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A Demonstration that the Solutions to Schrödinger |
Equations Pertain to the Time-Spent Probability Densities
Rather than Some Intrinsic Indeterminacy of Particles
In 1926 Erwin Schrödinger formulated an approach to Quantum Theory based upon partial differential equations. His approach, called Wave Mechanics, came to be the accepted theory of quantum level phenomena. The drama of the development of quantum theory at the time is still well worth reading.
Wave Mechanics seemed to give an accurate explanation of quantum level phenomena; the only problem was that the equations determined a variable, called the wave function, whose nature was uncertain. Schrödinger himself thought the wave function represented some physical quantity analogous to variables found in optics. However, Max Born of the University of Gottingen conjectured that the square of the wave function was a probability density. Bohr and Hiesenberg in Copenhagen replied to Born that they never considered the solution to be anything else. After Bohr and Heisenberg concurred this interpretation of the wave function came to known as the Copenhagen Interpretation. Schrödinger did not agree with this interpretation.
The probability density function, according to the Copenhagen Intrepretation, thus represented some disembodied probability reflecting the intrinsic indeterminacy of the particles. The Copenhagen Interpretation is often alleged to hold that the particle exist simultaneously in all of their allowed states. Particles assume a material existence only when they are subjected to observation and measurement.
In classical mechanics the motion of particles is deterministic, yet for periodic motion there is perfectly valid probability density function defined. It is the proportion of the time spent at the places along the path. It represents the probability density of finding a particle at the allowable places along the path at a randomly chosen time. This PTS (proportion of time spent) probability density function along with a minimum energy level satisfies the Uncertainty Principle. Thus the Uncertainty Principle, in itself, does not require any indeterminacy of the particles themselves.
Below the link between the classical PTS probability density distribution and the solutions to time independent Schrödinger equation is explored. The relationship is asymptotic, but the connection between the two concepts is nevertheless real. This should be no surprise. The Schrödinger is derived from the Hamiltonian equation for the system, which is the basis for determining the periodic path of the system. Thus the PTS probabiltity density function is, in a sense, a solution of the Hamiltonian equation in the sense of being one aspect of it.
Later in the analysis it is suggested that solutions of the Schrödinger equation imply a special nature of motion at a quantum level. This makes the PTS probabilty density distribution and the probability density distribution derived from Schrödinger's equation identical. But first the matter of the connection between the solution to the time independent Schrödinger equation and the PTS probability density function for a significant case must be shown.
A harmonic oscillator is a system in which the restoring force on a particle is equal to kx, where x is the displacement and k is a parameter called the stiffness coefficient. It is a system for which the Schrödinger equation can be solved explicitly. The time independent Schrödinger equation for this system is
where m is the mass of the particle, E is the total energy of the system and
h is Planck's constant
divided by 2π.
The solution depends upon an integer
n, called the principal quantum number.
Let φn(ζ) be the wave function for a harmonic oscillator with principal quantum number n. The solution can be expressed in terms of a dimensionless variable ζ, which is defined as x/σ, where x is the displacement from equilibrium. The parameter σ is a natural unit of length for a harmonic oscillator with mass m and stiffness coefficient k. It is defined as
The squared wave function for a harmonic oscillator with principal quantum number n is given by the formula
where Hn(ζ) is the Hermite polynomial of order n.
The probability density function in terms of the displacement x is then given by
The Classical dynamics of the system are given by its total energy E being constant. Its kinetic energy is ½mv² and its potential energy is ½kx². Thus
The particle follows a periodic time path s(t). The time spent in an interval ds of the path is ds/v(s). In effect there is a probability density function for the particle that is proportional to the time spent in any infinitesimal interval. It represents the probability density of finding the particle at any location at a random time. The integral of probability density over the total path must be equal to unity. This proportion of time-spent probability density function PC can be expressed as
where α is a constant that takes into account the factor of (m/2)½ and the normalization term ∫ds/v(s).
The classical probability density function can also be expressed as
where xm is the maximum displacement for the oscillator and its value is given by
where E is the total energy of the oscillator.
Both probability density functions depend upon the total energy E of the system. The quantum mechanical
value for E is
hω(n + ½).
When n is equal to 4 the two probability density functions are as shown below.
There are singularities at ±xm for the classical oscillator, which in the above case are at ±√5.
The ratios of the corresponding probability densities are shown below.
