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A Bose-Einstein Condensate (BEC) is often said to be a "fifth state of matter". But what exactly is it? In this video, I wanted to discuss BECs and the basic mathematical descriptions derived from the theory of quantum mechanics. #boseeinsteincondensate #quantumphysics #BEC

Firstly, we need to consider certain particles known as Bosons. (Both Bosons and the Bose-Einstein condensate are named after Indian Physicist Satyendra Nath Bose). Now as it turns out, Bosons are a class of "indistinguishable" particle. This means that if we have a system of two or more bosons, all we can know is the number of bosons in our system. They are identical to each other in every way (same mass, same charge, etc). And if we label each particle in the system and then leave the system to evolve, then some time later we have absolutely no way of knowing which particle is which.

In order for this to be actually true in our universe, and not just some mathematical fantasy, the wave functions of our systems need to behave in a specific way. A wave function contains all the mathematical information we can have about our system. And the square modulus of the wave function is directly related to the probability of certain getting certain experimental results on our system (such as finding our particles at given points in space). If the particles really are indistinguishable, then the square modulus of the wave function must be exactly the same regardless of what orientation we have our labelled particles in - thereby rendering the labelling of our particles incorrect.

So if the square modulus of the wave function must be identical under particle exchange, then the wave function itself must behave in such a way that it either remains the same under particle exchange, or it becomes negative. The first of these two possibilities is referred to as a "symmetric" wave function, and the second is "antisymmetric". Any kind of particle with a symmetric wave function is called a "boson", and any kind with an antisymmetric wave function is called a "fermion". The focus here is on bosons of course.

When constructing a wave function for a boson system, we find that it is possible for two (or more) bosons to be in the same energy level (or state) and still result in an overall symmetric wave function. This is different to fermions, for which there is no way to construct an antisymmetric wave function while allowing two or more particles to be in the same energy level. This can be thought of as a very important difference between bosons and fermions.

And as it turns out, the ability of bosons to all occupy the same energy level, is what allows Bose-Einstein Condensation to occur. A BEC is formed when a boson gas of relatively low density is cooled below a certain critical temperature. This critical temperature depends on the density of the bosons and the mass of the bosons. Once below this critical temperature, all of the bosons fall into the lowest energy state and often can even occupy a very very small region of space.

BECs are interesting because they clearly display quantum behaviour that would otherwise be restricted to the very small scale. For example, superfluidity (https://en.wikipedia.org/wiki/Superfluidity) has been observed in BECs. It's important to note that these quantum behaviours are not direct consequences of the BEC, but are easily observed in one.

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