Re: superfluidity and identical particles


Hi, Lyrical,


> > There is no
> >qualitative difference between the wavefunction of each individual
> >composite boson before or after Bose-Einstein condensation.
>
> This part i don't quite get.


Which part of 'no difference' don't you understand?

Bose-Einstein condensation is a thermodynamic phase transition.


> >The difference is that, after condensation, there are sufficiently many
> >bosons in the one level to enable you to detect coherent wave
> >properties at the macroscopic level.
>
> ....
> >However, a degree or so below the transition
> >temperature, there are still only 10 or so particles in level #2, but
> >perhaps 1e20 particles in the bottom level #1.
>
> It seems to me level #1 for one particle is different than the level #1
> for all the bosons.


Yes, it is different. It is singled out because it has the lowest
energy of all the levels.


> pre BE: This is because for one particle before BE condensation I
> thought the #1 level is localized.


??? If anything, delocalized, spread out over the whole system both
before and after.

>
> post BE :And for particles the #1 level is non-localized after BE
> condesensation, the wavefunction being spread out over a larger area,
> and nonlocalized.


Just like before the transition.


> If I am misunderstanding the situation , do I have the wf for the pre
> BE or the wf of the post BE situation wrong?


You are missing the crucial aspect of BE condensation.

BE condensation occurs in systems of conserved bosons that do not
interact too strongly. It even applies to conserved bosons that
apparently interact strongly, too, like Cooper pairs in metals, but
where for various reasons the bosons, decorated à la QFT with all
sorts of hanger-on virtual phonons and things, occupy discrete levels.

The theory only works for bosons that are conserved, and not for things
like photons that can be created and destroyed. It works for helium
atoms in liquid helium, for Cooper pairs of electrons in
superconductors, and for rubidium or neon atoms in UHV, all of which
are composite particles.

At thermodynamic equilibrium, the residual interactions between bosons
cause them to hop around between all the available levels. However,
averaged over time, each level has a well-defined expectation value of
its occupation number. Since it's a boson level, the occupation number
can be any non-negative integer. The expectation occupation f is given
by Bose-Einstein statistics for conserved bosons, and is a function of
the level energy E:

f(E) = 1/[exp{(E-E_f)/kT}-1]

This differs from BE statistics for photons by the addition of the term
E_f (the Fermi energy) in the exponent. I once knew how to derive this,
and could work it out again for you if you wanted me to. E_f cannot be
higher than the energy of the lowest level, otherwise you get
unphysical negative values for f.

In any given situation you have a total number of bosons N and a total
system energy E_tot. Once you know all the levels of the system, both
these quantities can also be expressed as a sum depending on the Fermi
energy E_f and the temperature T. These two conditions uniquely
determine E_f and T.

Now if the bosons interact only weakly, then they form a sort of gas,
and the energies of the levels are those of a quantum particle in a
box. The energy E = p^2/2/m and the allowed values of the momentum
vector p form a uniformly-spaced grid in momentum space. The lowest
level has zero energy. For a very large system there are lots of
levels, and it seems reasonable to treat them as if they formed a
continuous distribution, with a density varying as E^0.5. If that's
permissible, you can then express the total density of bosons as an
integral:

rho = integral_0^infty:E^0.5/[exp{(E-E_f)/kT}-1].dE

At any given temperature this value will be maximized by bringing E_f
as close as possible to zero on the negative side.

Now, the denominator goes to zero at E=0, so the integrand becomes
infinite. For small E you can linearize the exponential in the
denominator. The integrand is then proportional to T.E^-0.5. Now, even
though E^-0.5 diverges at E=0, its indefinite integral goes as E^0.5,
so the integral as a whole converges. And as T->0, its value goes to
zero. This is just crazy. These are boson levels, and everybody knows
you can fit any number of bosons into any of them. However the math is
saying that there is only a certain density of bosons that you can fit
in, and that it decreases with temperature, dropping eventually to zero
at T=0.

The resolution of the apparent paradox is that it is not permissible to
ignore the discrete structure of the levels for low E. The sum for all
the other levels may be approximated by a continuous integral, but you
must keep the expectation number of particles in the ground level E=0
as a separate additive term. E_f must be negative, but by making it
very close to 0, you can make the expectation number of bosons in the
ground level as large as you like. Hence you can get a situation where
an appreciable fraction of the bosons are in the E=0 level. When this
happens, you have a BEC transition. We are talking 1e23 bosons in all,
and there may be 1e20 bosons in the ground level. The next level up may
have more bosons in it than you have fingers and toes, but much, much
fewer than the number in the ground level.

All this theory is given on
http://www.siena.edu/physics/courses/phys220/cowen_Bose_einstein.pdf
and http://www.scielo.br/pdf/bjp/v34n3b/a05v343b.pdf.

Each quantum level is coherent within itself across the whole of the
space occupied by the phase. This is not unusual but just a consequence
of the wave nature of matter. However, usually, the particles are
distributed more-or-less uniformly between all the available levels,
and this coherence disappears in the average. If you took a million
lasers, all of different wavelengths covering the whole visible
spectrum, and mixed their outputs together, you would end up with
something indistinguishable from ordinary white light. In a BE
condensate, even though most of the particles may be in other levels,
there are far more particles in the E=0 level than in any of the 1e23
other levels. The coherence properties of all the other levels cancel
out in the average, but the coherence of the ground level does not.

Cheers,

Zigoteau.

.

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