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The Asymptotic Limit of the
Probability Distribution of a Particle
Moving in a Potential Field in 2D Space

Consider a particle of mass m moving in a two dimensional space subject to a potential function V(z), such that V(0)=0 and V(−z)=V(z) where z is the polar coordinates (r, θ) 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.

By eliminating the irrelevant constant factors 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. This is the time-spent probability distribution for the particle. Thus

P(z) = [(m/2)½/T]/(E½(1−U(z))½)

The constant factor (m/2)½ is irrelevant in determining P(z) because it is also a factor of T and thus cancels out.

The time-spent probability distribution is thus inversely proportional to to (1−V(z)/E)½, or equivalently (1−U(z))½.

It is convenient for typographic reasons to represent (1−U(z)) 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.

The Asymptotic Limit of the
Quantum Theoretic Solution

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

∇²φ = −(1 − U(x))φ
or, equivalently
= − J(x)φ(x)

with J(x)=(1−U(x))

Now define λ(z) by

φ(z) = λ(z)(J(x))−¼

The Laplacian ∇² of the product of two functions fg is given by

∇²(fg) = (∇²f)g + 2(∇f)·(∇g) + f(∇²g)


∇²φ =(∇²λ)(J−¼) + 2(∇λ)·∇(J−¼) + λ(∇²(J−¼))

Note that

∇(J−¼) = −(1/4)(J−5/4)∇J(z)
∇²(J−¼) = −(1/4)(J−5/4)∇²J(z) + (5/16)(J−9/4)(∇J(z))² − (1/4)(J−5/4)∇²(J(z))


∇²φ = − J(z)φ(z) = − J(z)λ(z)J−¼)
= − λ(z)J¾(z)


(∇²λ)(J−¼) − 2(1/4)J−5/4)(∇λ)·(∇J) + λ(z)[−(1/4)(J−5/4)∇²J(z) + (5/16)(J−9/4)(∇J(z))² − (1/4)(J−5/4)∇²(J(z))]
= − λ(z)J¾(z)

Multiplying through by J¼(z) gives

(∇²λ) − (1/2)J−1)(∇λ)·(∇J)
+ λ(z)[−(1/4)(J−1)∇²J(z) + (5/16)(J−2)(∇J(z))² − (1/4)(J−1)∇²(J(z))]
= − λ(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 (∇²λ) 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 λ(z) asymptotically approaches the solution to the equation

(∇²λ) = −λ(z)

This is the Helmholtz equation of two dimensions. Its solution is of the form

λ(z) = (AXn(z) + BYn(z))cos(z−b)

where Xn(z) and Yn(z) are the Bessel functions of the first and second kind, respectively, and n is a nonnegative integer.

Here are the general shapes of the Bessel functions.

So λ(x)² generally consists of a function which oscillate between relative maxima and zero values. The spatial average of that function is a constant. Therefore the probability densities are inversely proportional to J(x)½=(1-V(x)/E)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.

What was shown above is that the wave function that is the solution to the equation

∇²φ(x) = −J(x)φ(x)

can be factored as follows

φ(x) = λ(x)(J(x))¼

where λ(x) is a purely oscillatory function which is asymptotically equal to the solution to a generalized Helmoltz equation.

The time-spent probability distributions are for a particle that maintains its physical existence. There is no justification for the assertion by the Copenhagen Interpretation of quantum theory that particles do not have a physical existence until their characteristics are measured.


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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