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The Asymptotic Equality of a Spatial Average
of the Quantum Mechanical Probability Density
Distribution of a System to the Time Average
of its State Under Classical Conditions

In Quantum Mechanics (QM) the solution of the Time Independent Schroedinger Equation for a system provides a wave function. The squared magnitude of this wave function is the probability density function (PDF) for the system. This PDF depends upon the energy of the system. As the energy increases without bound the QM solution approaches the Classical solution for the system. This is what Niels Bohr deemed the Correspondence Principle. It guided the development of QM from its inception. Classical physics at the macroscopic level was empirically verified so if some proposed quantum mechanical analysis did not match the classical result when scaled up then the quantum mechanical analysis was wrong. The Correspondence Principle had the aspects of an axiom and a working rule.

The Copenhagen Interpretation of the results of QM analysis advocated (championed) by Niels Bohr is that the PDF for a particle indicates its intrinsic indeterminacy. According to the Copenhagen Interpretation a particle does not have a specific location and velocity until the probing of measurement forces the PDF to reduce to a single spike, a Dirac delta function.

The material here argues for a different interpretation. A macroscopic system may be deterministic but a probability density distribution can be constructed for one that operates on a cycle based on the proportion of the time it spends in the vicinity of any state. If v(x) is the velocity of a particle at point x of its trajectory then the time that it spends in an interval dx around x is dx/|v(x)|. For this to be a probability distribution the sum (integral) over the cyclic trajectory must equal unity. The probability density is then 1/(T|v(x)|), where T is the time required for the particle to traverse its trajectory. (There is a minor problem in taking into account whether the particle is located at point x at one or more times in traversing its trajectory.) This probability distribution density at point x is the probability density of finding the particle at point x at a randomly selected time.

The proposition argued here is that a spatial average of a QM PDF asymptotically approaches the classical PDF defined above.

Consider a harmonic oscillator. This is a particle of mass m that is subject to a restoring force of kx, where x is the displacement of the particle from its equilibrium position. The energy for the harmonic oscillator is given by

E = ½mv² + ½kx²
or in Hamiltonian form
H = ½p²/m + ½kx²

where v is the velocity of the particle and p is its linear momentum.

Classically the dynamics of a harmonic oscillator is given by

v(x) = [2E/m − (k/m)x²]½
or, equivalently
v(x) = (k/m)½[2E/k − x²]½

The oscillator cycles at a frequency of ω between the extremes of ±xm where

xm = (2E/k)½
ω = (k/m)½

The time T required for the particle to travel from −xm to +xm is such that the classical probability density function p(x) is given by

p(x) = 1/(π(xm²−x²)½

The shape of this function is shown below.

In contrast to this simple function the QM PDF Pn(x) is given by

Pn(x) = φn²(x/σ)/σ

where n is the principal quantum number, φn is the wave function and σ is a unit of length given by

σ² = h/(km)½

where h is Planck's constant divided by 2π. The details of the analysis are given in a previous study.

The principal quantum number n is closely related to the energy of the oscillator; i.e.,

E = hω(n+½)

The squared wave function is given by

φn²(ζ) = (1/(2n+2n!√π)Hn²(ζ)exp(−ζ²)

where Hn(ζ) is the n-th order Hermite polynomial.

The QM PDF for n=60 is shown below with the classical PDF for comparison.

The QM PDF oscillates between maxima and minima of zero. The average of adjacent maxima and minima serves as an adequate spatial average of the QM PDF. For this case of n=60 the average for the QM PDF corresponds apparently quite well to the classical PDF, at least in the mid-range of displacements. At the extremes the correspondence cannot be very close because the classical PDF has singularities at ±xm and the QM PDF has positive probability densities beyond ±xm.

However it was shown in a previous study that the average of the maximum and adjacent minimum QM probability density for x=0 (zero displacement) asymptotically approaches the Classical probability density for x=0. The ratio is given by

p(0)/P(0) = (1+1/(2n))½

Thus as n increases without bound the ratio goes to unity. This is as would be expected for generally as the scale of system increase the closer quantities approach the classical values.

Significance and Conclusion

By the Correspondence Principle if a spatial average of the QM PDF for a harmonic oscillator at one point approaches the Classical PDF for a harmonic oscillator at the same point then this must be true for all of the other points for a harmonic oscillator. Furthermore, again by the Correspondence Principle, if a spatial average for a QM PDF asymptotically approaches the Classical PDF for a harmonic oscillator then this must also be true for any other valid QM system. (Valid QM analysis was defined by Niels Bohr in terms of the Correspondence Principle.)

This means that the quantum mechanical probability density functions do not represent some pure indeterminacy of the particle as in the Copenhagen Interpretation but instead the proportion of the time a cyclically moving particle spends near the various points or states. The movement of the particle at the QM level is different in character, being more erratic than the classical case, but the particle still has an identity.

More simply put, Quantum Mechanical probability density functions are analogous to the blurred image of a spinning propeller and a not evidence of the intrinsic indeterminacy of the propeller.

(To be continued.)


Richard L. Liboff, Introductory Quantum Mechanics, Holden-Day Inc., San Francisco, 1980.

Barry Spain and M.G. Smith, Functions of Mathematical Physics, Van Nostrand Reinhold Co., London, 1970.

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