If multiple bosons can occupy the same state, does that mean you can put an infinite number of them in a fixed container at zero temperature without pressure.
2 Answers
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Being in the same state does not mean being at the same position.
For example, the ground state of a square well of side length $L$ goes as $\Psi \propto \sin(x/L)$. As you make $L\rightarrow \infty$, this becomes a flat distribution of infinite extent. So, indeed, they do not take zero space.
As far as pressure is concerned, the pressure $P$ of a weakly interacting Bose-Condensed gas with interaction strength $g$ is (at $T=0$): $$ P = \frac{1}{2}gn^2.$$
So a non-interacting BEC ($g=0$) will have zero pressure (at $T=0$).
An attractively interacting system in unstable as it will collapse ($P<0$), whereas a repuslively interacting system has a positive pressure so it costs energy to "stuff" particles in a box. NB at $T \neq 0$ there is always thermal pressure in interacting and non-interacting systems alike.
answered Sep 2, 2020 at 5:59
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$\begingroup$ Let me give an example to clarify my question. When helium is cooled to a critical temperature of 2.17 K, a fraction of the liquid becomes "superfluid". My question is "Does a discontinuity in volume occurs?" $\endgroup$ Commented Sep 2, 2020 at 10:27
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$\begingroup$ If non-interacting BEC at T=0 has zero presure, does that mean the volume can be infinitely small? $\endgroup$ Commented Sep 2, 2020 at 10:38
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$\begingroup$ I don’t know about discontinuity in volume. You cannot compress ad infinitum because when the system size is comparable to the de Broglie wavelength (and smaller) other effects start emerging. Eventually fermion degeneracy pressure is the atomic constituents stop the compression. $\endgroup$ Commented Sep 2, 2020 at 17:29
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$\begingroup$ Thanks for the response @SuperCiocia. My conjecture is that all matter take up space. Let's say classically you have a box which can contain 100 balls. If you make the balls identical particles (quantum), you can't put 101 balls in the box, regardless whether the balls are 'elementary particles'. The quantum view is different from the classical view only in that the position of the ball is a function of probability. So even photos have sizes. The ultimate limit is the size of a blackhole. $\endgroup$ Commented Sep 2, 2020 at 22:17
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Yes. An infinite number of Bosons can occupy the same state. But this does not mean that they will occupy the same location. The constraint that you applied (zero temperature and pressure) incorrectly assumes that there will cease to be any motion, so that all the particles are localised, and will not escape the container. This is incorrect, as even in zero temperature (and pressure) there will still be motion, due to quantum vacuum fluctuations. These fluctuations are a manifestation of the Uncertainty relation $\Delta E\Delta t \geqslant \frac{1}{2} \hbar$.
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$\begingroup$ For a BEC the de Broglie wavelength is much larger than the atomic size, you can forget about point particles and such. Also, an ideal non-interacting BEC has indeed zero pressure at $T=0$. $\endgroup$ Commented Sep 2, 2020 at 6:02
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