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The Asymptotic Limit of the
Probability Distribution of a Particle
Moving in a Potential Field in 3D Space

This is an extension to 3D space of the previous analyses for 1D and 2D spaces. It is a natural mathematical extension, but the question is whether it is physically relevant. When a planet is captured by the gravitational field of a star it executes a planar elliptical orbit. Relativistic effects only results in that ellipse rotating in the same plane. Thus while its motion is potentially three dimensional its actual motion is only two dimensional.

Nevertheless consider a particle of mass m moving in a three dimensional space subject to a potential function V(z), such that V(0)=0 and V(−z)=V(z) where z is the spherical coordinates (r, θ, ζ) of a point. The time-independent Schrödinger equation for the wave function φ(z) for this physical system reduces to

∇²φ = −μK(z)φ(z)

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

However, in the determination of probability distributions constant factors are irrelevant because in the normalization process they cancel out. Note that the above equation may also be expressed as

∇²φ = −μE(1−V(z)/E)φ(z)

This indicates that it is the variation in the energy E relative to the potential V(z) that is important. Let V(z)/E be denoted as U(z). Then instead of thinking of the issue being what happens to φ(z) as E increases without bound, it is what happens to φ(z) as U(z)→0 for all z. But first it is necessary to find a way to deal with the rapid oscillations in φ(z). Here is an example of φ²(z) for 1D space. It is for a harmonic oscillator, where V(z)=½kz².

What happens when E increases is not so much that the level of φ(z) increases but instead the density of the fluctuations increases. The range over which φ(z)² is nonzero also increases.

The equation for the wave function can be reduced to

∇²φ = −(1−U(z))φ(z)

where φ²(z) must be normalized.

The Classical Model


Consider again a particle of mass m moving in a two dimensional space whose position is denoted as x. The potential field given by V(z) where V(0)=0 and V(−z)=V(z). Let v be the velocity of the particle, p its momentum and E its total energy. Then

E = ½mv² + V(z)


v = (2/m)½(E−V(z))½

For a particle executing a periodic trajectory the time spent in an interval ds of the trajectory is ds/|v|, where |v| is the absolute value of the particle's velocity. Thus the probability density of finding the particle in that interval at a random time is

P(z) = 1/(Tv(z))

where T is the total time spent in executing a cycle of the trajectory; i.e., T=∫dx/|v|. It can be called the normalization constant, the constant required to make the probability densities to sum to unity. Thus

P(z) = [(m/2)½/T]/(E½(1−U(z))½)

Therefore the probability density function is inversely proportional to (1−U(z))½.

Later it will be convenient to represent (1−U(z)) as J(z). So it is noted at this point that the probability density function is inversely proportional to (J(z))½.

The Asymptotic Limit of the
Quantum Theoretic Solution

By eliminating the irrelevant constant factors the equation determining the quantum wave function can be reduced to

∇²φ = − (1−V(x)/E)½ = − (1−U(x))½

As noted above, for typographic convenience (1−U(z)) will be denoted as J(z). J(x) is proportional to kinetic energy and particle velocity is proportional to (J(x))½, as is also momentum p. Therefore the probability density function is inversely proportional to (J(z))½. Thus the equation to be considered is

∇²φ = − (J(x))½

Now define Ω(z) by

φ(z) = Ω(z)(1−U(z))−¼

The Laplacian ∇² of the product of two functions f(z)·g(z) is given by

∇²(fg) = (∇²f)g + (∇f)·(∇g) + f(∇²g)


∇²φ =(∇²Ω)(J(z))−¼ + 2(∇Ω)·∇(J(z)−¼ + Ω(∇²(J(z)−¼)

Note that

∇(J(z))−¼ = −¼(J(z))−5/4∇J(z)
∇²(J(z)−¼) = −¼(J(z))−5/4∇²J(z) + (5/16)(J(z))−9/4(∇J(z))²


∇²φ = − J(z)φ(z) = − J(z)Ω(z)(J(z))−¼
= − Ω(z)(J(z))¾


(∇²Ω)(J(z))−¼ − 2(1/4)J(z))−5/4)(∇Ω)·(∇J)
+ Ω(z)[−¼(J(z))−5/4∇²J(z) + (5/16)(J(z))−9/4(∇J(z))² ]
= − Ω(z)(J(z))¾

Multiplying through by (J(z))¼ gives

(∇²Ω) − (1/2)J−1)(∇Ω)·(∇J)
+ Ω(z)[−¼(J(z))−1∇²J(z) + (5/16)(J(z))−2(∇J(z))² ] = − Ω(z)J(z)

Note that

∇J(z) = −∇V(z)/E
∇²J(z) = −∇²V(z)/E

and ∇V(z) and ∇²V(z) are fixed as E→∞. Therefore all of the terms except (∇²Ω) on the LHS of the above go to zero as E increases without bound. They approach zero doubly fast because they have a derivative of J in their numerators and a power of J in their denominators. Furthermore J(z) asymptotically approaches 1 as E→∞. Thus Ω(z) asymptotically approaches the solution to the equation

(∇²Ω) = −Ω(z)

This is the Helmholtz equation of three dimensions. Its solution, using the separation-of-variables assumption, is of the form

Ω(r, θ, ζ) = Σi-0Σj=−ii [AjiXi(r) + BjiYi(r)]Hij(θ, ζ)

where Xji(z) and Yji(z)) are the Spherical Bessel functions of the first and second kind, respectively, and Hij(θ, ζ) are Spherical Harmonic functions.

Spherical Harmonic functions

Here are the general shapes of the Spherical Bessel functions.

The relationship between Spherical Bessel functions and ordinary Bessel functions is as shown below

Sn(x) = (π/(2x))½Bn+½(x)

where Sn(x) is a Spherical Bessel function, first or second kind, and Bn+½(x) is the corresponding ordinary Bessel function.

Spherical Harmonic Functions

Spherical harmonic functions satisfy Laplace's differential equation; i.e.,

∇²u(x) = 0

They serve for a spherical surface the same function that the trigonometric functions do for circular lines. Here is what they look like for the first few values of their parameters.

So Ω²(x) generally consists of functions which oscillate between relative maxima and zero values. The spatial average of those functions is a constant. Therefore the wave functions are inversely proportional to (J(z))¼ and their squared values, the probability densities, are inversely proportional to (J(z))½=(1−U(z))½, just as the classical time-spent probabilities are.

Here is an illustration of J(x), J(x)½, and 1/J(x)½ for the one dimensional case of a harmonic oscillator.


For the fundamental case of a particle moving in a potential field the spatial average of the probability densities coming from the solution of time-independent Schrödinger equation are asymptotically equal to the probability densities of the time-spent distribution from classical analysis.

There is no justification for the assertion in the Copenhagen Interpretation that particles generally do not exist materially. Effectively, except for its true believers, the Copenhagen Interpretation of quantum theory is demonstratively invalid.

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