Background

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

σ_xσ_p ≥ h/(4π) = ½h

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

σ_xσ_v ≥ h/(4π) = ½h/m

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.

Formulation and Analysis
of a Relevant Physical System

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

E = ½mv² + U(x)

Thus

v = (2/m)^½(E−U(x))^½

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

P(x) = 1/(Tv(x))

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

P(x) = [(m/2)^½/T]/(E−U(x))^½

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

P(x) = [(m/2)^½/T]/(K(x))^½

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

Var_P(x) = ∫(x−x)²P(x)dx

where x is the mean value of x, ∫xP(x)dx. For the case being considered

Var_P(x) = ∫(x−x)²/(T(K(x)^½)dx

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.,

dt = dv/(dv/dt) = dv/[(dv/dx)(dx/dt)] = dv/[(dv/dx)v]

Since K(x) is equal to E−U(x) and v(x) = 1/(T(K(x))^½) it follows that

(dv/dx) = −½(−dU/dx)/[T(K(x))^3/2]
which reduces to
(dv/dx) = ½(dU/dx)/[T(K(x))^3/2]
and hence
(dv/dx)v = ½(dU/dx)/(TK(x))²

Since the probability density function for velocity is proportional to 1/((dv/dx)v) this means that the probability density function for velocity is proportional to (K(x))² and hence its variance is proportional to (K(x))². Anything that reduces the variance of x increases the variance of v. Thus the product of the variances is subject to the contrary influences. Likewise the product of the variance of x and the variance of momentum is subject to those contrary influences.

The value of K(x) depends upon the total energy. If the kinetic energy has a minimum level greater than zero there will be a minimum level for the product of the variance of position and momentum that is greater than zero.

As the level of energy and kinetic goes to zero the system under consideration converges to a harmonic oscillator with a stiffness coefficient equal to (dU/dx) at x equal to zero. It has already been established sthat when the harmonic oscillator has a minimum quantum of energy equal to hω, where ω is the frequency of the oscillator, the product of the standard deviations of displacement and momentum for a harmonic oscillator is equal to h, Planck's constant. In order to sastisfy the uncertainty relation that product has to be greater than or equal to h/(4π). Thus the product of the standard deviations of displacement and momentum for a harmonic oscillator exceeds the required amount for a factor of 4π.

Conclusion

Since the variance of velocity involves kinetic energy K to the second power and the variance of x involves K to the negative square root power of K their product depends upon the three halves power of K the larger the levels total and kinetic energy the larger will be the product of their variances. Thus the Uncertainty Condition is easily satisfied by the proportion of time-spent probability density distributions. The particle in the system remains physical while in motion, contrary to the Copenhagen Interpretation of quantum phenomena.

Although the system considered here is one dimensional a particle in any system traveling along a trajectory is in a one dimensional space. The Correspondence Principle articulated by Niels Bohr in the 1920's holds that the only valid quantum mechanical analysis is that which when scaled up in terms of energy gives the classical result. Thus according to the Correspondence Principle the only valid quantum mechanical analysis is that which involves particles traveling in trajectories. What was shown here is that the Uncertainty Principle is no bar against particles traveling in trajectories and that the proper interpretation of the wave function in Wave Mechanics is as the square root of the probability density function corresponding to the proportions of time spent at the different points of the trajectory.

(To be continued.)