Ever heard of Quantum Operators and Commutators? (Explained for Beginners)!

hey what's up you lot parth here in today's video i want to be building up a framework so that we can understand what i think is one of the coolest theorems in quantum mechanics erin fest's theorem i'm actually going to be splitting this explanation up over two videos in part one that's this video we will be looking at quantum operators and commutators and right now this looks and sounds all weird and messy but don't worry using just a basic knowledge of high school mathematics we're going to be able to understand what operators and commutators are in part two of this little mini series we'll be learning about something known as expectation values something that's really important in quantum mechanics and something that you might be familiar with if you've studied probability theory and finally we'll be putting together everything we learn in this video and in the beginning of the next one to understand what erin fest's theorem is trying to tell us and like i've mentioned already you only need to be familiar with like high school level mathematics to understand what's going on here at least at a basic level oh and yes i realize my quarantine hair is getting longer and longer i'm gonna get a few comments about this so i thought i'd mention it straight off the bat but if before we start if you do enjoy this video then please do hit the thumbs up button and subscribe to my channel for more fun physics content okay let's finally get started okay so this is aaron fast theorem and like i said it looks really daunting but the really interesting thing about it is that it can be thought of as a link between classical physics and quantum mechanics classical physics of course being everything in physics that came before albert einstein's era and quantum mechanics being the weird strange stuff that we'll be looking at in this video but before i explain exactly what each term in this equation means we need to lay down some groundwork the first piece of information necessary to understand erenvest's theorem is a really important idea in quantum mechanics in general it's known as an operator if we happen to be studying a quantum system that is something that we're using quantum mechanics to study where that system could consist of for example a single electron then an operator is a mathematical object that we can apply to this quantum system or more specifically to what's known as the wave function of this quantum system now i realize i've thrown a fair amount of jargon at you so let's decode what i've just said we've said that our quantum system in this case is a single electron and a wave function is basically all of the mathematical information that we know about our system specifically the wave function is directly linked to something known as the probability distribution of our system which gives us information about things like how likely we are to find our electron at certain points in space and other things like that but basically it is a probability distribution so it gives us information about the likelihood of certain experimental results when we make a measurement in this particular case the measurement that we're making is a measurement of the position of the electron i've talked about this in a lot more detail in a previous video so i'll link that up here if you haven't seen it already but this brings us nicely onto the idea of an operator like i said an operator is a mathematical entity a mathematical object and we apply this mathematical entity to our wave function and in real life this is equivalent to making a measurement in order to make things a little bit clearer let's take a specific example into consideration let's talk about the position operator this operator is labeled with the letter x x for the x position of our electron along the x-axis and we put a little hat on top of this operator because all operators are labeled with little hats now if we were to make a measurement of where our electron is in space then mathematically this is equivalent to writing this we have applied the position operator to our wave function and this eventually ends up giving us some information we're not going to talk too deeply about that but let's look at x psi specifically first thing to think about is notation we've written our operator on the left hand side of our wave function ket and again i've talked about cats and bras already in previous videos so i'll link those up here as well but every time we see a wavefunction and then we see an operator to the left-hand side of it then we know that that operator has been applied to our wavefunction which is equivalent to making a measurement on our system and the fact that we write the operator on the left hand side by the way will become important shortly but at this point we should really talk about a fundamental difference between classical physics and quantum mechanics in classical physics making a measurement is pretty trivial in many ways for example let's say i've got a cricket ball we can make a measurement of the position of this cricket ball no problem i stick a ruler next to the cricket ball photons bounce off the cricket ball and off the ruler and i have information about where that cricket ball is in space whereas in quantum mechanics making a measurement is kind of a big deal because quantum mechanics tells us or at least the copenhagen interpretation of quantum mechanics tells us that before we make our measurement there's some probability of finding it here some probability of finding it here some probability of finding it here and so on and so forth before we make that measurement we can only work out probabilities about the location of this electron and it doesn't just apply to the position operator either this works for other things as we'll see shortly but when we make a measurement on our system this results in a fundamental change to our wave function this is known as the collapse of the wave function because before we made the measurement there was some likelihood of finding our electron here here or here and after we make the measurement we know at least for that very short moment in time that our electron is in one specific place so the wave function has fundamentally changed and this is because the probability distribution has changed it's important to note though that i'm not saying that the electron itself moves or collapses in any way what i'm saying is that the wave function of the system changes and making a measurement is what caused that change so making a measurement has genuine consequences for our system in quantum mechanics in a way that it doesn't in classical physics when we make a measurement for the position of our electron for example for that tiny sliver of time when we've made that measurement we know that the electron is here so the probability distribution is 100 here zero percent anywhere else but very quickly that probability distribution starts to almost spread out so we become less and less sure of where our electron is as time passes and this is in accordance with the schrodinger equation the schrodinger equation tells us how the wave function changes over time but actually making a measurement on our system caused our wave function to change in a different way not determined by the schrodinger equation now you should realize that i've skimmed over a huge amount of detail when it comes to talking about making measurements on quantum systems the most important thing for us to know here is that making a measurement has real consequences on our system through the wave function and mathematically making a measurement is written using an operator on the wave function and we've talked about the position operator already but a couple more examples of different operators are the momentum operator p and the hamiltonian operator h now the hamiltonian is the big boy in quantum mechanics you'll see this written everywhere you'll even see it in the schrodinger equation and this operator in most cases is actually the total energy of the system operator because it accounts for kinetic energy and potential energy again not super important for us here but it is good to know about the existence of the hamiltonian because it's going to crop up