﻿ The Asymptotic Limit of the Spatial Average of the Quantum Probability Densities of a Particle Moving in a Potential Field
San José State University

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The Asymptotic Limit of the
Spatial Average of the Quantum
Probability Densities of a Particle
Moving in a Potential Field

Consider a particle of mass m moving in a one dimensional space subject to a potential function V(x), such that V(0)=0 and V(−x)=V(x). The time-independent Schrödinger equation for the wave function φ(x) for this physical system reduces to

(d²φ/dx²) = −μ(E−V(x))φ(x) = −μK(x)φ(x)

where μ=m/(2h²) and K(x)=E−V(x), the kinetic energy of the system as a function of particle location. Here is an example of what the K(x) function might look like.

It is appropriate at this point to note that in the determination of probability distributions constant factors are irrelevant because in the normalization process they cancel out. Note also that the above equation may also be expressed as

(d²φ/dx²) = −μE(1−V(x)/E)φ(x)

This indicates that it is the variation in the energy E relative to the potential V(x) that is important. Let V(x)/E be denoted as U(x). Then instead of thinking of the issue being what happens to φ(x) as E increases without bound, it is what happens to φ(x) as U(x)→0 for all x. But first it is necessary to find a way to get rid of the rapid oscillations in φ(x). Here is an example of φ²(x). It is for a harmonic oscillator, where V(x)=½kx².

What happens when E increases is not so much that the level of φ(x) increases but instead the density of the fluctuations increases. The range over which φ(x)² is nonzero also increases. The range of x is from a maximum of xm where V(xm)=E to a minimum of −xm.

Suppose there is a Maclaurin series for V(x). Given the previous assumptions concerning V(x) it would have to be of the form

V(x) = (1/2!)V"(0)x² + (1/4!)VIV(0)x4 + ...

Limiting V(x) to just the first term corresponds to a harmonic oscillator with a stiffness coefficient k equal to V"(0). The unnormalized solutions of the time-independent Schrödinger equation for harmonic oscillators are in terms of the dimensionless variable ζ=x/σ and the principal quantum number n

φn²(ζ) = (1/(2nn!√π)Hn²(ζ)exp(−ζ²)

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

The unit of length σ is given by

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

The energy E of the quantum system is proportional to n; i.e. E=nhω. Therefore

nσ² = E/k and hence n = E/(σ²k)

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

Pn(x) = φn²(x/σ)/σ

It can be shown that in the limit as n→∞ the squared values of the Hermite polynomials Hn² approach

2(2n/e)ncos²(ζ(2n)½−nπ/2)exp(ζ²)/(1−½ζ²/n)½

where e=2.7218....

This means that the probability density functions for the harmonic oscillators asymptotically approach

(φ(ζ))² = 2(1/(2nn!√π))(2n/e)ncos²(ζ(2n)½−nπ/2)/(1−½ζ²/n)½

The average of cos²(z) over an interval from cos²(z)=0 to cos²(z+π) is 1/2. So the spatial average of (φ(x))² is essentially

(Φ(ζ))² = (1/(2nn!√π))(2n/e)n/(1−½ζ²/n)½

Replacing n with E/(σ²k) in (1−½ζ²/n) gives (1−½kζ²σ²/E) which reduces to (1−½kx²/E). Thus the probability density function for the quantum harmonic oscillator is equal to an irrelevant constant factor times (1 − ½kx²/E)−½. This is just (1 − V(x)/E)−½ for a harmonic oscillator.

The Classical Harmonic Oscillator

The classical time-spent probability density function for a harmonic oscillator is

P(x) = 1/(π(xmax² − x²)½ = 1/(xmaxπ)(1 − ½(x/xmax)²)½

where xmax²=2E/k. Thus

P(x) = (1/πxmax)/(1 − (x/xmax)²)½ = (k/(2E)½)(1/π)/(1 − x²/(2E/k)))½ = (k/(2E)½)(1/π)/(1 − ½kx²/E)½

Thus the probability density function for the classical harmonic oscillator is equal to an irrelevant constant factor times (1 − ½kx²/E)−½. This holds true for the asymptotic limit of the spatial average of the quantum theoretic probability density function for a harmonic oscillator. They will normalize to the same function.

The Correspondence Principle

Niels Bohr nearly a century ago observed that classical analysis for many areas of physics had been empirically verified. Therefore he asserted, for any quantum mechanical analysis to be valid its appropriate extension to the realm of classical analysis should agree with classical analysis. This is called the Correspondence Principle. In atomic physics the extension is in terms of scale and/or the level of energy. What comes out of the solution to the time-independent Schrödinger equation for one dimensional systems is that with higher energy there are more and more rapid (dense) fluctuations in probability density. Here is an example shown previously of the probability density function from the solution of the time-independent Schrödinger equation for a harmonic oscillator.

Although the analysis above was only for a harmonic oscillator the Correspondence Principle asserts that it should hold for all valid quantum theoretic analysis.

Conclusion

The asymptotic limit of the spatial average of the quantum probability density of a particle moving as a harmonic oscillator is the time-spent probability density function of a classical harmonic oscillator. By the Correspondence Principle the same should hold true for all valid quantum theoretic analysis.

(To be continued.)