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The Asymptotic Limit to the
Solution to Schrödinger's
Equation for Two Particles
Subject to an Attractive Force

This is part of a project to show that the solutions to the time independent Schrödinger Equation for physical systems are associated with the Classically based time-spent probability density functions. The probability density functions from Schrödinger's Equation involve very rapid oscillations. When these rapid oscillations are averaged out the result is the proportions of the time the system spends in its allowable states.

This has been proven for harmonic oscillators and particles moving in a potential field. It was established for one, two and three dimensional fields. Here it will be established for two particles subject to an attractive force and revolving about their center of mass.

Let m1 and m2 be the masses of the two particles. Let s be the length of the line between them. The attractive force between them might be

F = −G/s²

The constant G could be proportional to the product of their masses but does not have to be. In this case the potential energy V of the system would be given by

V = −G/s

Let r1and r2 the radii of the masses orbits. The center of mass is determined by

m1r1 = m2r2

Since s=r1+r2 it follows that

r1 = [m2/(m1+m2)]s
r2 = [m1/(m1+m2)]s

The kinetic energy is made up of radial and tangential components. The radial component is

½m1(dr1/dt)² + ½m2(dr2/dt)²

This can be reduced to


The quantity m1m2/(m1+m2) is equal to the reduced mass μ of the system. Reduced mass is defined by

1/μ = 1/m1 + 1/m2

The radial component of kinetic energy is then


The angle θ is the angle the line between the two masses makes with the coordinate system. The rate of rotation (dθ/dt) of that line is denoted as ω. The tangential velocities of the two masses are m1ω and m2ω, respectively. The tangent component of kinetic energy is

½m1(r1ω)² + ½m2(r2ω)²
which can be reduced to

Thus the kinetic energy K of the system is

K = ½μ(ds/dt)² + ½μ(sω)²

From this point the potential energy will be general, V(s). The Hamiltonian H (total energy) function for the system isthen

H = ½μ(ds/dt)² + ½μ(sω)² + V(s)

This is the same Hamiltonian function as for a single particle of mass μ moving in a potential field of V(s). The coordinate system is (s, θ).

The time-independent Schrödinger equation for the system is

[−(h²/(2μ))∇² + V(s)]φ = Eφ
which may be rewritten as
[−(h²/(2μ))∇²]φ = (E−V(s))φ = K(s)φ

where φ is called the wave function and |φ|² is the probability density.

However, in the determination of probability distributions constant factors like (h²/2μ) are irrelevant because in the normalization process they cancel out. Note that the above equation may also be expressed as

∇²φ = −E(1−V(s)/E)φ(s)

But even the constant factor of E may be ignored. Thus the relevant equation is

∇²φ = −(1−V(s)/E)φ(s)

This indicates that it is the variation in the energy E relative to the potential V(s) that is important. Let V(s)/E be denoted as U(s). Then instead of thinking of the issue being what happens to φ(s) as E increases without bound, it is what happens to φ(s) as U(s)→0 for all s. But first it is necessary to find a way to deal the rapid oscillations in φ(s). Here is an example of φ²(s) for 1D space. It is for a harmonic oscillator, where V(s)=½ks².

What happens when E increases is not so much that the level of φ(s) increases but instead the density of the fluctuations increases. The range over which φ(s)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 s. Let v be the velocity of the particle, p its momentum and E its total energy. Then

E = ½mv² + V(s)


v = (2/m)½(E-V(s))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=ˆ«dx/|v|a. 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 (1−V(s)/E)½, or equivalently (1−U(s))½.

It is convenient for typographic reasons to represent (1-U(z)) as J(z). 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(s))φ
or, equivalently
= − J(s)φ(s)

with J(s)=(1−U(s))

Now define λ(s) by

φ(s) = λ(s)(J(s))−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(s)φ(s) = − J(s)λ(s)J−¼)
= − λ(s)J¾(z)


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

Multiplying through by J¼(s) gives

(∇²λ) − (1/2)J−1)(∇λ)·(∇J)
+ λ(s)[−(1/4)(J−1)∇²J(s) + (5/16)(J−2)(∇J(s))² − (1/4)(J−1)∇²(J(s))]
= − λ(s)J(s)

Note that

∇J(s) = −∇V(s)/E
∇²J(s) = −∇²V(s)/E

and ∇V(s) and ∇²V(s) 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 numinators and a power of J in their denomerators. Furthermore J(s) asymptotically approaches 1 as E→∞. Thus λ(s) asymptotically approaches the solution to the equation

(∇²λ) = −λ(s)

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

λ(z) = (AXn(s) + BYn(s))cos(s−b)

where Xn(s) and Yn(s) 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(s) generally consists of functions which oscillate between relative maxima and zero values. The spatial average of those functions is a constant. Therefore the probability densities are inversely proportional to J(s)½=(1-V(s)/E)½ just the classical time-spent probabilities are.

The time-spent probability distributions are for a particle that maintains its physical 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. The case of two particles moving subject to a mutual attraction reduces mathematically to that of a single particle moving in a potential field.

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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