 Dwi'n tyfyr. Beth efallai cyn i'r bau sydd o'r ddiwethaf ar y dyfodol ble ar y cyfnod, yr hyfforddoedd dyfodol, ystydig i ddiwethaf, yn rhan ffant i ei byddo oedd ymwybr oleochol. Felly mae'n sicr o'r ddych chi'n dechrau sy'n ddwy'r rhan fydd o'r hyn. Dybin yn rhagol nhw yn unith yn sefydlu a phaith gyda'r angol yn pwyllhau yma, y dyfodol ymdishwyl yn ystod powerfa, mae'r pwyllt yn cael ei dyn, rydym yn gwybod i'r ddwyllwg mynd ar gyfer y lleff إadig, a'r hyn o chi'r twethaf hwyl fawr hwanaidd yn y cwysbeth y ffwrdd. Hwn oeddwn i'w ddweud y pwyllt yn y cwrdd Aberystwyll yw gwrdd ar gyfer y cwrdd. Felly mae'n blaen olygu ofyn yng nid, mae'n llwyddo'r pwyllt yn gwasio'r pwyllt yn dweith yng N. is going to be a half, and not minus a half, and you can only get those two answers, a half or minus a half. Even when you know it's plus a half, that doesn't mean the spin is really along the z-axis, there's still a substantial component of spin in the xy plane and you do not know its direction. So we use this loose torque, the spin is along some particular direction like the z-axis or this n, meaning it's a shorthand for you are certain to measure plus a half if you measure the spin in this direction. Right, so with that health warning, the state in which you're certain to measure a spin a half in the direction n is this linear combination of the state in which you're definitely going to get minus a half along the z-axis and this state in which you will definitely get plus a half along the z-axis. Similarly, the state in which you're definitely going to get minus a half along the n direction, the unit vector n, is this other linear combination of those same two states of well-defined spin along the z-axis. So I asked you to take that stuff on trust and then we did some stuff with that and I hope I persuaded you that these formulae are not completely implausible in the sense that what we did was we calculated the probability, if it's definite, if we know the spin is plus a half along the z-axis, we calculated the probability that we found plus a half along the n-axis, we found that that probability was in fact simply cos squared theta upon 2 and this behaved in a reasonably plausible way in the sense that when it was 1, when the n direction was the z-axis and it went to nothing, when the n direction was the minus z-axis and other such good stuff. Then I wanted to show you this, all I wanted to show you was what these formulae predict for what the state is of definitely having plus a half for the answer to what's your spin in the x-direction, right? It's easily done because we have the formulae here. I was thinking I was trying, for some reason it went into my mind that I had to derive these formulae and we didn't have the bits on the table to do it. All we have to do is plug into those formulae that theta's pi upon 2 phi's nought, that is, by definition of polar coordinates, makes n the x, the unit vector in the x-direction, which I'm calling ex, and if you put in pi upon 2, then you're looking at sine pi upon 4, cos pi upon 4, 1 over root 2, and pi being nothing means those two exponentials are nothing, so the state of having your spin definitely down the x-axis in that sense, right? Again, with that health warning, so the strict statement being that we are guaranteed to get plus a half if we measure the spin down x. It turns out to be just the sum, essentially the sum of these two states. That's not very exciting, I think. But now let's put in theta is pi upon 2 phi's pi upon 2, which, by definition of polar coordinates, makes the unit vector n the y-direction. Then what happens? Well, what happens is that those e to the i phi's upon 2 become e to the i pi's on 4, and if I take the first e to the i pi upon 4 out, then the second one, so this cosine is 1 over root 2 again, which we've taken out, but this one becomes e to the minus i pi upon 4 twice, i.e. e to the minus i pi upon 2, which is actually minus i, I'm slightly worried by this but never mind. The sine is of no importance. I thought it was a plus i, but it is looking like minus i at the moment, so maybe it is minus i, it's of no importance. What matters is that this state, which is physically quite distinct from this state, is also a linear combination of these two, and the probability of, if you are in this state, the probability of measuring your spin-along z to be minus is going to be a half because this 1 over root 2. The complex number which comes here has the same modulus as the complex number which comes here and ditto here, that this complex number is the same as the complex number which appears there in modulus but different in phase. The crucial point is that we are working with a formalism where we're saying the state of my system can be written as a minus minus plus a plus plus. We understand that these things are the probability amplitudes to measure spin down on z given that my system is in this state and this is the amplitude to measure spin up on z given that this is my state of my system. That's the formalism we're working with and you might think that it's only the modulus of these complex numbers that matters physically because the probabilities are obtained by doing mod square of them. But this example is showing you that that's not the case. The phases of these things are vitally important as well. That is the very quantum mechanical thing. The complex phase, the phase of the complex number encodes crucial physical information. Is the spin of this particle more or less down the axis or more or less down the y axis is controlled by the phase of this animal relative to the phase of that animal. Let's do another example of a physical system which is a two-state system. Let's talk about polarized light. This is an example which enables us to connect back to classical physics in an interesting way. Let's do classical physics. We know all about polarized light. You may not quite because it may be part of an upcoming EMAC course but you will recognise enough of it, I think. I can write the electric field supposing we have the y direction this way the x direction this way. Suppose we have polarized light with electric vector in this direction with that angle being theta then we can say the