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The Asymptotic Limit of theProbability Distribution of a Particle Moving in a Potential Field in 3D Space |
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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

where μ= (m/(2~~h~~²) 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

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

where φ²(z) must be normalized.

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

Thus

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

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

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}.

Quantum Theoretic Solution

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

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

Now define Ω(z) by

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

Therefore

Note that

and

∇²(J(z)

Since

= − Ω(z)(J(z))

Therefore

+ Ω(z)[−¼(J(z))

= − Ω(z)(J(z))

Multiplying through by (J(z))^{¼} gives

+ Ω(z)[−¼(J(z))

Note that

and

∇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

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

where X_{ji}(z) and Y_{ji}(z)) are the
Spherical Bessel functions of the first and second kind, respectively, and H_{i}^{j}(θ, ζ) are
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

where S_{n}(x) is a Spherical Bessel function, first or second kind, and B_{n+½}(x) is the
corresponding ordinary Bessel function.

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

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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