later side note by the way here's an exercise for you try and work out why the kinetic energy term in the hamiltonian is written like this with the knowledge that p is the momentum operator and m is the mass of the system in this case the electron so we've seen that making a measurement has a fundamental impact on our quantum system but before we go any further there is something i should clarify when i talk about making a measurement it's not the existence of a conscious being that's making a difference to our system it's actually the interaction between our system and whatever we use to measure it a lot of people like to very heavy-handedly talk about the importance of a conscious being to quantum mechanics but this is a very subtle idea the active measuring is very very important true but the existence of a conscious being doing the measuring is not really what we make it out to be there are a lot of very clever scientists working on this topic and it's not as simple as saying we are conscious and therefore everything of the universe exists because we are there to observe it anyway let's come back to talking about operators and the application of operators to quantum systems because we've seen that making a measurement can fundamentally change the behavior or properties of a system making two measurements is even less trivial it's even more complicated because now the order in which we make these two measurements matters going back to classical physics going back to our cricket ball let's say we were to measure the position of our cricket ball first and then we were to measure its momentum right now its position is x is equal to five from our origin and its momentum is zero but if we were to measure these two quantities the other way around momentum first and then position then we'd still find that momentum is zero and position is x is equal to five from the origin no big deal right well this is not necessarily the case in quantum mechanics because measuring the position of our quantum system changes the wave function of our system so when we measure its momentum afterwards we're actually making a measurement on a different wave function and so mathematically speaking if our initial wave function before any measurements was psi then px psi is different to xp psi in general not always but in general in other words measuring the position first and then the momentum of our system can yield different results to measuring the momentum first and then the position and we know which operation is done first by the way by seeing which one is nearest to our wave function so in this particular case the x the position operator is nearest to our wave function so that is done first and then we measure the momentum of the system now because p x psi is not the same as x p psi this means that subtracting one quantity from the other is not going to give us zero because these two values are different whatever those values may be like i said we're not going to go into too much detail about that here and based on this idea we can come up with a neat little mathematical trick if we first start out by factorizing this expression it's not quite the same as the factorizing that you do when studying algebra in maths but it kind of looks like it so we'll go with calling it factorizing and then we can say that this quantity inside the brackets is known as the commutator between x and p so first of all the commutator is written with square brackets around the two quantities and the reason that we define this commutator is because it tells us whether the order in which we make these two measurements actually matters in this particular case with position and momentum it actually does matter because our commutator is not zero which means that measuring position first does not give us the same result on our system as measuring momentum first and specifically for position and momentum this is actually true for when we measure these two quantities in the same direction for example when we measure the position along the x-axis and we measure the component of our electrons momentum along the x-axis we say that these two operators do not commute in other words the order in which we make these measurements does matter and we need to be careful which order we make these measurements in but we can think about two other generic operators let's call them a and b it doesn't matter what they are but we could have a scenario where measuring a first and then b gives us the same result as measuring b first and then a in that case the commutator would be equal to zero because a b applied to our wavefunction is the same as ba applied to our wavefunction so those subtract to give us 0 and hence the commutator is 0. and in this particular case we say that a and b do commute and this is quite similar to actually saying something like 2 times 5 is equal to 5 times 2. the order in which we do this operation 2 times 5 or 5 times 2 does not matter because it gives us the same end result and so we say that the multiplication operation is commutative because we can swap the order of the two things being multiplied but as we've seen for quantum operators this is not always true order matters now coming back to commutators that are not equal to zero in other words the order in which we make two measurements matters for our purposes we don't need to go into too much detail about what the right-hand side of this equation actually means we just need to know that if it's zero then the two operators commute and the order in which we make measurements doesn't matter but if it's non-zero the order does matter okay at this point let's pause so far we've learnt about operators and commutators we've seen that operators are mathematical entities that correspond to making a measurement on a quantum system in real life and we've also learned that commutators are a useful mathematical tool in helping us understand whether the order in which we make measurements on our quantum system actually matters this is where i'm going to end part one of this little mini-series in part two we'll be learning about expectation values in quantum mechanics and we'll be applying everything we've learnt in this video and expectation values to understand erin fest's theorem but if we take a quick peek at our invest theorem now you'll see that there's a commutator in there and you'll see that there's a few operators in there as well i'll tell you now that a is a generic operator which means we can replace it with operators such as position momentum or other operators and h as we've seen already is the hamiltonian and so we see that the hamiltonian indeed does play an important role in quantum mechanics as it does in erin fest theorem like i said though there's still a lot more for us to learn and we'll be doing that in the next part of the video and finally decoding exactly what this theorem means we'll be seeing how erin fess theorem sits very nicely in the weird and wacky world of quantum mechanics but links back to the safe but ultimately problematic world of classical physics so if you enjoyed this video then please come back for part two and hit the thumbs up button it really helps me out and subscribe to my channel if you're interested in more fun physics content hit the bell button if you want to be notified every time i upload and check out my second channel where i've recently released some music i think there's a lot that we've covered in this video so if i've not explained anything clearly enough or if i made a mistake let me know in the comments down below and we'll try and fix that as quickly as possible thank you so much by the way guys for 60 000 subscribers i'm always blown away by all your support you guys are so wonderful and like you leave me such nice comments so thank you so much for that anyway i've spoken on for long enough so i'll see you in part two of this mini [Music] series [Music] [Music] you

Channel: Parth G

Views: 32,329

Rating: 4.9785113 out of 5

Keywords: quantum operator, operator, commutator, quantum physics, physics, ehrenfest theorem, position operator, parth g, hamiltonian operator, advanced quantum physics, quantum mechanics, measurement, order of measurement

Id: so1szjHu7jY

Length: 13min 46sec (826 seconds)

Published: Tue Jul 21 2020