San José State University

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Thayer Watkins
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& Tornado Alley
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Probability Distribution of a
Particle Moving in a Potential Field
and Satisfying Relativity Dynamics

The time-spent probability distribution for a particle is just the proportions of the time it spends at the various locations of its path. This means the probability densities are inversely proportional to the absolute values of the particle's velocity at the different points of its path.

Newtonian mechanics indicates that velocity is proportional to the square root of kinetic energy. At speeds low relative to the speed of light this also holds true to a good approximation in Relativistic mechanics. At high speeds there is a different dependence. This material is to uncover that dependence.

The Analysis

Consider a particle with a rest mass of m0 moving in a potential field of V(z), where z is the position coordinates for the particle. The total energy E is composed of kinetic energy K and potential energy V. Thus

E = K + V

Energy is conserved so as the particle moves its potential energy changes and hence its kinetic energy must also change.

In Relativity the apparent mass of a particle depends upon the velocity of the particle relative to the observer according to the formula

m = m0/(1−β²)½

where β is velocity v relative to the speed of light c; i.e., β=v/c.

Under Relativity kinetic energy is given by

K = mc² − m0
= m0c²[1/(1−β²)½−1]

The series for (1/1−β²)½) is

1 + (1/2)β2 + (3/8)β4 (5/16)β6 + .. .

This means that

K = ½m0v²[1 + (3/4)β2 + (5/8)β4 + . . .]

Thus ½m0v² is just the first approximation for kinetic energy. In principle the analysis could be carried out by solving for velocity using the successive approximations for kinetic energy, but it is not too difficult to find velocity from the exact expression.

Since

K = m0c²[1/(1−β²)½−1]
1−β² = 1/[(K/(m0c²) + 1]²

For convenience let (K/(m0c²) be denoted as k.

Then

v = c[((k+1)² − 1)/(k² + 1)]½

Thus

(1/|v|) = (1/c)[(k² + 1)/(k² + 2k)]½

Limiting Cases

As k→0, k² becomes insignificant compared to k and 1. This leads to the above value for (1/|v|) approaching

(1/c)/(2k)]½

Thus for small energies the probability density for a particle is inversely proportional K½.

On the other hand, as K→∞ the above formula indicates that (1/|v|) approaches 1/c and consequently the time-spent probability density becomes uniform throughout the particle path.

(To be continued.)


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