Unfamiliar Aspects of Uncertainty Principle & "Instantaneousness" of Position and Momentum Probability Distributions

Gottfried introduces a series of questions to the betterment of the understanding of the process of position and momentum measurement.We consider them as below:






Consider a position measurement is being done on a particle.Say,the clock shows time t at the instant we measure the particle at x; if the uncertainty in the position measurement is δx and the corresponding uncertainty in the time-count is δt,then we are interested in the minimum of δt as t+δt is the degree of definiteness of the time measurement.The less is δt,the more definite is the time.

If we take a closer look,the photon which comes from x hits the detector at time t.But the photon coming from x+δx must hit the detector at time different from t; say,at t+δt.So,we are assigning δt to the scattered photon coming late or before from x+δx.
δx is some finite distance multiple of the photon wavelength λ' (wavelength of the scattered photon is the unit distance in this problem).So,the time error is δt~λ'/c ~>(ћ/mc2)





In Compton scattering the photon is scattered in a variety of angles (empirically produced probability distribution).So,it is a stochastic (random and only statistically predictable) drift in various angles.If we wish to reproduce the position measurement accurately,we must perform the measurememnt very quick so that the wave-function does not spread appreciably.As a result,it is found where it was found the last time.

Momentum is [mass*(L/T)] where L=|x1 - x2| and T=(t2 -t1).Measurement of both x and t involves uncertainty or error.With large L,δx1 and δx2 are small compared to L=|x1 - x2|.Similarly, δt1 and δt2 which we found ~[ћ/mc2] are also very small compared to (t2-t1).Hence,we are more or less justified with momentum=mass*|x1 – x2|/T.




A free particle state with well defined position will not persist in general.With time,it will evolve.If we remember the instance of Compton effect,the state will lose its localization by stochastic drift.As time increases more and more,τ(p+δp)/m [=distance]of different particles will increase more and more and the accuracy of position determination decreases.Clearly,as the time shrinks,the accuracy of position determination increases.






As we found earlier,momentum measurement is done by measuring the position of the particle at two different points: x1 and x2.So,again we wish to find the minimum δt to see how quickly momentum measurement can be done.

Let x1 be the point where 1st Compton scattering event occurs, and say we want to produce a state of so and so (momentum) magnitude and direction.Then we set x2 lie in that direction.Then,we select the photon scattered from x2.This will prepare (and hence,determine) a particle with desired momentum.Clearly,in this case, δt=λ/c can be made arbitrarily small.

Since, δt can be made indeed very small,(t+ δt) approaches t, the assumption of non-relativistic quantum mechanics that position and momentum probability distributions exist instantaneously is justified.