electric vector is equal to some number in front of cos theta times E x plus sin theta times E y times cos omega t. We are writing down the electric vector of a plane polarised electromagnetic wave travelling in the z direction in some plane. This is what it looks like. It oscillates with some angular frequency omega. Supposing we stick in this beam and hit some polaroid and let's imagine the polaroid blocks the electric vector so polaroid blocks one of the polarisations. Let's orient our piece of polaroid so it kills the oscillations parallel to the y axis and let's only through the oscillations parallel to the x axis. After polaroid we're going to have that E is equal to E naught cos theta cos omega t E x. It just wipes that out. The intensity of the radiation, the energy that it carries is going to be looking like E naught squared times cos squared theta. We should really do a cos omega t average value, which is in fact a half, but I didn't think we were really interested in that. The crucial thing is that the intensity of the light that gets through is going to be moderated by the square of the cosine of that angle. The angle between the electric vector of the wave and the direction that the polaroid lets through. That's what classical physics teaches us. How would we express this? Let's now think about this from a quantum mechanical perspective. What classical physics says is an electromagnetic wave, quantum mechanics says is a stream of photons. Each photon encounters that polaroid on its own, on its lonely own sum. Either it's killed by that polaroid, turned into something else, destroyed, or it's allowed through. It can't be half allowed through or cos squared theta allowed through. It's either allowed through or it's not allowed through. How does that look? The state of our incoming photon we can write as a linear combination we can say that this is equal to cos theta of a state in which it is going to get through, because in some sense its electric field is down the x-axis, plus sine theta of a state that is not going to get through. This is the state of certainly gets through and this is the state of certainly blocked, where we're taking the position that the polaroid is making a measurement on the photon. So what's the probability gets through? Well it's equal to, this is an amplitude, it's equal to the amplitude for getting through mod squared, which is equal to cos squared theta, and therefore the number of photons that gets through is proportional to cos squared theta, but the number of photons is the amount of energy that gets through. So it should be the intensity of the light that goes like cos squared theta and quantum mechanics recovers our classical result. We can go further than that because we know that, if we think about circular polarisation, so we know that classically we can write the electric field of a circularly polarised radiation. So in plain polarised radiation the electric field just oscillates up and down some definite direction. In circular polarisation at a given place the electric field always has the same value and it rotates in its direction. So now it's pointing in the x-axis, now it's pointing in the y-axis, now it's pointing in the minus x-axis etc. And it can go round clockwise or anti-clockwise etc. How do we write that? How do we write that classically? Well we can write that it's E naught over root 2 and then I would write the neatest way to write it is the real part of E x plus I E y times E to the I omega t and that's all inside this real operator. Let's think about that for a moment. Because what does that give me? This E x meets that cos plus I sin so we find when this real operator works and we're looking at E x times cos omega t and this I E y meets cos omega t plus I sin omega t so this I and the I that's sitting inside here make the real part of this minus sin omega t so this is looking like E x cos omega t minus E y is a unit vectors sin omega t and so that's what we get from this notation. So this indeed is a circularly polarized beam the mod square of this electric field is going to be E naught squared over 2 and it's in fact right hand polarized. In the plane it's going to go around that way because why is it going to become negative first? The component because of this minus sign. Similarly if we wrote, so let's call that E plus for E subscript plus for the electric field associated with the right hand circular polarized beam correspondingly we would have E minus for a left hand circularly polarized Johnny would be this. We get a change in the sense of rotation just by changing that plus I to a minus sign. This is a check that's true so this is left hand polarized. How would we do this quantum mechanically? What we would do is we would say that there's a state plus which is equal to the state that has this electric vector in the x direction plus I times the state which has this electric vector in the y direction and doesn't get through the pellaroid and this does get through the pellaroid and we would say, so this would be a state of right hand polarized state of our photon is a linear combination I should have a 1 over root 2 outside here that's that root 2 basically. So a state of circular polarization of a photon is a linear combination of two plain polarized states and similarly we would have that minus is equal to the left hand polarized state and we would be able to make statements like if we want a kind of statement we could make is we could add these two equations and we would discover that being polarized in the x direction is 1 over root 2 of being right hand polarized plus being left hand polarized and this is also a result that we have in classical physics that if you have a plain polarized beam consider it to be a linear superposition if you like an interference pattern whatever between two circularly polarized beams of opposite senses of polarization but there's a different but this has a different meaning sort of emotionally right this is saying that a particular state of one photon of a particular photon is this linear superposition of its two other possible states something else that you learned from this another thing that should be pointed out is that in classical physics I was