An Experiment Demonstrating A Fundamental Result

The LASER beam is producing linearly polarized light which can be thought of as to be produced by equal number of R state photons and L state photons.Therefore,the Hg+ ion has equal probability of interacting with +h spin photons or -h spin photons[Hecht,p-330:discussion on angular momentum picture of photon].Therefore, as a whole,the LASER light cannot possibly impart angular momentum to any of the ions and flip its spin.Hence,the probability of (resonant) scattering is small.This makes the experiment a succession of individual photon collisions by one OR the other of the two ions---ONE AT A TIME.The observed scattering is non-resonant scattering.

If the ions were point scatterers,they would have scattered the light (may be their unique P state would have changed---but as far as interference is concerned,state of polarization does not matter).Light scattered from the trap material would not interfere for lack of coherence. Whereas,scattered light from the two scondary point sources will maintain constant phase relationship and will interfere.This way it is equivalent to Young's double slit experiment.

Even though we started with linearly polarized light,it's not ideal.A few trace of elliptically polarized light will invole unequal count of R and L state.As a ressult,net angular momentum CAN BE imparted on the Hg+ ions (though probability is extremely small).This corresponds to resonant scattering where spin-flip occurs.More generally,where there is no spin-flip,non-resonant scattering occurs and scattered wave produce interference patterns.

LaTeX

This is just to announce that I have enabled LaTeX in this blog.

$\sum_{i=1}^nx_i$

$\ x^2\ -\frac{a}{b}\$

So,it's working nice...It took some time to understand that the black colour of the LaTeX was overlapping with the black background of my blog.So,I changed it.

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.

Gamma ray microscope and uncertainty principle in the light of Compton effect

In this post,I intend to write something on the position and momentum measurement of some particle in the light of uncertainty principle.Prototype example is the famous thought experiment by Heisenberg [Gamma ray microscope].I will show the picture from Scwabl's QM book p-21.

The text I am following now is written by Gottfried.It's a rather insightful book.In the connection of position and momentum measurement,the concept of Compton effect turns out to be so helpful.

I am quoting some parts from the text which looks interesting to me and I think you will also be interested.I thought of this as follows:

In case of Compton effect,if we use light of wavelength much below than λ_C,the change in the wavelength is still of the order of λ_C,and thus,λ'~λ_C; which means much of the energy of the incoming photon has been imparted to the electron.To reveal the details,or,to increase resolution,we have disturbed our system appreciably which is undesirable.So,we need to have a balance of effective minimum and meaningful maximum of the wavelength which is precisely of the order of λ_C.Hence,to have better resolution,we may decrease λ,but not beyond this limit.

Resolution R=λ/sin(φ) where θ=(90-φ) is the lab scattering angle for the photon.As photon scattering angle θ--->0, φ--->90.This happens when the photon is scattered away from the lens [fig: Franz Schwabl,QM-p21] and will not enter the microscope.This also means λ~λ'.As a result, the microscope is flooded with too much ineffective light.Precisely, this happens for longer wavelength limit: (δλ/λ) x 100% < 0.01%, for visible light for example and the Compton shift is non-discernible.Discernible Compton shift, therefore, correspond to photon wavelength of the order of λ__C.

Since,Resolution R=λ/sinΦ,hence, dR=dλ/sinΦ~λ_C/sinΦ~λ_C.

As λ_C--->0 in the non-relativistic limit, we are able to specify the particle's position with fantastic accuracy.

Any suggestion to the betterment will be gladly appreciated.
-Kolahal