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

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.

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As an experimentalist, I might not be the person best suited to answer this, but I'll give it a try.

The wavefunction is going to be difficult to visualize because in general it is a complex function. If you want to 'see' sqrt(-1), I suggest you resort to drugs, lots and lots of drugs. But as for physical interpretation, Born tells us that the amplitude of the wavefunction squared will give you the real probability distribution. So the wavefunction represents a probability amplitude, and the random nature arises in the measurement process.

When making a BEC, you extract energy from the atoms in a few ways. Usually, you start with laser cooling (in a magneto-optical trap, or MOT) that captures and cools about 1e9 atoms to about 100 microkelvin. From there, you often need to employ other tricks to beat the so-called doppler limit set by the linewidth or your laser. Another limit is the "recoil limit" that says you will never cool an atom to have less energy than the photon it (randomly) emits. So you turn the lasers off, and trap these very cold atoms in other ways only to cool them further. Either magnetic fields, or far-off-resonance, focused lasers (optical traps) are used to capture the atoms, but now we are not scattering lots of light off them like we were in the MOT. In both cases you are altering the energy landscape experienced by the cloud, and this causes it to seek the lowest energy. Lower the trap walls, and the hottest (fastest) atoms will escape, leaving the remaining (slower) atoms to thermalize to a lower temperature. This is called evaporative cooling. In magnetic traps, it is often done with radio frequencies which couple your atoms to different Zeeman sublevels, changing their potential energy, and ejecting them from the scene. In far detuned optical traps, they just lower the laser power.

As the atoms get colder, something remarkable happens. Their wave nature starts to emerge. The de Broglie wavelength gets longer as you get colder, until eventually, particle wavefunctions start overlapping with their neighbors. At this point, the atomic statistics are no longer well described by the good-ole Maxwell-Boltzmann distro, and we have to resort to Bose-Einstein statistics. (This of course assumes you are working with bosons, or integer spin particles.) What does that mean? Basically it means that particles start to favor occupying the same state, or 'mode'.

To determine the wavefunction of a BEC, you could start with the Schroedinger equation, where now you simply replace the single particle wave function with a many particle wavefunction that is built as a tensor product of the individual particles. This turns out to be a poor approach because the particles interact. They are constantly bumping into each other, and this manifests as a non-linear (i.e. density dependent) interaction term in the Hamiltonian. So we can far better approximate the situation with the Gross-Pitaevskii equation. If you want to model a BEC, the GPE might be a good place to start.

One of the striking features of QM in my mind is that the description of a particle is inextricably linked to its environment. It's right there in the GPE via the potential energy term, V. The solutions of this equation will be your wavefunction. When you confine atoms in a harmonic trap, one that looks like a parabola, your wavefunction will reflect this. Indeed it is not a gaussian you see, but an inverted parabola (see Thomas-Fermi approx.). It will look different in different traps. The residual gaussian is composed of thermal atoms, still described by the M-B picture.

I should note that when atoms are in the ground state, they still cannot be said to have "no motion". The uncertainty principle tells us that they will always smear out over some tiny volume of phase space. This is called the zero point motion.

The phase is a tricky subject. You are right that the book seems to have described it poorly. BECs are often said to be phase coherent, which is to say across the whole cloud, the phase will be the same. If it were not so, you would necessarily see striations since some of the atoms would destructively interfere with their neighbors. This is basically an excitation, and so energetically unfavorable. I might have just made a theorist lose her lunch with that explanation, but this whole thing is just meant to be a low level intro.

Good luck, and keep asking questions.

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wow, that's so good explanation, though I don't understand all concepts but at least I get the main point. Thanks – user1285419 Jun 29 '13 at 4:54
I read another book on quantum mechanics for beginner. I have one question regarding the wavefunction for an atom in free space at room temperature. In the book, it states that the wave packet for the atom is pretty much like modulated 'Gaussian' (I mean the shape just like Gaussian), like what you said, we need to square to tell the probability, so from that kind of wave function, does it mean the atom is more probable to be around it's original position but less probable to be found at far end? – user1285419 Jun 29 '13 at 5:03
That's exactly right! The atom is most likely to be found at the peak of the Gaussian, and less so further out. Incidentally, the square of a Gaussian is just a sharper Gaussian. The classical position of the atom will closely match the expectation value of the position operator, because at room temperature, the de Broglie wavelength is extremely small, and the wave nature obscured. – burgerking Jun 29 '13 at 13:50
+1 I like your answer. I do not agree with the statement that the whole cloud has the same phase. For example, if the BEC has linear momentum it will have a phase gradient. IF the BEC contains a vortex it will have a $2\pi$ phase winding, etc. – Joe Jun 29 '13 at 15:45
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.

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