using I here and up there as a sort of handy way to reduce algebra etc there was a real operator sitting in front of it the electric field was totally real and any appearance of the square root of minus 1 was merely as a shorthand as a trick as a device for compressing the algebra in the quantum mechanical case this I is I there's no real operator there's no nonsense with that this is inherently a a complex animal now maybe it's time to move across let's say a little bit about measurement we've already encountered these ideas really but let's let me take you back to what we did yesterday with the energy representation what I said was supposing I write up psi in terms of some basis vectors I because we had agreed that the quantum state of a system a ket was an inhabitant of a vector space vector spaces have bases therefore any ket can be written as a linear combination of basis vectors supposing these happen to be physically the amplitudes to measure a particular value of the energy say EI then I hope I persuaded you that the physical meaning of this Ith basis vector is the state in which you are certain to measure EI because because if if psi is the state EI in which we are certain to measure this energy then what does that mean it means that AI is one and AJ and every other A has to be nothing for J not equal to I and so we can look into this expression here under those circumstances under those circumstances of psi on the left here becomes EI this sum collapses just to I and that tells us that I is actually the state in which you are certain to measure EI so that's how we understood the meaning of these things now suppose that psi is some general thing it's some general state in other words loads of these AIs are non zero so it's some superposition linear combination of a non negligible number of these states of well-defined energy so suppose that psi is not a state of well-defined energy it is a sum AI EI with lots of non zero fine now suppose we measure the energy if we measure the energy according to our conception well obviously if we measure the energy we are going to find one of the allowed values one of the values in the spectrum of the energy we are going to come up with one of these EIs shall we call it EK so we do a measurement so we measure E and find EK having found EK we know what the energy is we know it's EK therefore we know the state of our system so now psi equals EK so after we've made the measurement psi is different from the cap that it was before we made the measurement it's changed into this which is just one of the terms that occurred in this series so this sum ran over many of these AIs and one of the AIs was K it just happened when we made the measurement bingo this is the one that popped out but having made the measurement we know what the energy is it's EK so the system is definitely in the state EK so the original wave function of the state quantum state is changed into a different quantum state on making the measurement and this different quantum state looks simpler than that one and what people conventionally say is that this quantum state is a result of our measurement is collapsed into this quantum state so this is the collapse of the quantum state traditionally known as the collapse of the wave function we haven't yet met wave functions but it's the same phenomenon now it's an extremely interesting question what's really happening this is a fundamental absolutely non-negotiable piece of this theory the matter is discussed rather more in chapter 6 of the book at some point to say in the vacation I would urge you to read that and you will find that it is this piece of the theory is fundamentally unsatisfactory it's clearly not right but nobody knows how there are various proposals many worlds and all sorts of things for fixing it but there is no known satisfactory fix there is no consensus there is no really persuasive fix the fundamental principle that I think everyone will agree on is A, it's non-negotiable it's absolutely essential for the working of the theory that we do some such collapse two, that when you make a measurement logically there are two possibilities logically it's just a thought process okay I was I wrote that down because frankly I didn't know what the energy was so it's just a thought process I wrote that down because frankly it's just a thought process so that covered my basis and it was probabilistic there were many possible values of the energy et cetera and I stuck in some amplitudes to reflect my uncertainty and having made a measurement I discovered what the energy was and so it's this and now everything's okay we've discovered something so I've updated my information and the state vector has proved information, it's a subjective change not a real change that interpretation proves to be untenable there really is a change it's there at the moment we're operating chapter 6 only chapter 6 introduces an apparatus that deals for muddle and uncertainty which is kind of worrying because in real life and real physics there's always masses of genuine uncertainty a genuine muddle but we are operating in an ideal world at the moment in which there is total clarity there is complete information there is nothing left to chance beyond what is inherent I mean so this is a completely well-defined state of the system it changes into some other completely well-defined state of the system it actually objectively changes and here we have a crucial thing that is being added in quantum mechanics to classical physics which is the concept that when you make a measurement you disturb the system that you are measuring I think this is totally reasonable it's obviously an abstraction that classical physics makes that you can make measuring instruments of arbitrary delicacy so that you can have these so when a measuring instrument interacts with a system the measuring instrument in classical physics is affected the needle moves over a light glows or whatever but the system carries on blithly without being changed in any way it's clear there's action and reaction if the instrument is affected the system is affected and since we're concerned with systems quantum mechanics is about systems which are very small it's very natural