Bose-Einstein Condensate: The Quantum BASICS - Bosons and their Wave Functions (Physics by Parth G)

big thanks to skillshare for sponsoring this video check out a free trial of skillshare premium by clicking the first link in the description below hey everyone parth here and in this video we are having a chat about bose einstein condensates many of you have asked me to make a video on this topic so today we're going to be doing just that if you enjoyed this video please do hit the thumbs up button and subscribe for more fun physics content let's get into it so first things first we need to have a look at a particular class of particle known as a boson named after satendranath boss after which one half of the bose einstein condensate is also named bosons are a type of particle that have some very specific characteristics that we'll look at shortly examples of bosons include the very famous higgs boson which you may have heard of it was discovered a few years ago now and photons as well photons are particles of light so what's so important about bosons well the first thing is that they are a particular type of indistinguishable particle that means if we have two of the same kinds of boson in a particular system we have absolutely no way of telling apart the two particles now this is a bit of a wacky idea and i've made a video covering indistinguishable particles specifically looking at bosons and fermions so check it out up here if you haven't seen it already but the fact that bosons are indistinguishable particles that is if we have two of the same kind of boson we can't tell which is which has a very important impact on the behavior of these bosons we can understand the impact that this indistinguishability has by considering the wave function of any system we happen to be studying now a wave function can basically be thought of as a mathematical function that contains all of the information that we know about our system for example if our system consists of just a single electron then from the wave function of our system we can calculate the probability of finding that electron at different points in space specifically it turns out that if we square any wave function technically we take its square modulus then that is directly related to the probability of certain experimental results occurring when we make a measurement on our system when we conduct an experiment on it i've also made many videos discussing this idea in a bit more detail so please do check out my quantum mechanics playlist for more but if we now consider two indistinguishable particles we can't tell these particles apart and we take this system's wave function and we square it then this wave function squared must behave in such a way that it's identical whether our two particles are found in this orientation or in this orientation in other words if we swap particles a and b the wave function squared should look exactly the same because we shouldn't have any way of knowing which particle is which whereas if the wave function changed when we swapped these particles then we would be able to tell there would be some experiment that we could do on our system that would tell us which wave function squared we had for that particular system and therefore which orientation we had our particles so basically the square of the wave function when we've got the orientation a b must be the same as the square of the wave function when we've got the orientation b a if we then take the square root of this mathematical expression then what we find is that the wave function itself for when we've got the particles in the orientation a b must either be exactly equal to the wave function of the orientation b a or it must be equal to minus the wave function of the orientation ba and so what we've done here is discovered the wave functions for two different classes of particle the particles for which the wave function behaves like this are known as bosons and the particles for which the wave function behaves like this are known as fermions with a boson wavefunction we basically say that it is symmetric under particle exchange what this means is that the wavefunction is identical when we swap particles whereas with fermions we say that the wavefunction is anti-symmetric basically means that when we swap the particles the wave function becomes negative the fact of the matter is that a symmetric wave function behaves differently to an anti-symmetric one and this is the crux of the difference between bosons and fermions and we'll be focusing in on the behavior of both suns in order to understand the bose einstein condensate now before we go any further i'd like to thank this video's sponsor skillshare skillshare is an online learning community where you can find a large number of inspiring classes focusing on topics such as productivity and lifestyle to building a business to learning creative skills many of you may know that one of my hobbies is creating music check out my music channel linked below and i've taken some classes on skillshare that have taught me some really cool skills for example i took a class called audio mixing on the go professional sound without the studio by king arthur which gave me lots of tips for improving my mixes without lots of fancy equipment and that's the key here skillshare has a large number of classes to choose from and it's all about learning so there are no adverts and skillshare costs less than 10 a month with an annual subscription but the first 1000 of you to click the first link in the description box below will get a free trial of skillshare premium please do go check it out and big thanks to skillshare once again for sponsoring this video now just to recap we saw earlier that bosons have a symmetric wave function and fermions have an anti-symmetric wave function to understand how we can go from this mathematical condition to the bose-einstein condensate behavior we'll be looking at a rather oversimplified description of what's going on but hopefully we can convey the idea appropriately let's now imagine that the two particles we're considering are two bosons and they can occupy two different energy levels the energy level labeled zero is the ground state energy level and the energy level labeled one is a slightly higher energy level if this is our system we can work out what a possible wave function would look like for our system it could for example look like this what this represents is particle a in energy level 0 and particle b in energy level 0. this is indeed a symmetric wave function when