally a particle does not exist in the physical world unless it is subjected to a measurement that forces its probability density function to collapse to a point value.

The alternate interpretation is that the solutions developed through the use of the separation-of-variables assumption are not valid quantum analysis.

Conclusions

  1. The solutions to Schrödinger's equations correspond to probability density functions. Their spatially-averaged asymptotic limits also correspond to probability density functions. According to the Correspondence Principle these spatially-averaged asymptotic limits must be equal to the classical solution. The only relevant probability density distribution for a deterministic classical situation is the time-spent probability density distribution.
  2. At any scale the solutions to Schrödinger's equations correspond to the time-spent probability density distributions and, in effect, correspond to the dynamic appearance of particles in motion. For one dimensional sitations the Copenhagen Interpretation is valid in terms of the dynamic appearance of particles.
  3. The conventional solutions for the two and three dimensional cases are derived from the Separation-of-Variables assumption. This assumption is incompatible with particleness and solutions derived from it do not satisfy the Correspondence Principle and therefore are not valid quantum analysis.


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