San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Asymptotic Limits of the Spatially
Averaged Probability Distributions from
the Solution to the Time-Independent
Schrödinger Equation as the Mass
of the Particle Increases Without Bound
are Equal to the Time-Spent Probability
Distribution from Classical Analysis

Consider a particle of mass m moving in space subject to a potential function V(z), such that V(0)=0 and V(−z)=V(z) where z is the location coordinates of a point. The time-independent Schrödinger equation for the wave function φ(z) for this physical system is

(−h²/2m)∇²φ(z) + V(z)φ(z) = Eφ(z)

where h is Planck's constant divided by 2π and E is the energy of the system. It can be reduced to

∇²φ = −μK(z)φ(z)

where μ= (2m/h² and K(z)=E−V(z), the kinetic energy of the system as a function of particle location.

It should be noted that m is dimensional and therefore by the proper choice of units m can take on any numerical value. It is not the numerical value which is important; it is the process of taking the asymptotic limit which is important.

Here 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 = −J(z)φ(z)

where φ²(z) must be normalized.

The Classical Model

Consider again a particle of mass m moving in space whose position is denoted as z. 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/(T|v(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 of (m/2)½/TK½ 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 (J(z))½..

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

The quantum theoretic wave function can be represented as

∇²φ = −J(x)φ(x)

with, as before, J(x)= μK(x)

Now let λ(z) be defined by

φ(z) = λ(z)(J(x))−¼

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

∇²(f·g) = (∇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)
∇²J(z) = −∇²V(z)

and ∇V(z) and ∇²V(z) are fixed as μ→∞. Therefore all of the terms on the LHS of the above except (∇²λ) go to zero as μ 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 ∞ as h→0. Thus λ(z) asymptotically approaches the solution to the equation

∇²λ(z) = −λ(z)

This is a generalized Helmholtz equation. In 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 proportional to the energy E of the system.

Here are the general shapes of the Bessel functions.

The singularity of Yn(z) at z=0 precludes it being a part of λ(x)².

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. 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 larger the parameter in the generalized Helmoltz equation the more rapid and dense the fluctuations. The parameter in the generalized Helmoltz equation is proportional to the mass of the particle. Thus at the quantum level the larger the mass the more zittery is its motion but that zittering is is wiped out by spatial or temporal averaging so the motion appears to be smooth. This is not what is conventionally believed; i.e., that the nature of the motion of a particle changes at the physical system is scaled up. What the above analysis indicates is that the nature of the dynamics of a particle in a potential field is the same at the quantum and macroscopic level. It is just at the macroscopic level the fluctuations are so rapid that any degree of averaging produces the familiar smoothness of macroscopic motion.

The time-spent probability distributions are for a particle that maintains its physical existence. Thus 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. In effect, the time-independent Schrödinger equation give the dynamic appearance of a physical system rather than its static appearance. The Copenhagen Interpretation treats the solution to Schrödinger equation as if it were the static condition of the system. This is like treating the appearance of a rapidly rotating fan as if it is a unchanging translucent disk which is a single particle.


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.

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins,