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The Wave Function from Schrödinger's
Equation is Related to the Time Spent by
the System in its Allowable States

The purpose of this webpage is to tie the probability density distribution derived from the time independent Schrödinger's (Schroedinger) equation to the probability density distribution derived from the time spent along a periodic path for a particle according to classical physics. According to the Correspondence Principle as articulated by Niels Bohr, for a quantum analysis to be valid it must betat when it is is scaled up by energy to a macroscopic level it must equal or asymptotically approach the results of classical analysis. What Bohr left out is that the macro level observation necessarly involves spatial and time averages. The correspondence is between the spatial and time average of the scaled up version of the quantum analysis.

Since the result from Schrödinger's Equation is a probability density distribution, scaled up it is still a probability density distribution. What probability density distribution is there for classical analysis that the quantum analysis could correspond to? The answer is the time spent probability density distribution. This is the proportion of the time a system spends in its allowable states while traversing a periodic path. For a particle traveling at a velocity v in a spatial interval the time spent in that interval is its length divided by the magnitude of the velicity. If the length of the interval is ds then the time spent dt is ds/|v|. If the total time spent traversing the periodic path is T then the proportion of the time spent in the interval is ds/(T|v|). This is the probability of finding the particle in the specified interval at a randomly chosen time.

The Classical Analysis

The system being considered is a point particle of mass m in a one dimensional potential field given by the function V(x). The potential function is assumed to be symmetric about x=0 and furthermore that V(0)=0.

The total energy E is given by

E = ½mv² + V(x)


|v| = [(2/m)(E − V(x)]½

The quantity (E−V(x)] can be represented as K(x), the kinetic energy of the particle as a function of location.


|v| = (2/m)½K(x)½

The factor of (2/m)½ is irrelevant for determining the probability density function since it multiplies everything.

The probability density function PC(x) for the particle is then

PC(x) = K(x)−½/T

where T = ∫K(x)−½dx.

Now consider the effect of K being scaled up by a factor of λ. The numerator is muliplied by (1/λ−½) and likewise for the denominator. Thus the effect of an increase in kinetic energy K operates only through an increase in the range of allowable states.

The range of allowable states is given by xmin≤x≤ xmax where xmin and xmax are given by

V(xmin) = E = V( xmax)

For a harmonic oscillator V(x)=kx² and thus xmax=(E/k)½.

The Proposition

The proposition is that as the energy of the particle in the potential field increases without bound the spatially averaged probability density function from Schrödinger's equation asymptotically approaches the time spent probability density function from classical analysis. This proposition has been established for harmonic oscillators, an illustration of which is show below. What is sought here is the establishment of the proposition for a more general type of potential function.

The light lines are for the probability density functtion according to the quantum analysis of a harmonic oscillator. The dark line is the result for the corresponding classical analysis. The spatial average of the quantum analysis closely approximates the results from the classical analysis of a harmonic oscillator.

The Quantum Analysis

Quantum analysis is based on the Schrödinger's equationfor the system, which is the based for it Hamiltonian function. For the particle in the potential field the Hamiltonian is just the total energy expressed in terms of its momentum p=mv; i.e., .,

H = ½p²/m + V(x)

This is converted into the Hamiltonian operator H^ by replacing the squared momentum p² with −h²(d²/dx²), where h is Planck's constant divided by 2π. Thus the time independent Schrödinger equation for the particle is

H^φ = −h²(d²φ/dx²) + V(x)φ = Eφ

where φ is such that its squared magnitude is equal to the probability density function for the particle.

This equation can be put into the form

(d²φ/dx²) = − (K(x)/h²)φ

where K(x)=E−V(x), the kinetic energy of the partcle as a function of location.

In a study of the solution to a generalized Helmholtz equation of the form

(d²φ/dx²) = −k²(x)φ(x)

it was found that the average value of φ²(x) is approximately inversely prportional to k(x); i.e.,

x−½lx+½Lφ²(ζ)dζ = α/k(x)

For probability density functions constant factors such as α in the above equation are irrelevant. They cancel in the normalizarion of the function.

This means that a spatial average of the probability density function from Schrödinger's equation is inversely proportional to K(x)½. This is exactly what results from the classical analysis.

The Copenhagen Interpretation

What the Copenhagen Interpretation tries to say is that solution to Schroedinger's is giving a probabilistic static structure of the system instead of the dynamic appearance of the system.

One element of the Copenhagen Interpretation is that a particle exists simultaneously, sort of, in all of its allowable locations. Consider a rapidly rotating propellor. It appears to be a blurred disk. If it is illuminated by a stroboscopic light there appear to be multiple propellors existing simultaneously.

The alternative to the Copenhagen Interpretation is that the solution to the time independent Schrödinger equation represents the dynamic appearance of the system, the blurred disk of the rapidly rotating propeller, instead of any intrinsic uncertainty of the static structure of the system, the propeller. This not to say that results of Quantum Theory are wrong. It is only the conventional Copenhagen Interpretation of the results of of them that is wrong.

When Erwin Schrödinger formulated the wave mechanics version of quantum physics in 1926 he did not specify what the wave function ψ represented. He thought its squared magnitude would represent something physical such as spatial charge density. Max Born suggested that its squared magnitude represented probability density of finding the particle near a particular location. Niels Bohr and his group in Copenhagen concurred and the notion that the wave function represents the intrinsic indeterminancy of the particle of the system came to be known as the Copenhagen Interpretation. What was shown here is that the wave function relates not to any intrinsic indeterminancy of the particles but instead to the proportion of the time the system spends in its allowable states.

One of the pillars of the Copenhagen Interpretation is Heisenberg's Uncertainty Principle; i.e., that the product of the standard deviations of location and momentum must be greater than or equal to ½h. This condition is easily satisfied by the time-spent probability distributions for a harmonic oscillator.

The potential function for a harmonic oscillator is V(x)=½kx². This is not of a form most relevant for quantum level physical systems. The crucial function is

K(x) = E − V(x)

This is the kinetic energy of the particle as a function of location. For a harmonic oscillator it is as shown below.

For the electrical field of a point particle with a charge of q the potential function is V(x)=−Kq/x. There is a singularity at x=0. But point particle with a field are unrealistic because their fields have infinite energy. It is more realistic to presume the potential is create by a particle that is a spherical shell of charge of radius. The the potential function is given by

V(x) = −Kq/x for |x|≥ R
V(x) = −Kq/R for |x|< R

The kinetic energy function for a particle in such a field is as shown.

For another spherical shell particle of radius r<R the potential function from the particle of radius R would have the sharp edges rounded off.

The kinetic energy function would be as shown below.

For the smaller spherical charge at low energies within the larger spherical charge the kinetic energy function would be similar to that of a harmonic oscillator. Thus at low energies the charge within a charge would behave as a harmonic oscillator behaves. In particular its time-spent probability distribution would satisfy the uncertainty principle.


At low energies the spatial averages of the probability distributions from the time-independent Schrödinger's equation for a particle in a one dimensional potential field are equal to the time-spent probability distributions from classical analysis.

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