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.

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 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 P_TS is then given by

P_TS(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 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.

Consider a particle of mass m moving in a central potential field V(r). Its total energy E is

E = K + V(r) = ½mv² + V(r)

Let K(s) be the kinetic energy of the particle expressed as a function of its position s on its path; i.e., K(s)=E−V(r(s).

Then

½mv² = K(s)
and hence
v(s) = (2/m)^½K^½ -

Thus the time-spent probability density function can be expressed as

P_TS(s) = 1/(SK^½)

where S=∫ds/K^½

Now consider the quantum theoretic analysis of the same system. Its Hamiltonian function is

H = p²/(2m) + V(r)

where p is the momentum of the particle.

The time-independent Schrödinger equation for the system is

−h²/(2m)∇²φ + V(r)φ = Eφ
which can be expressed as
∇²φ = −k²φ
with
k² = −(2m/h²)(E−V(r)) = −(2m/h²)K(r)

where φ, the wave function, is such that |φ|² is the quantum probability density function, P_Q.

Generally k² is a function of particle position, but if were constant over some interval the solution over that interval would be

φ = α·cos(ks)
and hence
P_Q = α²cos²(ks)

The wavelength λ of the fluctuations in the solution is such that

kλ = 2π
and hence
λ = 2π/k

If there are n lobes to P_Q each one would have to account for 1/n of the probability. The value of n would be given by

n = L/λ = Lk/(2π)
and hence
1/n = 2π/(Lk)

Where L is some measure of the span of the system, such as the difference between the extreme values of s. The extreme values are the solutions to K(s*)=0.

The average value of P_Q over a lobe is ½α² so the total probability of the system being in a particular lobe is\\

(½α²)λ = 1/n = λ/L
which means that
½α² = 1/L

(UNDER CONSTRUCTION)

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.