What does the wavefunction of atom look like at low temperature?

Question

I am reading an introduction material on Bose-Einstein condensation (BEC) at low temperature and it stated that when the temperature approaches zero kelvin, almost all atoms are degenerated into the fundamental level so all the wavefunctions for all atoms add up to one big wavefunction.

I am not good at quantum mechanics but I know that the wavefunction doesn't have the 'regular' meaning like that for mechanical wave. Instead, it's probabilistic nature tell us the probability of finding the particle in a certain position only. So in the introducing material on BEC, it reads that million of atoms are initially moving in a vacuum chamber and once the temperature gradually decrease, more and more atoms become less 'active', when temperature drop to so low, those atoms are not moving at all. So I am thinking to illustrate that process in animation but I have a few questions

1. If the wavefunction of atom is probabilistic, so does it mean we could not visualize the wavefunction at all? What about the movement of atoms? I create an animation to show the movement of atoms in that chamber at rather high temperature (room temperature), but I consider all those atoms are moving around randomly. Does that sound right? So how does the motion of atoms (direction and speed) changed when temperature changed?

2. I googled it and I found that in some material, the wavefunction will be plotted as a wave packet with envelope taken as gaussian function. I don't understand why guassian, but is that why the atom cloud looks like gaussian? and if this applies to atoms in a vacuum chamber for my case too? Again, how does temperature change the profile of the atom cloud?

3. Finally, the book said when temperature close to $0$ $K$, all atoms are at rest with lowest energy so all wavefunctions for all atoms are collapsed into one wavefunction. Well, actually, even all atoms are in the lowest state, but what can we tell about the phase of wavefunction of each of them? Must the phase be all the same (why)? Since I read in different text on QM that the phase in wavefunction doesn't matter, but if the phase are all different (or random) for atoms (even they are in the same lowest state), when we add all the wavefunctions, what will we have? Plane wave? It is confusing because the book said the BIG atom is not moving, so if the corresponding wavefunction is plane wave, so the atom should be in motion.

Answer 1

1. The wavefunction is, as the word suggests, a function, usually a complex one. You can always visualize a function. The question is: does the visualization have a physical meaning? In the case of a wavefunction, what has more meaning is visualizing its amplitude squared, since this corresponds to the probability of finding a particle at a given position. Regarding the random movement of atoms, if you imagine the gas of atoms as solid balls moving around, bouncing off the walls of the container, then that visualization is wrong. It may be more illustrative to say that it's the atoms' wavefunction that moves around and this wavefunction is spread out in space, interferes with itself when it hits a wall (like a wave) and so on. Changing the temperature changes the distribution of atoms in the available energy states. To simulate how this redistribution happens, one would have to model the dissipation of energy that leads to the temperature drop. This depends on the cooling mechanism (refrigeration? laser cooling?).

2. The actual profile of the atom cloud will depend on the trapping mechanism. This happens to be Gaussian for the mechanisms involved in trapping BECs. The atom profile changes with temperature depending on the statistics of the trapped particles, here Bose - Einstein statistics. In pictures you usually see this as a smearing and expansion of the initial profile.

3. When one says that they are all in the same state, they mean exactly the same state, including phase factors. Also, you're not supposed to sum wavefunctions, so the phase is indeed irrelevant (again though, the phase is unique and global for the condensate). The wavefunction of the condensates realized in experiments is not a plane wave, it is localized by a trapping potential, usually a (magneto-)optical trap.