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The Relationship between the Quantum Mechanical Probability
Density Distribution for a Harmonic Oscillator and the Time
Average of its Location for Low Principal Quantum Numbers

A previous study found that the spatial average of the quantum mechanical probability density distribution for a harmonic oscillator near the point of zero displacement is at least asymptotically equal to its classical probability density distribution defined as the time average of its location. This study looks at the relationship between the two probability density distributions over the entire range of displacements.

Let φn(ζ) be the wave function for a harmonic oscillator with principal quantum number n. The variable ζ is dimensionless and is defined as x/σ, where x is the displacement and σ is natural unit of length for a harmonic oscillator with mass m and stiffness coefficient k. It is defined by

σ² = hω/k = h/(mω)
where the frequency ω is
ω = (k/m)½

The squared wave functions for a harmonic oscillator with principal quantum number n are given by the formula

φn²(ζ) = (1/(2nn!√π)Hn²(ζ)exp(−ζ²)

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

The probability density function in terms of the displacement x is then given by

Pn(x) = φn²(x/σ)/σ

The classical probability density function is given by

p(x) = 1/(π(xm² − x²)½

where xm is the maximum displacement for the oscillator and its value is given by

xm = (2E/k)½

where E is the total energy of the oscillator. The quantum mechanical value for E is hω(n + ½).

The units of the dimensions can be chosen so that h=1 and the values of m and k likewise may equal 1. This means that ω=1 and likewise σ=1. The energy of the oscillator is then equal to (n+½). This makes the maximum displacement for the classical oscillator equal to (2n+1)½.

When n is equal to 4 the two probability density functions are as shown below.

There are singularities at ±xm for the classical oscillator, which in the above case are at ±√5.

The ratios of the corresponding probability densities are shown below.

This ratio would be sensitive to any errors involved in the computation of the probability densities, particularly systematic ones. However since both functions are probability distributions the areas under the curves must equal unity. The computation of the area involves an approximation but the curves utilized in computing the ratios essentially satisfy that requirement. Since the quantum mechanical probability extends outside of the range allowed for the classical oscillator the quantum mechanical probability densities are necessary less than the classical harmonic oscillator values.

Thus for case of n=4 the relationship between the spatial average of the quantum mechanical probability densities and the classical ones is not close although the general shapes match for low displacements.

The correspondence is much closer for the case of n=60 which is shown below.

The correspondence at either end of the range appears not to be close because of the singularities there for the classical case. However in the above graphs the spatial averages for the lobes of the quantum probability density distributions are plotted as red dots. These red dots fall almost exactly on the curve for the classical probability density distribution. The averages for the quantum probability density function do not have to match the singularities of the classical function. It is the classical function whose values must match the quantum function values at isolated points near the end of the range of the oscillation.

However it is known from the previous study that the spatial average of the quantum mechanical probability density near zero displacement is asymptotically equal to the classical probability density at that location. It is quite likely that as the principal quantum number n and hence energy increase without bound the spatial averages of the quantum mechanical probability densities go to the classical values. This would be as would be expected for generally as the scale of system increase the closer quantities approach the classical values. 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 moving particle spends near the various points.

(To be continued.)

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