 Dwi'n gweld, fyddwn ni'n gweld, mae'na gynllun o'r sainopsys ar y website sefydlu'r ffordd o'r ffilm hefyd. Felly mae'n bwynt yn gweld yn cael ei ddweud, yma, a bwynt i'n gweithio a gwyfynu'r newydd, y newydd, y newydd yma, oherwydd y llwyth ar y llwyddiad, bobl yn gweithio y ffrifwyr o'r gweithio yma, mae'n gweithio i'r ffilm hefyd. ond we have this week, and I'd like to make sure we do this thing properly. Now this topic is of syllabus, right, but it is actually very important. It's the core of quantum mechanics, and it's the core of 20th century physics, and I think you'll find it illuminating because it should explain why the time-dependent Schrodinger equation takes the form that it does. It should explain why the momentum operator takes the form that it does, why the canonical commutation relations take the form that they do. So I think it explains many things, but the sort of historical reasons it's not actually on the syllabus. Okay, so we know that this thing is a function, this thing is a function of x where x is now going to be a position vector. This is being the amplitude to find your system, your particle, whatever, at the location x, right? So it's because it's an amplitude which depends on x, it's a complex valued function of x. So we can tell a series, expand this. Physicists always assume you can tell a series, expand everything. So we tell a series, expand this, and we say that if we evaluate this at x minus a, then that is going to be essentially, well, to keep the notation simple, we call this psi of x, right? So this is going to be psi of x minus a dot d by dx of psi plus a dot d by dx of psi squared minus blah blah blah blah. This is just the Taylor series expansion in three variables. It's covered in some prelims course, right? And we've, yeah. So this is just the Taylor series expansion. And we now make an observation that this can be written with our now, now that we understand how to take a function of an operator and we realize that d by dx is an operator, we can write this as e, well, the exponential of minus a dot d by dx operating on a psi, where we're defining this exponential to mean this thing raised to the north power, namely one plus this thing raised to the first power plus this thing raised to the second power on two factorial and so on and so on and so forth. And that's what we mean by the exponential of this operator, okay? But we notice that this can also be written as x, this is working on psi of x. This can be written by the rules of operators and the definition of p as the exponential of minus i p upon h bar a, sorry, dot p over h bar operating on the ket of psi. Let me just remind you what my authority for that is. My authority for that is the observation that, or the definition of p, which was that x p of psi was by definition minus i h bar d by dx of x of psi. This animal here could be rewritten as x of psi, okay? So I can just make a change of notation here because of this. And where I've replaced p now by the function of p that you see there. And then we have a function of this operator. So what have we discovered? What we've discovered is that x minus a, this is the bottom line on this little piece of calculation, which is really only Taylor series expanded into psi, is equal to x on u of a of psi, where u of a is a new notation for, it simply means the exponential minus i a dot p over h bar. So that's what u of a means. This could also be written as x minus a on up psi is equal to x on up psi primed, where up psi primed, the ket is by definition u, the operator operating on up psi, right? We just call this thing psi primed. So let's think about, so let's just mathematics that Taylor series expanding, and a little bit of slight sophistication in taking functions of operators, but we've begun to do that. We understand that that comes with the territory. What does this physically say? It says that the, so there is, if you use this operator on up psi, you get a new state of psi primed. What's the point about this new state? Well, if your system is in this state, the new state, then the amplitude to be at x is the same as it was when we were in our old state, somewhere behind our current location, back at x minus a. So here's a visualizer. Here's meant to be a picture of this. We can get it to come back. Yes, we can get it to come back. If I could find a pointer, which I probably can't never mind, but so if psi were that sort of, if the probability density associated with psi with that spherical blob on the left, the lower left, and a is that vector displacement up there, then the amplitude to be at some point, take any point x in the sphere of psi primed, if you move back by a, you come to the corresponding point on the spherical density associated with psi and the amplitude in psi matches the amplitude in psi primed. That's sort of visualisation. This statement of what this is telling us, it's telling us that psi primed, the amplitude to be at psi primed is the same as the amplitude over here at a point back, which means psi primed is the state that our system would be in if we were able to just shove it down the vector a to translate it by a. Then we would get a new state with these properties. So what have we done? We have discovered what the operator u of a does. u of a shoves the system by a displacement a. Now a is just an ordinary boring vector. This is an operator, but this is an ordinary boring vector. It's a set of three real numbers and we can differentiate, we can do d of psi primed. The state