Crowding of bosons and the density of Bose condensate

Question

A separate posting from http://physicsoverflow.org//22089/applicability-quantum-statistics-essence-bosons-fermions ; suggested by @Arnold Neumaier.

Wikipedia did not treat well the question why a Bose condensate of atoms has the density of the same order as ordinary (non-quantum) matter. I tried to address the problem myself at https://en.wikipedia.org/wiki/Boson#To_which_states_can_bosons_crowd.3F

Possibly there is no need to include such stuff into the “boson” article (since there is https://en.wikipedia.org/wiki/Bose_condensate ), but is the section encyclopedic? Are there professional sources that consider the problem of condensation of bosons interacting in such way as atoms and molecules do? Actually I remember that tried to search and read some papers, but wasn’t satisfied.

Comments

This is already answered on stackexchange regarding the repulsive forces experienced by bosons composed of fermions when the wavefunctions of the Fermionic constituents start to overlap. It is a well-known subject at the undergrad level--- the main result is that there is a new effective force when you place two bound states of Fermions close together, due to the increase in the energy levels of the composite system (quantum composite systems are described collectively by all the particles together, not individually)
 

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I am not satisfied with you answer, and not because you neglected to provide a link. I certainly know that bosons may be composed of fermions. But it’s not the only case of a possible repulsion. Are W^± composed of fermions, really?

The question is not why can a boson be repulsed from its another copy, but which states can form a Bose condensate, how it can be described. I looked into papers that considered some particular cases, but we started to discuss Wikipedia that should address the problem in general.

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Fundamental bosons will collapse to arbitrary density, there is no limit. So W+ bosons have no upper density bound of the order of ordinary matter. The same goes for light, you can focus light to concentrate arbitrarily much, any density limits (if they come) start at the Compton wavelength of the electron, when you have one electron per Compton wavelength, which is a much higher density than ordinary matter.

The answer to this question is well known at the undergrad level and it should be closed.

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Isn’t anything wrong with the fact that W⁺ carries an electric charge? ☺

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http://physics.stackexchange.com/questions/45644/ is the SE reference.

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I read it. They stuck to Pauli exclusion principle, following Ron Maimon’s suggestion, and diverted away from the topic interesting for me to this electrono-nucleono-trivia.

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I agree with Ron that this is rather undergrad-level physics. Voting to close.

Answer 1

The question ''To which states can bosons crowd?'' has the only possible answer ''to a boson condensate''. This is a family of collective states of the bosons, determined (if they form) collectively by the interactions and the temperature.

The internal composition of a boson (elementary or composed of constituents, e.g., fermions, as in the case of He-4) is irrelevant for the problem of boson condensation. When one talks of boson condensation, one already chose to ignore the substructure and replaced it by a bosonic model with effective interactions. These residual interactions are of ''usual'' sizes, so the density resulting is also ''usual''. 

Comments

You certainly understand what do I mean. Is my conclusion that states with a high probability density (derived from the wave function) are energetically prohibitive for condensing correct? That’s the thing I added to Wikipedia as my own opinion (that generally goes against the rules).

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''states with a high probability density'' is meaningless. A state describes the probability density everywhere in space. One cannot say that a state ''has a high probability density''. It simply has a probablility density, which is high in the region where the system is localized, and low elsewhere.

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OMG… isn’t obvious what do I mean yet? A state where with 90% probability a particle is located where the state’s probability density is greater than 1/nm³ is a localized state; it has a high probability density. A state with the probability density everywhere less than 1/cm³ is a delocalized state; it is a low probability density.

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What you want is not obvious at all since your language is very vague. If you want to communicate with physicists you need to work on your capability to express yourself clearly, and in established terms.

In equilibrium, both a boson gas and a boson condensate have a uniform probability density everywhere except outside their container. Thus you cannot use it to make distinctions.

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Okay, I’ll try to get what such authors as Bogoliubov wrote on this topic (half-century ago) rather than to argue here.