﻿ The Time-Spent Probability Distribution for a Particle Under Relativity
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The Time-Spent Probability
Distribution for a Particle
Under Relativity
,

## Time-Spent Probability Distribution.s

Let dt be the time a particle spends in an interval ds of its trajectory. Then the probability of finding it in that interval is dt/T where T is the total time the particle takes to execute its periodic trajectory. But dt=ds/|v| where v is the velocity of the particle.

## The Classical Case

For a particle of mass m in a potential field V(x)

#### E = ½mv² + V(x) so v(x) = [(2/m)(E−V(x)]½

This can be rewritten as

#### v(x) = [(2/m)K½

where K is kinetic energy.

Therefore the wavefunction ψ(x) associated with the time-spent probability density function PTS(x) is given by

## The Relativistic Case

The total energy of a particle is mc² where m is relativistic mass m0/(1−β²)½. Therefore kinetic energy K is mc²−m0c².

In the derivation below the dependence of K, v and β on particle position is ignored to simplify the algebraic expressions.

#### K/(m0c²) = 1/(1−β²)½ − 1 (K + m0c²)/(m0c²) = 1/(1−β²)½ (1−β²)½ = (m0c²/(K + m0c²) 1−β² =[(m0c²/(K + m0c²]² β = [1 − ((m0c²/(K + m0c²))²]½ v/c = [(m0c² + K) − m0c²]/(m0c² + K) v/c = [K/((m0c² + K)]½ v = cK½/(m0c² + K)½

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Note that as K→∞ v→c as it must under Relativity.

There a further derivation of v based upon factoring m0c² out of the denominator of the above fraction; i.e.,

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## The Time-Spent Probability Density Function

Since the probability density function P(z)=1/(Tv(z)).

#### P(z) = (m0½/T)(1 + K/(m0c² ))½/K½

Thus when kinetic energy K is small compared with (m0c² ) density is inversely proportional to K½ just as in the classical case.

## The Wave Functions

If the wave function ψ(z) is such that ψ(z)²=P(z) then