that you get is a function of a. We can do d by d of psi primed of a sub i, a sub j, should we say, to avoid confusion between the index i and the square root of minus one. We consider the rate at which this thing changes when we change the parameters that appear in here. When we differentiate this exponential, as everybody knows, when you differentiate an exponential, you get the exponential back. That's just by the magic of that particular power series that defines the exponential. Then we need the differential, so that's going to be u. So differentiating u, we're going to get back u, but we're also going to get the derivative of this with respect to a sub j, which is going to be minus pj over h bar. Then, of course, psi will stick around because psi is not a function of a. If we would now set... Now we can just recall that this thing is psi primed, so I now have that d of psi primed by the a sub j is equal to minus i... Let's multiply through by i h bar, and then we have that this is equal to pj of psi primed. This now answers a question which I forgot to ask at the beginning of the lecture, which is, what actually does the operator p do? An operator associated with an observable. So, with each observable, observables, we have associated an operator. We did it originally by saying that q, the observable associated with q, was by definition qj, qj, qj. And this operator, we're taking advantage of the fact that an omission operator is uniquely characterized by its eigencets and eigenvalues. So, if you specify these, you specify these, if you specify this, you specify this. There's a relationship here, which we found useful. We've discovered that the expectation value of q, for example, is equal to this mathematical animal and other things, and other useful things. We found the rate of change of expectation values depends on the commutator of q and on the operator, which is the operator associated with the energy, et cetera, et cetera. But we haven't actually addressed or answered the question of what these operators that we're introducing actually do to states, because an operator turns a state into a new state. So, for example, so the operator q turns a psi if we expand a psi in its eigenstates, right? So, if we write it like this, so we know we can expand any psi, thus, in the eigenstates of this operator, and then we know how to use this on this. So, this is equal to the sum qj... qj... oops. So, when we use the operator q on a psi, we get this stuff here, which is some long gobbledygook. But if we measure q, then a psi goes to qk for some k. It doesn't go to this long list of stuff. It goes to one of these things, and the one of these things is chosen at random, somehow by nature, not discussed by theory, no answer offered by theory, merely probability distribution under which we get one of these things is predicted. But we know that the state of psi on making a measurement collapses into one of these states here. So, the operator q is not doing measuring, that's the point. And we have discovered, apropos of the operator p, what is it doing? What p does is give you the rate of change of your state when you shove something along, so px gives you the rate of change, px of psi gives you the rate of change of your state if you shove it down the x-axis. So we're learning what the operator px does, and what it does is not measure, but displace. Let's, for a piece of practice, let's check this out on let's check this out on this state. Let's, for fun, apply ua to this state, which is the state of definitely being at x, and make sure that we can produce x plus a the state of being at x plus a, because if it's true, if you take the state x and you displace it by a, you must have this state, right? Let's make sure that this is the case. So what we want to do is use ua on x. Now this operator here is a function of the momentum operators, right? It's that exponential a dot p. The natty way to do this is to decompose this into a linear combination of states of of well-defined p. So we write this as d cubed p of px p. So basically I've subbed an identity operator in front of the x. This is a boring complex number. This is the complex conjugate of the wave function of the wave function associated with being of having well-defined momentum. So we know what it is. It's e to the minus ip upon h bar dot x over h bar to the three-halves power. We discussed that when we talked about generalisation to three dimensions. That's what this complex number is. This operator ignores that complex number because it's a linear operator and goes straight to its target, which is this. Then all the p operators in here meet their eigenstate p and get transformed simply into their eigenvalues. So this becomes, when this thing hits this, in the p's in here are operators, but when they meet that because that's its eigenstate, they simply become eigenvalues. So we get an e to the minus i a dot p over h bar times the ket, the eigenket left behind, and still we have to do a dqp integration. So this is no longer an operator because it already worked on that and produced its eigenvalue. So we can rearrange this. We can put those two exponentials whose arguments are mere complex numbers. We can gather them together, and this becomes the integral dqp over h bar. Sorry, that isn't barred. That's unbarred. Excuse me, three-halves power. H Planck's naked constant. E to the minus i p. Well, it doesn't matter what order we write these in. You see because this is a number and that's a number. So