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vlaiv

Airy disk size and Uncertainty principle

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I was trying to derive airy disk size from Uncertainty principle, but I have numeric mismatch that I can't really explain.

If we use following formulae:

1. p = h / lambda

2. deltax * deltap >= hbar / 2

We derive following formula :

angle = lambda / (2 * pi * aperture)

(where aperture is deltax and using small angle approximation sin(x) = x)

Now even if we interpret Uncertainty principle to be standard deviation in measurement, meaning value of deltap is just standard deviation, and relation to gaussian approximation to airy disk is R ~ 3 x sigma, meaning that radius of airy disk is 3 times sigma of gaussian profile that approximates it, we still get

angle = (3 / (2 * pi)) * (lambda / aperture)

where classical formula is 1.22 * lambda / aperture

So there is numeric coefficient mismatch: Uncertainty principle gives 0.4775 while classical approach gives 1.22

Can anyone explain why is this so?

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from what I see you are calculating a formula for the uncertainty disc from uncertainty princip saying tht the momentum/wavelength is known. This gives a formula for a 'blob' of light passing through an aperture..but does not account for any diffraction. The airy disc is a diffraction feature and the 1.22 comes from the zero in the Bessel function. You might want to calc the 'smudge' of the airy disc due to the uncertainty princip. ;)  

 

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I was under the impression that two are related in that by restricting position of photon - letting it pass thru aperture, momentum is spread out - and that is what is "causing" diffraction (well not causing per se, but related in the same way as geometry of allowed trajectories influences path integral - by presenting an obstruction in path we are changing path integral. Similarly obstruction is changing "allowed" values of position from "allover" to "narrower context").

So I was just trying to see if applying uncertainty principle would let me calculate effects of diffraction (or some aspects of it). I've found couple of references to this exact thing, but authors seem to run into same numerical mismatch which then try to circumvent by using "more relaxed version": delta_x * delta_p >= h/2 (as opposed to hbar)

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12 minutes ago, andrew s said:

So if I understand correctly, only thing that can be deduced from deltax * deltap >= hbar/2 is just what it says :D

Meaning that exact uncertainty depends on case but can never be less than hbar/2. It can't be used to estimate airy disk diameter, but it can tell us for a fact that airy disk diameter will not be smaller then .... calculated value.

In this particular case deltax * deltap = h, and of course h>=h/4*pi.

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