# How does an atom look like in momentum space?

I am reading an article in a popular science magazine about atoms in extremely low temperature (condensation?). It's interesting and I've got few questions regarding this phenomenon.

1) In the article, the author show an figure of how the atom looks like in momentum space (like the top figure here). I'm pretty confused at how people measure the momentum of the atom? By light? I remember in the text, it introduced in one chapter how people shine a line on a material, and collect the emitted light, compare how the light intensity is reduced so as to be able to estimate the number of atoms there are. But does this method work for momentum estimation too?

2) I remember in the lecture of quantum mechanics, even at absolute-zero temperatures, the atoms won't be still, it has something called zero-point energy, so does it mean that the atom will be vibrating even at zero kelvins? So in the top figure here, we can observe that the momentum distribution has width about the peak, is that width telling us there will be some atoms moving at different momenta or the width is due to the uncertainty? So what does the distribution of momentum look like in absolute zero temperature?

3) The last question is in the article, the author said the momentum distribution is a Gaussian. But I searched in the library, they said for Boson, the condensation satisfies Bose-Einstein distribution. So what is the physical origin to use Gaussian distribution instead?

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## 1 Answer

I'm not quite sure about your background so I'll try to give you a gentle introduction: The article you were reading was probably about Bose-Einstein-Condensation. This is a phenomenon where all the atoms (with bosonic statistics) sit in the ground state of some external confining potential (e.g. a magneto optical trap) at finite temperature. The important part is the one about finite temperature. This is non trivial since we would expect that at finite temperature some fraction of the atoms were excited by thermal fluctuations. This state of matter was predicted by Satyendranath Bose and Albert Einstein in 1924 and realized in 1995 by several groups at JILA and MIT.

ad 1) What the people do in order to measure the momentum distribution of an ensemble of atoms (not just one) is the following: They trap their ensemble in an external potential which is suddenly switched off. Particles with higher momenta will fly further than particles with lower momenta. By shining light on the expanding cloud and checking how much was absorbed they find the velocity distribution of the cloud. (see your pictures http://jila.colorado.edu/bec/) This is also called TOF imaging (time-of-flight imaging). Thus one has measured the momentum distribution via the spatial density distribution.

ad 2) The BEC is a remarkable phenomenon occuring in ideal quantum gases relying only on the bosonic statistics of the atoms. Yet if the atoms interact weakly, the gas still exhibits a condensation into its ground state. The temperatures when this occurs are about $T_c\approx10^{-7}K$. If we look at TOF images at increasingly colder temperatures we observe a Gaussian distribution of the particle density (with width proportional to the temperature). Ones we cool the ensemble below the critical temperature (BEC) we observe a so called Thomas-Fermi-profile (an upside down parabola growing out of the thermal Gaussian distribution http://phys.strath.ac.uk/laser/BEC.jpg) given that the trapping potential was harmonic. The width of the density distribution of a BEC is merely due to the interactions between the gas atoms. At $T=0$ the distribution would look the same.

ad 3) The Bose-Einstein distribution for "hot" gases almost looks like the Boltzmann and thus in momentum space like a Gaussian distribution. (You can see this by doing a high temperature expansion) Differences arise, roughly speaking, at low temperatures and give rise to the condensation phenomenon.

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thanks for your explanation. It really shows me a general picture of the phenomenon and your words a easy to understand too :) –  user1285419 May 27 '13 at 0:57
Following your description, I read another article about BEC. Here come to my question, at finite (low) temperature, though weak interaction, the BEC manifest Gaussian momentum distribution and you said the width (FWHM) tells the temperature. But for each atom, uncertainty principle holds, so if you use the TOF (as you mentioned above) to measure momentum distribution, is the momentum distribution accurate? I know this question might be confusing. I have background of engineering and learnt undergraduate physics. In terms of measuring, I always think of uncertainty principle. –  user1285419 May 27 '13 at 1:09
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