that the impact on the system should be kind of substantial so it's totally reasonable that we should be working with the theory where every measurement is associated with the disturbance of the system and leaves the system in a configuration different from the one that it founded in so that's not the problem the problem is that the theory doesn't describe the process of getting from here to here but that's a topic which I can't discuss at this stage or indeed in this course you can find something about it in chapter 6 it's all highly off syllabus I want however to point out something else which is that we we started with a basis up there remember we started with i and the mathematicians already taught us to associate with i the ket i a bra i such that j equals delta ij so in our physical example this maps into ej ei equals delta ij so this was just mathematics but it has it has a deep physical meaning as follows I think I made the point yesterday that if you want to know what a k is the way to find it is to do e k that's broadly speaking why we introduce these bras we introduce these bras because we want it out of an object like a psi to extract the amplitude for something to happen because you know amplitudes are the things whose mod square make predictions and we you know that's what we take down to the lab to test against nature so so let's ask ourselves in this context let's look at this formula this is the amplitude to find energy e k if the system is in the state of psi so what's this this is the amplitude to find e j if the system is in the state e i well if the energy is e i it can't be e j can it if j is different from i so the reason this thing vanishes when i is not equal to j is because it reflects well it reflects the fact that if your energy is e i it's e i it's not e j so this this orthogonality condition is a logical necessity the mathematicians have given it to us but we need it for physical reasons we need it it's associated with it's a requirement of our fundamental principle that this gives us the amplitude to measure e k and details like the cover of this point is suppose we've got psi psi is equal to sum a i sum on some basis i might be the energy to stay at my point and suppose we have some other state we have some other quantum state which is the sum j so these these two states of two different states because they have different they are associated with different amplitudes this is associated with a set of amplitudes a i numerical values a i and this state is associated with numerical amplitudes b i b j whatever and let's calculate the number phi psi so we have to take the complex to make the bra out of that and use our rules that we introduced yesterday so this is the sum j of b j complex conjugate j times the sum a i i but when i meets j we get a delta i j which means this and then when we conduct the sum over i we get nothing except there's only one term that contributes because of the delta i j that's when i equals j so this becomes the sum b j star a j so that tells us how to work out this complex number in terms of these quantum amplitudes that turns out to be very useful the thing I want to say at the moment is supposing I worked out the other thing I worked out phi sorry, psi onto phi instead of phi onto psi the thing would be the same here except that this would be an a j star and that would be a b i and we would be looking at the sum over j of a j star b j but this is the complex conjugate of this by the rules of complex because this is just a sum of complex numbers so we know what the rules of complex conjugation are so this is the sum b j star a j star which is phi psi star so this is an important equation to remember that psi phi is equal to phi psi complex conjugated we'll need that many times I think that's all we want there let's now introduce the next topic which is operators and their connection to observables things we can measure so what we're interested in linear operators what does that mean I guess you probably know but let me just write it down anyway so let's if q is a linear operator well first of all q is an operator what does that mean? that means it turns kets into kets it produces a ket that is to say phi if I do q the operator on a psi the ket I get another ket phi that's what an operator is it's something which turns kets into kets what's a linear operator if I have q on a linear combination of alpha psi plus beta say of chi this is just two any old two kets I take alpha times one and add beta times the other because I know I'm allowed to do that well what is that that's equal to alpha q operating on a psi plus beta of q operating on chi that's the linearity property so we're only going to be interested in these linear operators now write down an operator there's an operator a very very very important operator like this if we have a basis of kets I I can form this creature here this is the ket I somehow multiplying the bra I just like that the first thing I have to do is persuade you that this is an operator right let's consider this and I need to persuade you that this is an operator how do I do that I show you how it operates if you know how this operates on any ket then it's an operator so let's have a look at this supposing I do I at psi what does that give me it gives me the sum of I of I of psi now this is a complex number it's even an interesting complex number with emotional appeal because it's the quantum amplitude for something but let's not worry ourselves about that at the moment this is a complex number this is a ket so this is a linear combination of kets ergo it is a ket it is something we can call it phi if we want to so that means that I does turn up psi into some state phi now let us let us replace psi with no let's replace this psi with its expansion sum aii so I can write this as the sum I on this is going to work on the sum over j of a j j so this is another way of writing up psi we've done it time and time again now this I is going to meet that j and produce a delta I j when I do this sum over j every term will vanish except the term where j equals I and then when j does equal I will have I on I which is 1 so this is equal to the sum I which is up psi so I this operator this thing