we swap the particles because the wave function remains exactly the same under particle exchange and what this actually means is that if our two particles are indeed bosons they can occupy the same state as it turns out for this particular system there are also two other wave functions that we could construct they look like this but we will take a look at those in a moment instead let's consider what would happen if our two particles were fermions not bosons if the wave functions had to be anti-symmetric remember the condition here is that when we swap the particles the wave function must become negative of what it used to be and for this particular kind of system that wave function ends up looking like this what we've got represented here is a superposition between the state where particle a is in the energy level zero and particle b is in the energy level one and the other state which is when particle a is in energy level one and particle b is in energy level zero now superposition is an important idea in quantum mechanics it's when a system can supposedly occupy multiple states before we've actually made a measurement on it and when we do make a measurement the system will collapse into one of those states again more discussion on this in my quantum mechanics playlist now the reason that our fermionic wave function looks like this is because if we swap the two particles now if we label particle a as b and b as a then we see that the wave function becomes negative of what it used to be which is exactly what we need for fermions we need an anti-symmetric wave function but interestingly it's not actually the superposition that makes our wave function anti-symmetric as we can see from one of the bosonic states we've also got a superposition but in the fermion wave function it's the negative sign here specifically that makes it an anti-symmetric wave function now this negative sign has got something to do with the phase between the two possible superposed states but it's not really relevant to us here the important thing is that when we swap the particles the wave function becomes negative and as we've already mentioned this is what we expect from a fermionic anti-symmetric wave function but importantly notice the fact that the two particles cannot be in the same state in this wave function when we make a measurement on our system either we'll find particle a in energy level 0 and b in energy level 1 or we'll find particle a in energy level 1 and b in energy level 0. this is different to what we saw earlier with our system of two bosons they could be in the same energy level and this is a really important difference between bosons and fermions we can extend this logic to a multi-boson system and a multi-fermion system and all of these bosons can basically collapse down into one energy level whereas fermions cannot do that they must occupy different energy levels an example of this fermionic behavior is the electron shells around atoms electrons don't all collapse down into the lowest available energy level or for that matter into any one of those energy levels electrons are fermions and we see that as the number of electrons around our atomic nucleus increases the electrons have to occupy higher and higher energy levels because the lower ones are fully occupied now of course i've avoided the small matter of spin which is actually what allows two electrons of opposite spins to be in the same energy level rather than just one which is what our basic mathematics was predicting but that's a slightly different issue and i want to continue this in a video about the pauli exclusion principle but if we return back to our bosons then the fact that they can all occupy the same energy level is what allows both einstein condensation to occur usually when we have a fairly low density gas made up of bosons and we cool it to a sufficiently low temperature all of these bosons will collapse down into one energy level usually the lowest energy level and this is a bose-einstein condensate now the temperature below which our gas of bosons becomes a bose-einstein condensate is given by this equation here tc is the critical temperature below which will have a boson stand condensate and it depends on a couple of different things like the number density of the bosons in our boson gas and the mass of the bosons now when we have a bose-einstein condensate when all of these particles fall into the same energy level we tend to see some rather interesting effects scientists often like to describe this as being able to see quantum mechanical effects on a macroscopic level on a large scale whereas normally quantum effects are difficult to see they're restricted to being at a much smaller scale but for a bose-einstein condensate many quantum mechanical effects become readily apparent for example superfluidity has been observed in both einstein condensates a superfluidity is when a fluid flows with near zero viscosity which is a ridiculous concept if you think about it a fluid that will essentially flow forever now it's important to note that superfluidity isn't a direct consequence of the bose-einstein condensate it's just very easily seen in a bose-einstein condensate and this is also true for many other quantum mechanical effects both einstein condensates are really good for experimenting on but my aim for this video is to hopefully show you that some rather basic mathematics gives rise to the prediction that both sons will behave in this weird way and i hope we've been able to achieve that so if you enjoyed this video then please do hit the thumbs up button and subscribe for more fun physics content please hit the bell button if you'd like to be notified when i upload and do check out my patreon page if you'd like to support me on there huge thanks once again to skillshare for sponsoring this video do check out the link in the description below for a free trial of skillshare premium and lastly as always i want to say a huge thank you to you for watching and for supporting my channel i really appreciate it thank you so much for watching and i'll see you very soon [Music] you

Channel: Parth G

Views: 34,739

Rating: 4.9682899 out of 5

Keywords: bose-einstein condensate, bose einstein condensate, bose einstein, boson, fermion, parth g, quantum mechanics, wave function, symmetric wave function, antisymmetric wave function, critical temperature, bose gas, satyendra nath bose, indian physicist, indistinguishable particles, energy level

Id: lBpxQdikm0w

Length: 11min 27sec (687 seconds)

Published: Tue Apr 06 2021