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The Satisfaction of the Uncertainty Principle by the Proportion of Time Spent Probability Density Distribution of a Particle in a Potential Field |
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In 1926 Erwin Schrödinger published six articles establishing Wave Mechanics as the foundation for quantum theory. Wave Mechanics involved a variable, called the wave function, which was of an uncertain nature. Max Born of the University of Gottingen conjectured that the square of the wave function was the probability density of finding a particle at a particular location. Niels Bohr and Werner Heisenberg in Copenhagen agreed with this interpretation of the wave function and it became a key element of what became known as the Copenhagen Interpretation (CI) of quantum theory.
In 1927 Heisenberg formulated the Uncertainty Principle; i.e., that the product of the standard deviations of the probability distributions for location and momentum for a particle must be greater than or equal to Planck's constant divided by 4π. In symbols
where x stands for location and p for momentum, which is mv, mass times velocity.
In terms of the location and velocity of a particle the Uncertainty Principle is
where v is the velocity and m the mass of the particle. Given that the masses of nucleons (protons and neutrons) are more than 1800 times greater than the mass of an electron, what is true for electrons is not necessarily true for nucleons
Considering the scale of atoms Heisenberg concluded that electrons could not have definite trajectories and therefore an electron could only exist as a probability smear unless some action of an observer forced the probability distributions to collapse to a definite value. This also became a key element of the Copenhagen Interpretation.
What follows below is an element of an alternative interpretation of the solutions to Schrödinger's equation. The probability distribution involved is simply the proportions of the time a particle spends at the various locations of its trajectory. Thus the solutions to Schrödinger's equation correspond to the blurred disk of a rapidly rotating propeller and do not reflect any intrinsic uncertainty concerning the propeller itself. What is covered below is the demonstration that for a particle moving in a potential field the proportions of time-spent probability distribution satisfy the Uncertainty Principle and that therefore Uncertainty Principle imposes no restrictions on the physical reality of a particle.
Consider a particle of mass m in a potential field given by U(x) where U(0)=0 and U(−x)=U(x). Let v be the velocity of the particle, p its momentum E its total energy. Then
Thus
For a particle executing a periodic trajectory the time spent in an interval dx of the trajectory is dx/|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
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. Thus
It is convenient to represent (E−U(x)) as K(x), the kinetic energy expressed as a function of the location. T is the total time required to complete the periodic trajectory. Thus
The constant factor (m/2)^{½} is irrelevant in determining P(x) because it is also a factor of T and thus cancels out. Hereafter T will denote the cycle time with the constant factor eliminated.
The variance of P(x) is defined as
where x is the mean value of x, ∫xP(x)dx. For the case being considered
The Uncertainty Principle also involves the variance of momentum and that momentum is determined by the variance of velocity. The time the particle spends in a velocity interval dv is the interval length divided by the acceleration of the particle; i.e.,
The acceleration of a particle is equal to the force upon it divided by its mass; i.e.,
For a small displacement x, F(x) = kx and thus |a(x)| = (k/m)x. and the frequency of oscillation ω is equal to (k/m)^{½}. (To be continued.)
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