I'm going to write this a plus x dot p upon h bar up. But if I now ask myself what is x in this notation, I probably should have written this down originally, it was dqp over h three-halves power e to the minus i x dot p over h bar p. This is just a standard expression which I've essentially used above for decomposing a state of well-defined position as a superposition of states of well-defined momentum, where this is, this thing here is nothing but px. So since this is the general formula, this state that we're producing ua on x is given by the same formula but with x replaced by x plus a. Because the only difference between this formula and this formula is that here we have an x and there therefore we have an x and here we have an x plus a, so we should have an x plus a. So this establishes indeed that x plus a is equal to u of a on x. So that's just a particular extra very vivid example of basic principle. So what we want to do now is generalise this to any continuous transformation. We always require proper normalisation. We require apsi apsi is equal to one. Why? Because this tells us that the total probability to get some measurement to find something is one. That's why we're completely weighted to that normalisation. So we're interested in transformations that preserve this property. This shoving it along transformation was one example. In a minute we'll talk about the transformations associated with rotating our system around some axis. But there are many transformations we might make. So what we require, what we're going to say is that apsi goes to some newfangled state which is some operator u on our old state and the restrict in light of this we're going to restrict ourselves to one is equal to apsi primed apsi primed. If we take our new states they've got to be properly normalised which means that we're looking at apsi u dagger u apsi right? We require this is one but this is by definition u apsi so if we take the mod square of this we're looking at that where u is this as yet undetermined operator and the thing is so this has to be true for all apsi for any quantum state this has to be true that this thing is one and there's a technical detail about establishing that this is one, there's a box in chapter 4 of the book doing this which I don't propose to go through it's very straightforward and simple but I don't want to take the time to do it because it's mere mathematics from this from the fact that this has to be one for any apsi we can deduce that u dagger u is in fact the identity operator from this statement this follows fairly straightforwardly but I'm not actually proving it right now so operators of this sort as I expect you know from Professor Esla's course are called unitary so unitary operators are precisely those operators which leave the length of our states unchanged and in the present case for physical reasons the length is one now let's so we're dealing with one such earlier on but let's suppose that u is a function of theta in that case u is a function of a let's theta just be some parameter where so theta is a parameter which we can make small well shall we say which can go to zero so the idea is that theta is the amount by which you transformed there a was the distance which we had displaced so a is analogous to theta here theta is just stands vaguely for the amount which you can do something and we want to be able to say that we can we can reduce this amount continuously down to nothing when we're doing absolutely nothing so we're going to have that u of nought is the identity operator because that's the operator that does nothing so we want to have this parameter and now we're going to argue that if theta is small we should be able to tailor expand I said physicists assume you can tailor expand everything so we're going to tailor expand this so we're going to have that u of theta which is now small is u for theta equals nought which we've said is one the identity and now we're going to write the first order term in a slightly funny way we're going to write minus I theta tau and then we'll have terms order theta squared so this is a tailor series expansion only the first two terms the zero term and the first derivative term and all the other terms we just got wrapped up under order theta squared not saying what they are and this is an operator it has to be an operator because this is an operator this course is an operator that is a mere number that is a mere number, a real number so therefore this has to be doing the operating but we've just chosen a particular way of writing the first order, the first derivative term in a tailor series so this is a tailor series and relies only on the idea that there's a whole family of transformations which could be reduced to the identity transformation as theta goes down to nothing when you don't do anything ok now we want to look at this condition that we want to have a look at the condition that the identity is u dagger u so let's write what's u dagger if this is u u dagger is going to be i dagger which is i and then we'll need the dagger of this which is going to be plus i theta tau dagger plus order theta squared which we're going to ignore and that has to be multiplied on i minus i theta tau plus order theta squared which we're going to ignore so when you multiply these two brackets together ginormus job in principle because they're all this infinite number of terms and this and that but we won't need to bother with much algebra we must get the identity operator and we must get the identity operator completely regardless of what theta is because this is meant to be this is a unitary transformation regardless of theta so let's work this out so what do we have the lowest order term is this on this then there are first order terms which you get this on this and this on this so we're going to have plus i theta tau dagger minus tau and then we will have terms like this on this which will be order theta squared this on this will be order theta squared we'll have this on this will be order theta squared so plus order theta squared we've accounted for everything through linear order so this is supposed to be true for all theta doesn't matter what theta we take should be true if it's going to be true for all theta then we can equate powers of theta on both sides so the coefficient of theta to the north namely the identity should be the same on both sides well it is that's a relief the coefficient of theta to the first power should be the same on both sides on this side of the equation there is no power is nothing so it better be nothing on this side too so this implies that tau dagger is equal to tau that is to say tau is Hermitian Hermitian operators we suspect are associated with observables so the argument here is that every such transformation is going to be associated with a Hermitian operator and the reason this i was put in here this was totally gratuitous um sorry in right up there the reason that i was put in there which was a totally gratuitous decoration but it went in because that ensures looking forward it ensures that tau is a Hermitian operator rather than an anti Hermitian operator which it would have been if the i had not been put in so there's a suspicion that this tau and it will always turn out to be the case that this tau will be associated with an observable this is how observables become associated with operators in both classical mechanics and quantum mechanics or should have said in quantum mechanics and in classical mechanics it turns out that it's true in any mechanics right and if we if we write the equation psi primed is equal to u of theta times of psi is equal to 1 minus i theta tau plus dot dot dot psi and we do d theta primed sorry d psi primed by d theta we find that this is equal to minus i tau psi plus order delta squared so if if we put theta equal to naught then the delta squared goes away sorry the order of theta squared and multiply this equation through by i and we get a very important equation which is that i d psi primed by d theta is equal to tau so this observed the operator tau the Hermitian operator tau which we suspect is connected to some observable well will turn out to be connected to some observable in every case is the thing what does it do what it does is it measures the rate of change of your states when you change the parameter theta so this is a generalisation of where are we this equation this equation here yeah so this is a concrete example of this now this equation has a tiresome H bar here why is it got a tiresome H bar here because in that exponential there's a tiresome H bar on the bottom right so here we had the exponential of minus i a dot p over H bar and if you do the Taylor series expansion of that you get 1 minus i a p over H bar so that the role of tau in our conceptual apparatus here is played by p over H bar there and it's an unfortunate historical accident that momentum that this operator which we call the momentum operator has been defined with a wretched H bar so we have to divide through by H bar to get rid of what we shouldn't have put in the first place so it's one of these many cases in physics where history forces us into a bad notation and even a degree of intellectual muddle that H bar would have been better left out but the reason is that momentum came to Isaac Newton's attention before quantum mechanics or this stuff was thought about and so it came to mean something which is really a derivative thing which is really something which follows on from momentum's fundamental role which is something which shoves your system which spatially translates your system and if we want to do so we've defined tau, tau came in here through a formula for u of theta when theta is small we would like to know how to do u of theta even when theta is large so for large theta what we should say is take a transformation through large theta n steps so if we are told to find out what u is for a large value of theta the way to go is to make many transformations one after another through small steps of length theta over n then if n is big enough no matter what the value of theta we're given we can write that what we can do is we can say that psi primed which is u of theta of psi of course is equal to u of theta over n u of theta over n u of theta over n n of these terms will multiply together operating on up to psi so we make a transformation by an amount theta over n and then another one theta over n so there are n terms and each one of these we can use that natty formula up there because for each one of these theta over n is small so this can be written as i theta over n tau plus stuff which we're going to be able to neglect this is raised to the nth power because there are n of these terms on up psi and now we take the limit as n goes to infinity to be completely sure that this plus dot dot dot stuff can be neglected this plus dot dot dot stuff is order theta over n squared so to be sure it can be neglected we can go to the limit n to infinity and then we have a theorem of calculus for what this one plus a bit plus something over n raised to the nth power goes to an exponential so this mathematics now tells us that this is the exponential of minus i theta tau operating on up psi so tau as the first order Taylor series term but this apparatus tells us that that's all we need to know in order to find out what u of theta is for any theta which is I think slightly surprising you don't need to know anything in the higher orders what do we say we say that tau is the generator of both the unitary transformations unitary operator rather and the transformations psi goes to psi prime we say it's the generator this is badly written tau but the generator is the operator you stuff in up here in the exponential it's always times minus i for conventional reasons and then a parameter theta that tells you how much you've generated so for example p over h bar not p sadly but p over h bar is the generator of translations that's just jargon so now let's think about time to move to a new board, think about rotations this is where it becomes slightly more interesting because we will discover that in quantum mechanics rotations seem rather more complicated they seem a bit different they are actually significantly different quite amazingly different from rotations in classical physics and I think this is not fully understood even now alright so to generate translations we in fact need three operators we need px, py and pz why do we need three operators because to define a translation we need to specify a vector because we have to say in what direction we're going to go and how far we're planning to go and those three numbers define a vector alternatively we can say ok you know that so we should expect that there are there's more than one generator of rotations because in order to specify a rotation we have to specify a rotation axis and how far around that axis we're going to go if you know the axis around which so here's a solid body I can rotate this in a whole variety of ways to specify one rotation I specify the axis I'm going to rotate around and I specify how far around that axis I'm going to rotate so we expect three generators of rotation because we specify a rotation with three numbers now there are many ways just as there are many sets of three numbers I can use to specify a translation because I can orient my x, y and z axes in any which way I like there are many ways in which I can specify three numbers that define a rotation and those of you who've done s7 the classical mechanics option will have heard of Euler angles of which there are three Theta Phi and a Psi but the handiest way to specify three rotations is actually through a vector we're going to use alpha so that's alpha x, alpha y comma, whoops, comma alpha z so an alpha hat the unit vector operator, it means a unit vector whoops unit vector parallel to alpha is axis of rotation and mod alpha ouch the modulus of the vector alpha is the angle through which we plan to rotate by so these three numbers are a handy, convenient system for specifying which rotation you wish to refer to and I now say that there must be, so the rotations form a continuous set of transformations of my system because I can rotate my system by a little bit or a lot and when I, so there must be a state of the system which differs from my previous state only in being rotated and this state must be reachable by some unitary operator u of alpha and this apparatus here tells me that u of alpha can be written as an exponential of minus i alpha dot j where this is playing the role of tau this is the operator it's a set of three operators as promised because this means alpha x jx plus alpha y, jy plus alpha z, jz so it's a set of three operators jx, jy, jz they must exist because and it's going to be Hermitian it's going to be Hermitian because this operator is going to be unitary and we've shown the connection between Hermitian operators and unitary operators so I think this, I hope that much is absolutely self-evident it will be when you think about this through again there must be this operator it's going to be Hermitian so it's going to be a candidate for an observable and the question arises what observable is it going to be the operator of the operator associated with translations which had to exist it was a logical necessity that it existed and we have shown that the operator is actually a it's actually the momentum operator divided by h bar so I hope it won't come now as a great surprise that this operator is going to be the angular momentum operator we're not proving this I'm saying it will turn out to be angular sorry, this is terrible angular so the angular momentum operators are the generators of rotations in the same way that the momentum operators are the generators of translations but we will we will have to build confidence that that's the case as we go along I'm saying that this will turn out to be the case I hope it will be clear at the moment I just hope that that's a plausible conjecture that it is the angular momentum that we're talking here about the angular momentum operators and of course the reason there are three of them is that angular momentum itself is a vector so you can have an angular momentum around the x-axis an angular momentum around the y-axis and angular momentum around the z-axis because you have those three numbers you have three operators and we're going to have the analogue of this formula here is going to be that d by d alpha the modulus of this angle d by d alpha of psi primed i times this is going to be the unit vector alpha dot j on a psi you might want to just check the algebra on this if you do the derivative so why is this a function of alpha only because this is u which depends on alpha on a psi which does not so we're talking about the derivative of this exponential with respect to the modulus of the vector but this exponential could clearly be written as mod alpha times the unit vector alpha do the derivative and you'll find this important relationship here so what does the operator this is just a single operator if you take the unit vector alpha and you dot it into the three operators j you get a linear combination of these three operators which is an operator and what does this thing do for you it measures the rate of change of your state when you rotate it around that axis that's specified by this that's what it's physically doing for you and this is an important relation we'll come back to we're not going to quite finish this but let's get going so we've talked about translations and rotations and they have this in common that they have a free parameter how much you do of them which you can turn right down to zero when you do nothing and the operators just become the identity but we have to use in physics we have important use to make of transformations which are discreet you cannot turn them down to nothing you either do it or you don't do it and the classic example is the parity or reflection operator so if I have an ordinary vector then p turns the position vector x into minus x there's a transformation you can make where you start with a thing and you choose a point which you call the origin and you move every every part of your thing through that origin into another thing so if the origin is here if I move every part of my lower hand through the origin by a certain amount to this equal distance opposite from the origin it should become my upper hand my right hand left hands and right hands are related in this way through reflection through the origin if you put the origin symmetrically between the two so this is a transformation you can make this is a mental transformation it's not a real physical transformation it's a mental transformation you can ask yourself suppose I had a system which was obtained from my real system right here in the lab by this operation a question you can ask is would the dynamics of this system so as my real system around here is moving around imagine this thing is a solar system or my hand wiggling would the system that you get above by mirroring each of these points through the origin would that behave like a real thing in the universe and in classical physics that's the case what you see down here reflected through the origin produces a wiggle up here which could happen all on its own there would be a dynamical system that would produce that wiggling one of the great discoveries of the 20th century was that that's actually not true in all physics when weak interactions are involved if you make a model by taking your real system and playing this silly game with it you get a thing up here which you can distinguish from a real system because the real system up here couldn't behave in that way so there is this operation of taking your system and mirroring it through the origin and there's a classical operator p which does this it just changes the sign of all your of all vectors of all components of a vector it turns x into minus x it puts a point to the opposite point across there that's the real analog of this well it's this animal so I'm going to have a long tail on p to imply a classical operator which simply changes the sign this is classical and it's just a change of sign of all vectors we want a quantum analog and the quantum analog is the thing this what's this this is the amplitude to be at x if you're in the state i should have written this separately this is the amplitude so ppsi is going to make a new state what state is it going to make this state what's the point about this state the point about this state is if you're in this state the amplitude to be at x is minus is the amplitude to be at minus x if you're in the original state, right? So this is our original state, and the amplitude to be at minus x in this state is equal to the amplitude to be plus x in this state, which we've gotten by using p on a psi. So p takes the state of my hand here and makes this state. That's what it does. Can we say about what interesting statements can we make about this? Well, one very obvious statement that we can make is that if we look at x p squared of psi, then that's x p p of psi, by definition, obviously. We use this rule here on this state. So that's equal to minus x on p of psi, right? Because this p and that x, using that rule on this state, gives me this. I'm replacing psi in this formula by p of psi. And now I can play the game all over again. So by using the same rule, I find that this is equal to minus, minus x on up psi, which, of course, is equal to x on up psi. So if you use, what does this tell me? It tells me the amplitude to be at x when I'm in the state p squared of psi is the same as the amplitude to be at x if I was in up psi. In other words, these two states are the same. So that implies that p squared is the identity operator, which implies that p inverse is equal to p, p is its own inverse. And what I should do next is show that p is Hermitian, but we'll have to do that tomorrow. And therefore, p is going to have the properties of an observable.