here is not only an operator it is the identity operator because it turns up psi any psi gets turned back into itself so we have that this thing here is the identity operator and I've told you nothing about what these things here are except they form a basis or a complete set and we use this representation of I sometimes called a resolution of the identity it's not a phrase expression I will use but we use this representation of the identity operator time and time and time again it's tremendously valuable now let's introduce a sexier operator in fact the most important operator in the universe we're going to introduce H which is by definition the sum EI EI so these are the states of well defined energy and the numbers EI are the possible energies they're the spectrum of the energy operator this operator here is called a Hamiltonian after William Rowan Hamilton who lived in the first half of the 19th century and introduced the classical analogue of this and I think it's I hope it's clear from what I did above that this is an operator let's just say if we do H we will get this stuff times E this is a complex number which multiplies this real number times some kets so we will get back a ket and it's obviously an operator associated with the energy and the general scheme is going to be that we can measure we're going to associate an operator and we're going to do it in just this way but let's just focus on this particular one for a moment because it is the most important operator in quantum mechanics and let's work let's see one thing that we can do with this supposing we work out PSI H PSI so what is this this is a number a complex number why is it working on PSI makes some state shall we call it PHI and PSI the bra working on PHI produces a number so we can say straight off that this is some complex number let's find out which complex number by putting in for PSI it's expansion some AIEI so what we're going to do now is replace both of these by their expansions so on the left here I'm going to have AJ star EJ so that is PSI the bra then I have H and then I have EJ sorry then I have the sum AI EI an age itself oh dear this is getting complicated is the sum over K of EK K sorry EK EK so every term in this expression here has been expanded in terms of the basis states the states of well defined energy states where measurement of energy is certain to yield a particular result now what happens with summing over every everything J is being summed over I is being summed over and K is summed over this EK oops yes right sorry let's work on this EK is going to this EK is a linear function on this so this passes through the AI by the linearity of this function and produces me a delta KI so when I sum over I I get nothing except when I is K so that is going to this sum is going to collapse and in the next line we're going to be looking at the sum over J of A J star EJ oops which way it's pointing is pointing this way sum over K EK EK and the action here is that is that this sum of I is going to make that into an AK so that's looking this way and that is an AK because of the orthogonality of that and that now we repeat this trick we say when we do this E J is going to pass through this by linearity and etc meet that and produce a delta JK so we'll get nothing on this sum except when J is equal to K so this is going to become the sum over K of EK this A star the J is going to be the only term that's going to survive when this meets this is when J equals K so this is going to become an AK star this AK star is going to meet with that AK AK mod squared but AK mod squared is the probability of getting this energy so this is equal to the sum of PK the probability of getting EK times the value EK in other words is equal to the expectation value of the energy so that's one reason why these operators why operators associated with observables in this way are so useful if you want to know your status of psi and you want to know what the expectation value of the energy is you take H the operator associated with energy and you squeeze it between the brar of psi and the ket of psi so you give me any observable call it Q for that observable we have agreed there will be a spectrum so there will be numbers QI the spectrum, the possible values that the measurement can return and there will be a complete set of states QI these are the states for which the result of observing Q is certain and what can I do I form the operator Q and we might give it a hat because it's an operator which is a mathematical object whereas Q is a sort of concept like momentum or angular momentum or position or something we got a natural operator which is defined to be the sum of a K of QK QK QK and by exactly the logic up there we will find that the expectation value psi Q psi so that's just repeating what I've done for the energy operator the Hamiltonian making the point that it's going to carry through for any observable now there's more than we can do more than that we can now ask ourselves okay let's look at this what is Q operating on QI what does Q is an operator what does it do to one of the special states associated with that observable well by definition this is the sum of a K of QK QK QK QI right so this is by the definition of the operator Q there is what we're operating on this is going to produce a delta KI so when we do the sum of a K we get nothing except when K is equal to I so what we get only one term in this series survives and the answer is it's QI QI so what Q does to this state of well-defined Q of the well-defined value of the observable is it turns it makes a scale model of what we started with with a scaling factor QI so in the mathematical language which I guess you've met this says that QI is an eigen ket of Q you have met that right you've met eigen ket eigen vectors whatever of operators and the eigen value so that the states of well-defined observable which are really the primitive physical thing are going to turn out to be eigen states of this operator that we've introduced and the eigen values and the eigen values are the possible values of the elements of the spectrum the possible numbers you can get if you measure the observable Q well that probably is the right moment to stop