Since the quantum mechanical probability extends outside of the range allowed for the classical oscillator the quantum mechanical probability densities are necessary less than the classical harmonic oscillator values.
Thus for case of n=4 the relationship between the spatial average of the quantum mechanical probability densities and the classical ones is not close although the general shapes match for low displacements.
The correspondence is much closer for the case of n=60 which is shown below.
The correspondences at either end of the range are not close and could not be so because of the singularities there for the classical case. However, as is shown below, the spatial average of the quantum mechanical probability density near zero displacement is asymptotically equal to the classical probability density at that location.
The quantum mechanical analysis for a harmonic oscillator finds that
By Stirling's approximation of n!=(2πn)½(n/e)n, for n even
The appropriate quantum mechanical probability density function for comparison with the classical value is Pn(0)=½φn²(0)/σ where
Pn(0) may be put into the form
The value of the principal quantum number n is related to the energy of the oscillator by the formula
On the other hand the classical probability density function for the harmonic oscillator at zero displacement is given by
The ratio of the two probability density functions is then
Thus as n increases the ratio goes to unity.
Although it can be shown that as the principal quantum number n and hence energy increase without bound the spatial averages of the quantum mechanical probability densities go to the classical PTS values, the case above for x=0 is sufficient to establish the connection between the two concepts. This connection is as would be expected for generally as the scale of system increase the closer the quantum theoretic quantities approach the classical values. This is consistent with the Correspondence Principle for quantum theory articulated by Neils Bohr in the 1920's. Thus quantum mechanical probability density functions, as found as solutions to the Schrödinger equations, do not represent some pure indeterminacy of the particle, as in the Copenhagen Interpretation, but instead correspond to the proportion of the time a moving particle spends near the various points.
The notion under the Copenhagen Intepretation that somehow the particle of a quantum system exists simultaneously in all of its allowed locations has an interpretation in terms of the dynamics of a physical system undergoing periodic motion. If such a system is illuminated with a stroboscopic light at the proper frequency the particle can be made to appear to be existing simultaneously at multiple locations.
The above analysis could be extended for the about one half dozen quantum mechanical systems such that the time independent Schrödinger equation can be solved explicitly for the wave function. However there is something that comes out of the analysis that makes further analysis along the lines of the above irrelevant. That something concerns the nature of the motion of particles in a quantum systems.
For the classical analysis the relationship between probability densities and particle velocity is
where T is the time period of the particle's motion and serves as a normalization factor.
The relationship also works the other way around. From the probability densities the particle velocity may be found; i.e.,
where S is a factor to adjust the velocities to satisfy the energy level of the system.
From the case of the harmonic oscillator it is seen that quantum motion is not smooth as is the case for a classical oscillator. Instead quantum motion is in the form of slow-slow-fast movement. The particle is always moving. There is no real differentiation between allowed and disallowed states. The so-called allowed states are merely the ones through which the particle moves relatively slowly. The so-called disallowed states are not really disallowed; they just happen to be the intervals through which the particle moves relatively fast.
In the Copenhagen Interpretation if a particle changes from one allowed state to another it does so by making a quantum jump and during the time the particle is jumping it does not exist as a particle in space. There was a famous exchange between Schrödinger and Bohr on this matter.
Schrödinger said to Bohr
If we have to go on with these damned quantum jumps,
then I am sorry that I ever got involved.
But the rest of us are extremely grateful that you did. Your wave mechanics has contributed so much to the mathematical clarity and simplicity that it represents a giant advance over all previous forms of quantum mechanics.
The presentation of the solution to Schrödinger's equation as corresponding to the probability density function associated with the proportion of time spent in path of the particle does away with the notion of quantum jumps. Motion at the quantum level and at the macroscopic level are quite different but the spatial or temporal average of the quantum motion coincides, as least asymptotically, to the classical motion. This fits in with Bohr's Correspondence Principle that the quantum behavior must coincide with the classical macroscopic behavior as the scale of the system increases.
In classical analysis the probability density function for a system is found from the solution to its Hamiltonian equation. In quantum theory the probability density function is found as a solution to its Schrödinger equation and the motion of the quantum system can be constructed from that probability density function. This in the nature of an alternative interpretation of quantum theory.
In this table vertically there is asymptotic equality of spatial averages but horizontally there equality by identity. The solutions to Schrödinger's equation pertain to the dynamic appearance of a quantum system. The Copenhagen Interpretation mistakes the blurred disk of a rapidly rotating fan for an indeterminacy of the fan itself.
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