San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

A Classical Interpretation of the
Wave Mechanics 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 Gottingen University were developing the New Quantum Theory of physics. Heisenberg, Born and Jordan were in their early 20's, the wunderkind 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 wunderkind 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 Mechanicswas easier to use and became the dominent 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 simular to such variables involved in optics. However Niels Bohr and the wunderkind had a different interpretation. Max Born at Gottingen 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 predeliction to emphasize the puzzling aspects of quantum theory. He said

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

But Bohr also articulated the Corresondence Principle. He said that the validity of classical physics was well established so for 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 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 can be 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 probabilty 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 probability density function.

The classical harmonic oscillator is deterministic but there is still a legitimate probability density function which is the time-spent probability density function. 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 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 correpond 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 exercuted at untold billions 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.

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins