Lecture 1 | Modern Physics: Quantum Mechanics (Stanford)

this program is brought to you by stanford university please visit us at stanford.edu i just want to do one thing actually before we start the uh the material of the class a whole bunch of people a lot of people particularly people from europe who actually follow these things on the internet seem very puzzled they send me email even some of my friends in europe who are physicists who are following it uh are very curious about what's going on here they don't see you people they only see me and they know that i'm teaching a class in this or that special uh special topic in classical mechanics or whatever and they can sense that there's something a little bit different about it they can sense that it's not a standard undergraduate or graduate course in these subjects and they ask me what is this what is this about or who are these people you're teaching to they can tell it and most of them don't know what continuing education means they didn't know what it meant and so they asked me i thought i would tell them over the uh over the internet what this class is it's continuing education which means education for people not from stanford people not oh they can be from stanford some of the people probably are either employed by stanford maybe were students at stanford but um almost everybody in this class is a little bit too old to be an undergraduate student even a little bit too old to be a graduate student they're people from the community from palo alto from mountain view from uh who lives further than mountain view okay what's the furthest there you come for this where foster city okay so that's 20 miles away so people from within a certain radius i don't know whatever the radius is are allowed to come and take courses at stanford they pay a bit of money for it and a professor teaches them now this is not it is most definitely not your standard freshman physics there's another guy on the internet who teaches freshman physics i'm told he does it very well that's not what this is this is the real mccoy theoretical physics at the full scale level we use equations we not only use equations we sometimes use hard equations but we tend to try to use the simplest equations that will do the job basically we try to keep it minimal uh just out of curiosity i i more or less know the answer just by looking but what the age distribution of people here is by comparing is there anybody here under 40 yeah there's uh there's maybe one handful of people under 40 incidentally for those who can't see the audience there's probably about 100 people in the room i'm not sure it's a it's a small theater uh with a lot of seats it's uh a lecture theater more or less filled up and um maybe four or five or three or four people were under 40. let's see let's go to the other end how many people here are over 70 a good deal more maybe uh 10 or 12. anybody over 80 i got a couple of people over 80. anybody over 90 this time oh there have been people over 90 they're having people over 90 in this class so you see the age distribution is such that basically people want to get to the basic ideas fast they don't have a hell of a lot of time they want to get there fast so i try to tell them in some minimal way what the basic things you really have to know in order to get on to the next thing sometimes the basic things that you need to know are a little more difficult a little more elaborate we do them anyway we do them as efficiently and as straightforwardly and with the minimal amount of stuff uh to get them right to get them really right not metaphor not the not the analogy but equations when necessary what else can i say about this class i guess it has various names quantum mechanics the physics 25. i'm inclined to call it quantum mechanics for old people including myself anyway the first course in the series let me just uh say what the outline of the whole course of courses is these series of courses will consist of about six ten lecture series the first of which was in classical mechanics the classical mechanics is the basic basis of all classical physics classical means before quantum mechanics anything that doesn't involve quantum mechanics or which ignores quantum mechanics or which is in a range of parameters where quantum mechanics can be replaced by basic classical logic that's called classical physics so the first class was classical mechanics which is in some sense the basis for all of physics the motion of objects the energy of objects the momentum of objects what the characteristic behavior of systems is as they evolve with time and so forth we discussed that last quarter and anybody who is thinking of following these classes should begin with that the next class which is already on the internet was called i believe quantum entanglement now this is also a class on quantum mechanics and it will be self-contained but i would strongly advise anybody who's following to go to course number two was it called quantum entanglement does anybody remember i think it was called quantum entanglement it is also on the internet it was the first one that was on the internet and uh to get that under your belt first before attempting to do the full-scale quantum mechanics that we're going to do here although what i'm going to do is pretty self-contained i'm going to start with this evening with some basic thoughts about the deep differences in the logic of classical mechanics and quantum mechanics or classical maybe even mechanics is too strong a word classical physics and quantum mecha and quantum physics there are some very queer phenomena in quantum physics that don't exist in classical physics now one of them is the fact that quantum mechanics is based on statistical thinking randomness a certain degree of um not imprecision that's not the right word a certain degree of uh non-deterministic or indeterministic behavior unpredictability that's the word i'm looking for unpredictability but it's a special kind of unpredictability einstein famously said god does not play dice niels bohr told him einstein don't tell god what to do but in fact in a sense god i don't know if there's a god when i i use the word god in the same uh sense that physicists always use it meaning uh the laws of physics or something like that the laws of physics don't play dice at least not in the standard sense let me imagine for you a theory that's based on a bit of statistics which is not the way quantum mechanics works and then i'll illustrate it for you by showing you some of the differences imagine that there was a element of randomness in newton's law whatever which law f equals m a together with the law of um of gravity all right so f equals m8 together with the law of gravity tells us how objects move let us say how the moon moves around the earth and it predicts with infinite precision if you could work out the equations with infinite precision and if you could account for every detail of the earth and the moon and all the material that's in between and so forth it would predict with enormous detail deterministically the motion of the moon which means if you know where the moon starts and you know how the moon is moving in the beginning then you can predict forever after exactly how the moon moves around the earth that's the deterministic classical mechanics now let me imagine a modification and the modification involves a little bit of randomness for example okay so let's imagine god sitting on his throne throwing dice every tenth of a second god throws the dye and if he gets snake eyes what he does is he gives the moon a little extra push in one direction if he gets a seven he pushes the moon a little bit in the other direction a degree of randomness based on the random throwing of dyson really random a really random number generator uh being used to put a little bit of fluctuation into the motion of the moon that sounds like it introduces the kind of uncertainty the kind of thing that one talks about in quantum mechanics uncertainty um non-predictability unpredictability but it is not anything like the randomness and unpredictability of quantum mechanics the randomness and unpredictability of quantum mechanics is exceedingly special exceedingly special and quite different and it's that that we want to get our head around and learn and understand the difference between these things by the time this class is finished but also we want to learn how to use quantum mechanics a little bit to calculate some things let's notice one thing about this law which includes a little bit of randomness the throwing of dice in order to uh to either kick the moon a little bit or inhibit the moon's motion a little bit one of the things that would do is to add a little bit of energy or subtract a little bit of energy from the motion of the moon randomly if you randomly give the moon a knock this way and then a kick that way and a bunk that way and keep doing it over and over again each time you do it this on the average it may not change the energy but each little increment is randomly going to either increase the energy or decrease the energy of the moon and if you do that randomly eventually that will build up to a statistical randomness in the energy in other words energy would not be exactly conserved in a world in which the laws of motion included a little bit of classical randomness i call it classical randomness to distinguish it from quantum randomness in quantum mechanics you prepare a system the same way that you might prepare the moon in an initial situation you let it go for a while and then you look at it and indeed you discover at the end of it that what you measure is a little bit unpredictable but you also find that energy is exactly conserved no hint no remnant at all of energy being knocked this way and not that way if you start it with a given energy with a given precise energy and you let it evolve for a while and you measure the energy later the energy is exactly the same as you started with so there's something funny about this randomness it seems to affect some things and not other things and doesn't work the way you might expect uh randomness and classical physics to work let me give you two oh i think three other examples of the oddness of this randomness the first comes from an experiment which to my mind shows the weirdness of quantum mechanics in the easiest and most straightforward least difficult way much easier for my money than bell's inequalities and all that sort of stuff it's called a two-slit experiment all of you know about it or most of you know about it if not you learn about it right now but it is extremely odd when compared with classical uh randomness right so classical randomness just being this idea that every now and then you give the system a little knock the two-slit experiment involves a source of particles those particles could be photons they could be electrons they could be neutrons they could be bowling balls except the effect for bowling balls is so minute that you'd never be able to measure it so when we speak about particles we think about things which are very light and because they're light the quantum effects associated with them are significant and measurable all right so we have a source of some kind it could be a laser shooting out photons but imagine the photons are coming through one at a time very small number of them uh quanta of light one at a time one at a time one every five minutes if you like i don't know just to go to some extreme situation these are the photons coming out of the photon gun and the photons pass through a obstacle with a little hole in it all right this is a two dimensional diagram of a three-dimensional situation this blue object over here is intended to be a disc with a tiny hole in it so the photons can go through and they come out the other side of the hole and when they come out the other side of the hole they eventually get to a screen over here that screen records the photon by a flash a flash of light at the screen not the flash of the original photons but another flash of energy appears at a point on the screen and records or it could be just the blackening of a photographic a photographic plate something that records the position of the photon when it goes through here now first let's think classically uh but classically with a bit of randomness the bit of randomness that we can imagine is that when the photon goes through here a little random kick might influence the photon and either kick it upwards or downwards in fact by a random amount what would we expect then well if there was no randomness then the photons would go straight through and illuminate a single point be completely deterministic the photons would always arrive at exactly the same point if the hole was small enough and if the beam of photons was narrow enough but now we can imagine a random kick what does the random kick do well it changes the direction of the photon of the photon not the whole photon beam but one photon at a time it might kick the first photon up a little bit the second photon down and so forth and so on eventually what you will see is a blob of illumination on the screen over here on the average the photon might go straight ahead so the blob might be most intense at the center it might be highly improbable to knock the photon through uh 60 or 70 degrees so the signal would fade as you moved away from the center you would see a blob with a maximum intensity near the center and thinning out as you move far away and you might describe it by a probability function the probability function being the probability that the photon arrives in different places okay now we go oh now we do the same thing in real quantum mechanics in other words in the real world we see essentially the same thing the photons go through the hole with no ability to control the situation we find out that the photons again create a blob like this but now we're going to do something a little different and i think everybody here more or less more or less everybody knows what i'm going to do i'm going to open a second hole okay but let's think about what classical randomness would do classical randomness would simply mean that we and we'll also imagine that this beam of photons is at its origin a little bit uncertain and a little bit random so that some photons begin a little bit upward some photons a little bit downward some photons go through the upper hole some photons go through the lower hole if a photon goes through the upper hole it may or may not get a random kick and get knocked off of course if it goes through the lower hole it also gets a random kick now we're imagining the photons come through one at a time we could even imagine a okay i do that all the time where was i yes the photons get a kick we're imagining that the photons come through one at a time very sparsely and so what one photon does doesn't influence a later photon because the later photon comes through so much later that whatever gave the first photon a kick has already finished happening and it's waiting for the next uh throw of the dice in fact the next photon may come through a hundred throws of the dice later and so we expect the next photon uh to be random statistically independent of the first photon under those circumstances what we would expect well let's uh let's let's decide what we would expect supposing only one hole is open if only one hole was open then we would see a blob of illumination like that with a profile it might look something like that supposing we closed up the first blob the first hole and opened the second hole close the first one open the second one all right so first we begin with just one hole then we close the first hole and open the second hole what would you expect to see what you would expect to see under those circumstances is a different blob slightly displaced from the first one the blue blob wouldn't be there because we closed the first hole the green blob would be there because we opened the second hole now what happens if we open both holes what happens if you open both holes in classical mechanics is the probability for a photon to get to the screen at any given point is the sum of the probabilities for it to get there by either root the photon photon can either go through the upper root or it can go through the lower route if both holes are open the probability to get to this point over here let's call it the green point over here the probability to get over there is the probability to go through the upper hole and arrive at the green point plus the probability to get to go through the lower hole and arrive at the green point so the result is in classical physics you would always see the signal over here the profile over here just being the sum of the two probability distributions and it would look i wish i had another color they never leave me enough colors all right uh we would see something which would look like just a higher i don't know it would be much bigger than that it would look like that and in particular if there was any point over here such that the photon could arrive either from one hole or from the other hole or both then we would find illumination at that point for sure by opening both holes that's what classical logic let's let's call it that's right name classical logic that's what classical logic classical statistics classical probability would dictate for a series of particles coming through here one at a time when they come through one at a time they make blip blip blip blip but the average probability of the distribution of blips would be a distribution which would be the sum of the two distributions what happens if you really do this experiment you find what's called an interference pattern the interference pattern looks like this well let's see let me get it right in particular well first of all what is this figure this figure is a probability distribution and it tells you the horizontal axis here tells you what the probability of a photon getting to a particular point at a particular height here but in particular it says that there are no photons which arrive at that point there are no photons which arrive at that point there are no photons which arrive at that point this is odd if you opened only one hole then you would find a probability distribution which wasn't zero in other words you would find illumination at that point you open the other hole you still find illumination at that point you open both holes and all of a sudden no photon gets to that point even though they're coming in coming through once every 20 minutes or once every 20 years and therefore how can they know about each other nevertheless if you open both holes there will be places where no photons can get to despite the fact that photons arrived at those points when only one hole was open that is um you might be able to sit down and work up some in some interesting but rather elaborate mechanism to make this happen you could imagine elaborate complicated mechanisms where somehow the this this screen here some degrees of freedom inside the screen remember how many photons went through and they remember what they're supposed to do but it would be a rather elaborate mechanism just for this one purpose this phenomenon of interference of destructive interference this is called destructive interference that the probabilities cancel instead of adding at certain points that's a very generic property in quantum mechanics and so it requires a kind of explanation which is not some detailed mechanical complicated explanation it requires a a broad new idea about how statistics works and how about how the logic of um of quantum mechanics works so that's the first really weird thing that happens in quantum mechanics now let me give you another example if you remember in the last course we talked a little bit about reversibility we talked about the laws of physics in particular we talked about the laws of physics for discrete systems for example uh we discussed the possible laws of physics deterministic laws of physics for a coin which can either be heads or tails now of course flipping the coin that introduces a level of of uncertainty a level of statistics probability into things i want to think about the deterministic laws the truly deterministic laws are the ones that whatever the coin is doing it will tell you what the coin is doing next so when we discussed this last quarter we talked about two possible laws of physics the first was that if you find heads in the next instant when you look at it after an instant of time you'll find heads again if tails then you find tails again then your laws of physics just permit two possible um evolutions heads heads heads heads heads heads heads or tails tails tails tails tails and the other possible laws of physics or law of physics was that when you see a head in the next instant microsecond or whatever your unit of time is you will see the opposite tails then the two possible laws of physics are the two possible not laws of physics the two possible evolutions one of them begins with heads it goes heads tails heads tails heads tails and the other one begins with tails and goes tails heads tails heads tails heads whether you know it or not those two different things were different they began one with heads one with tails uh that was that was the basic idea of a deterministic law of physics now you can have of course more complicated systems in this just two state systems we imagine for example a six state system a six state system was a die a die that can be one two three four five six but let's just take a simpler thing let's take a coin with three sides if you can't imagine a coin with three sides then you'll have to do this using abstract mathematics okay a coin with three sides has heads tails and whatsoever else feet heads tails and feet all right heads tails and feet all right and there are two all right let's let's consider a simple law of physics a simple law of physics could tell you that whenever you have heads it goes to tails whenever you have tails it goes to feet whenever you have feet it goes to heads again and then wherever you start you just cycle around endlessly forever and ever now there's a certain sense in which this is revers in which this is what does it mean to say it's deterministic what does it mean to say information is not lost in this process what it means is that no matter how long the system evolves let's suppose you started with heads and you let it go a million units of time it will just go around and around and cycle if at the end of that you reverse the law of physics now reversing the law of physics just means having it go in the opposite direction if you could somehow press a button or turn a knob which had the effect of reversing the law of physics in other words changing the direction of every arrow and think about actually if you could really do this by pressing a button change the law of physics then if you allowed the system to evolve for any length of time at the end of that time reverse it and let it evolve for the same length of time again guess what it magically comes back to the same original configuration that's what it means to say that physics is reversible or that's what it means to say that information is never lost in physics that no matter how long you keep going if you reversed if you could find a way to reverse the laws of physics and run them backward for the same exact length of time you'll come back to the starting configuration now to do that you might not even need to know the laws of physics you might need to know very little the only thing you would need to know is how to reverse the law you need you needn't even know what the law is whatever the law is if you can find a button to push that reverses it then you can test the determinism of physics by simply starting some place letting it evolve for a long period of time and then letting it evolve with the reverse law of physics if you come back to the same place every time then your law physics is deterministic now what about a little bit of classical probability a little bit of deity playing dice so let's suppose with some very very small probability the deity does something different that's not prescribed by the laws of physics for example in some random way where the probability of one in a million the law might say don't move instead of moving the way the diagram says stay still if you allowed the system to evolve for a short period of time then and then run it backward it will go back to the same point but if you allow it to run long enough that there's a significant statistical probability for a fluctuation for something to happen which is not deterministic in other words if it was one in a million that you stand still but you let the system run for 10 million units of time then guess what no chance or not no chance you have a chance that you'll come back to the same point but you'll also have an equal chance that you'll come back to any other points in other words this test will fail it'll fail uh one third of the time you will come back to the same point two-thirds of the time you will come back to two different points so a little bit of classical randomness destroys the um what i called last quarter the conservation of probability no the conservation of information information gets lost what about quantum mechanics is information lost well there is an element of statistical things in quantum mechanics when a electron goes through a hole like this it has a probability for getting kicked up let's just take the case let's simplify now this is the one slit experiment we don't even need the two-slit experiment let's start with the one-slit experiment the electron goes to the slit it's aimed toward the slit sometimes it more or less goes straight through sometimes it gets kicked up a little bit sometimes it gets kicked down a little bit and you might think that this is more or less like throwing dice in fact if you look for the electron afterwards if you look for the electron out here after you've given it time to pass through at the moment now let's just send one electron through one electron if we send one electron through it may get kicked up it may go straight through or may get kicked down if we do the same experiment repeatedly which is of course is the same thing as sending many electrons through but if we do the same experiment repeatedly we'll find some go up some go down some but now let's ask the following question supposing after a period of time we send an electron through one electron after a certain period of time having given it enough time to get to the other side but not but let's remove the screen over here let's remove the screen send the electron through it goes through comes out someplace else but we don't look at it we don't we don't bother detecting where the electron is instead we just reverse the law of physics we do what we did over here reverse the direction of time if you like can you really do this can you really is there really a button that you can push is there really something that you can do to a system that will reverse the motion yes in many systems there is in many systems we actually know how to manipulate the system how to change magnetic fields how to do things to a system of electrons so as to run it backward so let's take it as a as a given that after a certain amount of time somebody can press the button that reverses the law of physics that reverses the direction of time so to speak and runs the system backward for the same length of time what happens does that electron we're not going to look at it but do we later after after the end of the experiment we allow it to evolve for a time t and then we allow it to evolve backward for another time t 2t altogether do we find the electron moving backward along the original trajectory or do we find the probability the fluctuations compounding and that after we turn it around there's even worse fluctuation particle comes through gets knocked up we run it backward it either gets knocked down and locked up which is it does it does it reverse precisely along the original trajectory every single time or does the statistical fluctuation the imprecision the the unpredictability in one direction add to the unpredictability in going backward and make it even more unpredictable afterwards than it was to begin with the answer is very curious the answer is that if we don't look at the system in the intermediate stage after it's gone through the hole over here don't look at it means don't interfere with the electron in any way take that screen away that might have converted that electron into a little pulse of light just remove any influence on the electron over here completely do not look at it do not interfere with it do not do anything to disturb it but just run the law of physics backward then we will exactly every single time detect the electron running back along the reverse trajectory on the other side over here in particular we'll find the electrons just going right back into the gun what if somebody does look at the electron when it goes through to the side over here for example supposing somebody sets up an electron detector which detects where the electron is and then lets it go having reversed the law of physics we look at it and then reverse the law of physics now in this case over here the case where we're talking about classical coins looking at a thing doesn't disturb it very much if i have a coin where's my coin i've lost my coin it's okay it's only a chilean peso i lost my chilean peso i think it was actually a hundred pesos okay now here's my chilean peso i put it down and i put it down heads don't look at it now i look at it it would be rather amazing if just the process of looking at it was capable of flipping it from heads to tails now of course if i look at it with an intense enough uh high frequency light beam a beam of very very energetic photons sure enough an energetic photon could uh could hit the coin knock it into the air and spin it over that's true but you could but in classical physics you can look at an object and determine its state determine the heads or tailsness of it with an arbitrarily gentle interaction and so by looking at a system in classical mechanics and looking at it with a very very gentle apparatus or very gentle photon or whatever you like it does not entail disturbing the system so if you look at it after you allowed it to go a million times and then reverse the law it will have no effect no detrimental effect on the experiment and you will come back to the same uh to the same point that you started with all right so just looking at the system in between has no effect no does not necessarily have an effect on it even though whoever looks at it can determine after a million units of time where the system is here looking at it has no effect on what happens after you run it backward exactly the opposite in quantum mechanics if you detect the object if you do anything to detect the object electron in this case and then run the law of physics backward what you'll find out is that the probabilistic character of it gets compounded so the fluctuation that's centered up here when you run it backward and you now know that it came out up here and you run it backward it's likely to come out down here or up here and it will disturb the system in such a way that the test of reversibility will fail so this is curious that that whatever quantum logic is the questions of the kind we're asking are deeply dependent on a ridiculous question did somebody look at the system during the course of its evolution and as i said of course looking at a system can disturb it but in classical physics we can look at a system without disturbing it we can look at a system detect the system measure the system as gently as we like arbitrarily gently and have arbitrarily small effect on it so that running it backwards will be exactly as if we didn't look at it not so in quantum mechanics determining the state of a system is never a small thing to do to the system and this is an example it completely destroys the uh the experiment the two-slit experiment has also a similar uh a uh a similar story that goes with it the story that i told you a moment ago about the interference pattern and the destructive interference is only true if nobody records which way the electron went through now when i say no body i don't mean a human being necessarily i mean that nothing in the environment of the experiment records and remembers which way the electron went through in other words after the experiment there is nothing in the environment of this experiment which has recorded which way the electron went through nothing in here only no this hasn't this also hasn't recorded where the electron went through if nothing records where the electron goes through then there's an interference pattern but if you were to put a little demon over there a real physical demon a real physical demon could be another electron it could be some some gas in the apparatus some gas molecules in the apparatus whatever it is in such a way that something records which way the electron went through something remembers it afterwards perhaps a molecule over here gets disturbed gets excited if the electron goes through the upper hole gets excited and changes the character of that molecule if it goes through the lower hole it changes the character of a different molecule so looking afterwards after the electron went through you can tell which hole the electron went through then the interference pattern is destroyed and the answer is exactly the same as the classical answer namely the probabilities add just exactly as in classical physics so again there's no way to record whether the electron went through the upper hole or the lower hole without seriously disturbing the experiment without so seriously disturbing the experiment that you drastically can change the conclusion that's a quantum that's roughly speaking the general character of quantum mechanics that you cannot do measurements on systems without disturbing them and disturbing them can change completely the character of a yuck question yeah that's a very very good question that's an excellent outstanding question uh i wish i knew the answer no i know the answer um yeah i will uh the answer has to do with the probability distribution for the position of the uh of the um of the let's call this the detector let's call this the detector okay now if the detector is very well localized in space as it would be if it were a heavy massive classical detector then by the uncertainty principle which we haven't talked about yet if we're going to let me skip ahead and imagine that we have talked about the uncertainty principle that's a sophisticated question so i suspect you you've thought about it a little bit if the position of this detector the up down position of this detector delta x is very very small that means that the uncertainty in the momentum of the detector is large okay that means if i were to plot the probability distribution of the momentum of the detector let's plot the probability distribution of the momentum of the detector it's rather broad because for a heavy detector its location is so well defined all right now the electron comes through and kicks the plate kicks this thing a little bit it gives it a small kick and what does it do it shifts this probability distribution a small amount but unless the probability distribution has been shifted by something approximately equal to its width then you can't tell afterwards whether it got a kick or not right you have to be in order to be able to be certain that it got a kick you'll have to kick it by an amount large by comparison with the uncertainty so it's the uncertainty principle that comes in and rests you rescue rescues you or rescues me and makes sure that the interference pattern is not destroyed the uncertainty principle itself so let's go to the uncertainty principle since it's been raised since i raised it the uncertainty principle there's another factor in quantum mechanics which is extremely different than physic than classical physics conceptually very different i know we haven't talked about the what the uncertainty principle is yet but let's uh let's discuss it anyway let's jump ahead my main motivation now is to explain the strangeness of not to explain the strangeness of quantum mechanics but the points and fingers at the strangeness of quantum mechanics so that you see that it really is fundamentally logically different than classical mechanics logically different whole logic of quantum mechanics is different okay it's easy to imagine a bit of uncertainty in classical physics as i said the uh the coin throwing or the dice throwing deity who sticks his finger into the system and gives it a push this way i push that way and so forth and that way things after a short period of time become uncertain and it's easy to imagine that that uncertainty can affect both position and momentum and if you wait a little while both the position and the momentum may be uncertain in fact if you wait a little bit or more than a tiny fraction of a second for a for a particle you might discover that inevitably both the position and the momentum get jostled about and so that there's a good deal or certain amount of uncertainty and position and uncertainty and momentum but you would be unlikely to say that that uncertainty and position and momentum is um what's the word i'm looking for uh well first of all that's fundamental in any sense you would say it was a result of hitting the system randomly but i think most of us would agree that under those circumstances it was a bit of laziness that didn't allow us to watch the particle carefully enough to see what exactly the momentum was and the position was we always imagine in that kind of classical context an example of that classical context would be the random walk of a brownian moving particle right so if we if we watch a whole bunch of particles a whole bunch of particles might form a cloud and that cloud might spread and perhaps even the velocity of the cloud or the uh the cloud describing the velocities might also spread but every one of those particles if we cared to we could look at with better precision better and better precision we could look at it gently as gently as we liked in classical physics and determine both its velocity and its position simultaneously certainly that would be true of a classical brownian motion particle that was being knocked around by uh by ordinary collisions with uh some gas or something like that in principle we could just get ourselves a better microscope better better accuracy better precision better resolving power and do it very gently so as not to disturb the system when we measure the position we don't want to jostle the momentum when we measure the momentum we don't want to do something funny to the position and we just measure it gently enough so that we measure both the position and the momentum that would be expected to be true for a brownian particle so the uncertainty in position and momentum is something which in a certain sense is due to our own laziness and not spending enough money and buying a good enough detector and so forth to to be able to detect both the position and the velocity at the same time on the other hand in quantum mechanics there's a really fundamental obstruction a logical obstruction a deep obstruction to knowing that ever being able to measure both the position and the velocity of a particle i'm going to work it out for you and show you how it works i'm going to show you how heisenberg first thought about it well he first thought about it entirely through abstract mathematics for us that's going to come later but then when questioned by boar what are you talking about that you can't measure the position and the velocity simultaneously your mathematics is a is a crock of bologna don't tell me that x times p is not equal to p times x come on it's like saying 3 times 5 is not 5 times 3 did i write it right yeah not equal to don't tell me such nonsense stories give me some physics and so heisenberg cooked up his experiment to show uh to illustrate the fact it's not show but illustrate the fact that there was a good deep uh consistent reason why it's impossible to ever ever under any circumstances simultaneously determine the position and the momentum of an object so let's go through that a little bit it has a similar character to some of the other illustrations that i've given here what heisenberg and bohr and einstein and all those people knew around 1926 was the property of photons in fact whenever they thought about measuring the properties of a particle they were always thinking roughly speaking of looking at it under a microscope detecting it by bombarding it with photons now it doesn't matter whether they were photons or not it's just that they were thinking about microscopes and they were thinking in a language where you optically look at things it doesn't have to be optical but this is what they were thinking so heisenberg imagined putting his particle under a microscope and detecting its position and its velocity and seeing what he could learn okay the measurement involved interacting with the particle with a photon a photon would be the thing which would be used to determine the position of momentum in other words looking at it really meant hitting it with some light letting the light scatter off and then focusing the light waves focusing the light waves in order to see exactly where the particle was just as you would do it for a billiard ball or for anything else focus the light on your retina focus the light on some screen and reconstruct the position of the particle that was what they were thinking about okay now here's what einstein and the broglie and others have taught them about photons first of all einstein had told them that the energy of a photon is equal to planck's constant times the frequency of the light describing the photon this is equivalent to planck's other constant h bar times the angular frequency where f stands for the frequency of a wave number of cycles per second measured in hertz okay usually a frequency of a light wave is a lot of hertz how many hertz how many 10 to the 15th for ordinary light that's just a number 10 to the 15th hertz omega is the same as the frequency except multiply by two pi it's the angular frequency instead of the instead of the number of cycles per second all right and h bar is just h divided by 2 pi so these are the same expressions i'll use this one over here for the moment energy is equal to h times frequency that's something that was known by einstein now here's another fact about a beam of light beams of light have not only energy but they also have momentum you can take a beam of light and shine it on something and it will warm it it will heat it that tells you it has energy but you can also take a beam of light and you can shine it at that door and if the beam of light has sufficient intensity it'll just push the door open in other words the beam of light has momentum uh so that when it when the door absorbs the beam of light the door gets a kick just as if uh just as if you threw a ball at the door and the door colliding with the wall it has momentum okay now what is the relationship between the energy of a beam of light and momentum i'll just tell you what it is this is this is classical maxwell theory of why uh if a beam of light moving in a particular direction assume it's moving in a particular direction has energy e then it also has momentum and the relationship is that the energy is the speed of light times the momentum incidentally let's just check the units of that equation in another way compare that with newton's uh theory of momentum and energy for an ordinary particle the energy is p squared over 2m that's non-relativistic the momentum b squared over 2m which is equal to p times p over m and an extra one over two one-half momentum times momentum divided by velocity now what's momentum divided by velocity sorry momentum divided by mass velocity so this is equal to one-half momentum times velocity well it's almost the same for a photon there's a it's not the same because we have to use the theory of relativity for a cosine but the correction from the theory of relativity is not so uh so enormous this is one-half the velocity of the particle times its momentum this is just equal to the velocity of the particle times the momentum so for highly relativistic particles the formula is very similar except the half goes away among other things this argument tells you that the units are right energy is velocity time to a minimum let me just solve that momentum is energy divided by the speed of light we can now plug in and find that the momentum of a photon or momentum of any little piece of light let's do a single photon now the energy of the photon is h times f uh so the momentum is h times the frequency divided by the speed of light now if you have a wave that's moving with the speed of light you have a way of moving with the speed of light it's moving down and it has a certain frequency and a velocity there's a connection between the frequency and the uh between the frequency and the wavelength think about it for uh for a moment here's a wavelength let's call the wavelength lambda lambda is a standard uh there's a standard notation for the wavelength of light the wavelength of anything how far does the wave move in one cycle the answer is lambda that's what lambda is it's the distance that the wave moves in one cycle if you stand there with your nose watching that light ray and it's moving past you all right it moves past you one wavelength per cycle one wavelength per unit per cycle how long does a cycle take how long does one cycle take what's the period that goes with one cycle the inverse of the uh of the frequency so the time that it takes the time that it takes to move distance lambda is one over the frequency so let's just write that down time to go one cycle is equal to one over the frequency the distance that it goes in that same time is just lambda so what's the velocity of the wave the velocity of the wave is the distance that it goes divided by the time that it travels distance all the time is velocity c is equal to lambda divided by t which means lambda times the frequency so this is a general formula relating velocity lambda and frequency and let's see let's plug it into here now let's get rid of the frequency the frequency is c divided by lambda so frequency is c divided by lambda and then there's another c down here and we get the broglie's equation that the momentum is planck's constant divided by the wavelength the smaller the wavelength the larger the momentum if you want a beam of particles with very high momentum give it a small wavelength or to say it the opposite way if you want a very short wavelength particle it's at the cost of having that particle have a large momentum momentum and wavelength are inverse to each other so now let's let's go back to once the measure wants to get a roughly speaking he wants to get a photograph of the electron where the electron is uh not fuzzy on scales larger than delta x just wants to take a photograph of the electron oh it doesn't have to be an electron it could be a golf ball doesn't matter what it is once they get a photograph of it and wants the photograph to be non-fuzzy on a certain scale delta x well every photographer knows that or anybody who understands anything about waves and images and so forth knows that to form an image which is precise or non-fuzzy the size delta x you have to use wavelengths that are shorter than delta x if you're trying to make an image of a golf ball with a radio wave a radio wave of let's say 10 meters the golf ball will look fuzzy on the scale of 10 meters if you're trying to do it with a wavelength of a tenth of a tenth of a golf ball a tenth of a centimeter the golf ball will look pretty good so the rule is that lambda the wavelength of the light must be less than delta x if you want to get an image with precision delta x now i erased an equation the equation that i erased is that the momentum is equal to planck's constant divided by lambda so now heisenberg was caught in a bind if he wants to measure the position to a high accuracy delta x he's got to use a um he's got to use a high momentum electron that's gonna use a short wavelength electron if he wants delta x to be smaller than a centimeter then he's got to be using a wavelength smaller than a centimeter but if he's using the wavelength smaller than a centimeter that means the momentum of the photon has to be larger than h over one centimeter so the smaller delta x the larger the momentum of the photon that he has to use to make the image the shorter the wavelength of the photon that he has to use to make the image if lambda is small then p is going to be large well now what does that mean that means that we're going to wind up bombarding this object with a high momentum photon the high momentum photon may make a very good image of the position but then it's going to collide with this and knock it off in some random direction knock it off in a random direction with a momentum uncertain uncertain momentum of order of magnitude of this momentum here this particle will come in and just like the photon hitting the slit it will get knocked in some random direction and so the conclusion will be that immediately after you try to measure the position immediately afterwards the momentum has become very uncertain it has been kicked hard having been kicked hard if you measure its momentum afterwards it will have nothing to do with the momentum beforehand so you cannot determine both the position and the momentum at the same time measuring the position necessarily imparts a random momentum a kick a random momentum kick for the particle result is whatever the momentum is before was beforehand it won't be that afterwards and that's another example of the fact that there's no such thing as a gentle determination in quantum mechanics well you can do a gentle determination but it will be a very imprecise determination in classical physics incidentally oh keep in mind uh why what about classical physics why is it different in classical physics the reason is because light doesn't come in discrete packets in classical physics we've used the fact that that light comes in discrete indivisible quantum discrete indivisible protons and if we have a wavelength then there's a minimum amount of energy that can be that can go with that minimal amount of momentum that can go with that wavelength namely one photon you can't have less than a single photon in classical physics energy does not come in discrete multiples of some basic unit and so you can do the same experiment with as small an energy as you like you could it's as though in classical physics you could subdivide that photon into arbitrarily small units of body take just one of them with the same wavelength and form an image with it in quantum mechanics you're always stuck by the fact that the energy of a light wave comes in these discrete packets and that discrete packet that it has wavelength lambda will have a momentum h over lambda and therefore give this an inevitable kick of a mountain overlapping now that's this is several examples of the same kind of thing that doing experiments in quantum mechanics is different than doing experiments in classical mechanics you can always imagine classical mechanics doing a very gentle experiment that doesn't disturb a system and then just go on from there and do a later experiment and do a later experiment the earlier experiments not having influenced the outcome of the later experiments so for example you could measure the position and that will and that not affect the outcome of a later detection of the velocity which is not true in quantum mechanics these are a lot of examples but the examples add up to a so a if you like a notion that the basic logic of classical mechanics is incorrect basic underlying logic is not sufficient to understand the measurement process in quantum mechanics um the whole setup is wrong the whole setup not just not just particular it's not just each experiment you can go and analyze it and try to figure out what's wrong with it and try to correct for it no the whole underlying structure of classical physics is inadequate to discuss quantum mechanical phenomenon uh let's take a break for five minutes and then start formulating its proper okay uh somebody asked me about how do you measure the velocity of a particle uh the answer is gently first of all uh here's a simple here's a simple conceptual way to do it velocity or momentum momentum let's suppose we know the mass of the particle so but by uh that by measuring its um velocity we also measure its momentum a simple conceptual way to measure velocity is to measure um location at two different times and then take the difference of location that's how far it travels and divide by the time between measurements and that's the velocity that's the way you would measure velocity now you have to be careful you want to measure the you want to measure the position twice in succession but you don't want to measure the position with such with such good with such a good determination that it gives a random kick to the velocity which is what you're trying to measure you wouldn't want to measure the velocity by beginning the experiment with a random kick which randomizes the velocity and sends it off in some direction very different than it was moving with to begin with direction and velocity so your two measurements of velocity sorry of position should be very gentle measurements that don't change the velocity very much or that don't change the momentum very much that means that they must be done with photons of very long wavelength if you don't want your first initial detection of the location of this particle to to to give it a good whack then you want to do it with a very long wavelength photon a long wavelength photon will then tell you only that the electron is in some region of size lambda okay so what you know then is the position of the electron x let's say plus or minus lambda meaning to say that there's an uncertainty of magnitude lambda then you wait a really really long time until the electron has moved a long long ways it has some velocity it started out with some velocity you've you've changed the velocity only by a very small amount by using a very long wavelength photon so we take the wavelength of the photon to be so long that there's been an unappreciable change in the momentum and then at a much later time we discover that the particle is at x plus or minus lambda maybe even plus or minus 2 lambda plus vt is velocity times the time between the two measurements so true there's a little bit of sloppiness in the measurement of the positions a large sloppiness excuse me in the measurements of the positions and that's going to lead to a sloppiness in the distance be that the particle moves what's the sloppiness the sloppiness the distance that the particle will move will be vt plus a sloppiness of order lambda and we take lambda to be very big so that so that we don't disturb the velocity very much then how do you find this is this is how far the particle moves let's call it d distance that it moves in order to find the velocity we have to divide the distance that it moves by the time so let's divide it by the time this is a this is now a measurement in the sloppiness of the velocity the sloppiness of the velocity measurement is lambda divided by t no matter how big lambda is if we wait long enough if we let t be very very large then the sloppiness and the velocity can be made small so the way we measure the velocity is by taking a long time to do it do very very gentle measurements of the position so that we don't whack the velocity and in that way we can measure the velocity with great precision however it's been at the cost of knowing where the particle is the particle is not been determined to a precision better than lambda the velocity has been determined to a precision of something like lambda over t so whatever you do you'll never be able to determine both the position and the velocity in the same experiment to do one and not the other one or the other kind of quantum mechanics yeah uh let's see you wait a very long time so that uh that's that's because you're taking this to be the uncertainty and the velocity but there's also an uncertainty from here that has to do with the original setup let me think about it you i understand your question let me think about it and come back to you with an answer um yeah it's a good let me let me come back to it yes i see i see yes you can um you can determine the position of this wall by averaging over a long period of time with photons many many photons banging off it uh yeah yeah i'm uh i'm um trying to remember what i was going to talk about we'll come back to the uncertainty principle for sure but i'm i've lost the threat of my thought all right so i i forgot what i was going to say but let's just move on so the fundamental logic of quantum mechanics is not the same as the fundamental logic of classical mechanics and that shows up at the earliest possible stage namely what do you mean by the state of a particle what do you mean by the configuration what do you mean by knowing everything that there is to know about a system or everything that can be known about a system in classical mechanics the space of states what we call the phase space we had some various versions of it in one version it was just a set of points i use the word set and i use the word set on purpose a set of points the states of a system heads tails or one two three four five six for a die uh in particle mechanics or mechanics more that's a little bit less primitive than this very simple system here it's phase space the states of the system are phase space p's and x's p's and x's uh points in a phase space are the states of a system but again the collection of possible states of a system form a set they form a set they form a set of points and the basic logic of classical physics of classical phase space is set theory sets of points describe the possible states of a system and transitions or motions from one point of in that set to another point in that set describe the mechanics or the dynamics of a system in phase space it's the flow through phase space what exactly is a state a state is a point in that set a member of the set an x and a p for particle a point h or t for heads and tails or a bunch of p's and x's for a general system of many particles but a point or a member of a set now you could think of something a little more general and possibly call it a state in which you introduce a bit of statistics a bit of a probability or a bit of uncertainty from the beginning you might say look i my apparatuses do are not sufficient to determine with infinite precision the position and velocity of a particle so instead of doing that i will say the particle is somewhere in some little region here or i might assign a probability distribution a probability distribution as a function of x and p which might be peaked at the center here and fall off as you move away from it and so forth a probability distribution on phase space might even be a more general version of what you mean by the state of a system but if i told you that the notion of a state of a system is a probability distribution i think most of you will come back and say yeah but you know if i look more carefully at the system if i look more carefully i can always reduce that probability distribution and i can always get as close as i like by doing delicate experiments and again in classical physics doing experiments which don't disturb the system terribly much doing experiments on the system which will get it to be closer and closer to a point so when you speak about a statistical distribution you're as being a state you're not talking about the maximum amount of knowledge that you could have about a system you're talking about an impr a a practical limitation because of the coarseness of your apparatuses and so forth might force you to use a probability distribution and only talk about things to within a precision that you can measure them but you wouldn't think that that was very fundamental and eventually you would say with sufficient accuracy in your apparatuses that uh that the maximum you can know about a particle corresponds to a point in the phase space and in that sense states in classical physics are points in a set points in a set and you can know everything that's implied by knowing a point in the set in particular in this case knowing a p and an x in quantum mechanics states are not sets do not form sets the natural way of manipulating states and asking questions about them is not set theory as not state is not set theory states are vectors in a vector space in quantum mechanics quite a different mathematical object than a set a vector space is mathematically extremely different than a set and in order to understand quantum mechanics this is not some abstraction that is really um sort of unnecessary to understanding the subject this is so central that that to not talk about it we would completely miss the basis of the basics of quantum mechanics what in order to understand the logic of quantum mechanics we have to understand the mathematics of vector spaces we're just forced to it there's no way around it if i tried to do things without it i'd be faking so let's talk about vector spaces now the use of the term vector or these are vector spaces linear vector spaces every vector space is linear so linear is a redundant word but they're vector spaces over the complex numbers that may mean nothing to you right now but it will as we go along vector space the space of states the state space space of states is not a set but it is a vector space now the use of the word vector can be in vectors a vector space over the complex numbers over c c stands for the complex numbers and i'll tell you exactly what this means if you don't know what it means don't worry because you will know okay first of all most of us are familiar with the notion of vector from the classical notion of a pointing of a thing pointing in a direction an ordinary three-dimensional space with a certain length what i'm talking about is not vectors in space in that sense these are abstract vectors in abstract vector spaces which have nothing to do with your naive concept of a vector in space as an arrow pointing in some direction in space with a given length when i teach this i usually make a linguistic distinction between vector spaces and pointers in space which are the things you usually think about as vectors uh i i can never quite figure out how to do this without getting everybody confused including myself i will try as much as possible when i'm talking about a three-dimensional vector in space a kind of vector that could correspond to velocity or position or momentum and that sort of thing i'm going to use the term pointer whenever there is any ambiguity about what i'm talking about a pointer meaning a direction an ordinary three-dimensional space together with a length i'll use the term pointer in order to not get confused with a completely different mathematical concept or not a different mathematical to a much more general mathematical concept called a vector space a vector space is a space of vectors but of course these vectors are not pointers in ordinary direction of ordinary space pointers in ordinary space are a special case of vector spaces but not the special case we're going to be interested in okay so let me tell you what a vector space is first of all it's a collection of mathematical objects called vectors again i'll emphasize over and over not vectors nor just abstract objects called vectors maybe i should change their name and not the name of uh the vectors in three-dimensional space we could go we could call them vectors or uh or schmecters but uh but uh one way or another we've got to uh make some distinctions all right it's a collection of objects which we can label let's label them we can also label sets by points and give the points labels like a b and c and so forth but now i'm labeling vectors and to indicate that they are vectors of the right kind namely the kind that come into quantum mechanics we're going to draw a symbol like that this is called a ket vector it's called a ket vector by dirac there's another kind of vector called a bra vector and if you put them next to each other they make a bra cat or a bracket that's where the name came from but we'll come to brackets in a little while it's half a bracket a bracket being what happens when you put a bra next to a cat is that clear yeah okay okay so there's a collection of objects collection of objects called vectors okay now what does it mean to say it's a vector space over the complex numbers what it means is that first of all you can multiply vectors by complex numbers now the ordinary vectors in ordinary space are a vector space over the real numbers if i have a vector for example three units of length pointing in the direction north by northwest that are 45 degrees to the to the horizontal that would define a pointer if you like all right so ordinary three-dimensional pointers uh have a length and a direction you can multiply them by numbers if you multiply a certain vector this one here this sorry a certain pointer by 2 it just gets twice as long in the same direction if you multiply it by -1 it's the same length but pointing in the opposite direction if you multiply it by seven and a half it becomes seven and a half times as long so you can multiply of a pointer a pointer by an ordinary real number positive or negative mathematicians would say that pointers form a vector space over the real numbers vectors over the real numbers meaning to say that you can multiply them by real numbers and when you multiply them by a real number you get another pointer back we're going to make a generalization of that idea where vectors can be multiplied by complex numbers now eventually we'll see some examples there's nothing like an example to to make it clear but what that means is that given any vector you can multiply it given a you can multiply it by any complex number let's use for complex numbers let's use the greek alphabet so if alpha is any complex number every i assume everybody here is familiar with complex numbers i'm going to assume that i'm not going to teach you complex numbers here you can multiply any vector by a complex number alpha and it's the and it's a new vector not the same vector it's another vector so the operation of multiplying by a complex number is well defined in a complex vector space we can call this b for example multiplying by alpha maps every vector into another vector alpha being any complex number that's the first property of a vector space over the complex numbers sometimes we can call it a hilbert's i mean sometimes it's just called the hilbert space vector space is over the complex number a hilbert space hilbert was a mathematician so that's the first property given a vector you can multiply it by any complex number and it's a new vector second property is that if you have two vectors any two vectors let's call them a and b you can add them and you get another vector so adding vectors give new vectors now first of all here's something which is really these both of these things are really new when you go from classical mechanics to quantum mechanics in classical mechanics the notion of a state is a point in a in a space of points in other words in a set it doesn't make any sense to multiply uh points in a set by a number for example let's take heads and tails what does it mean to multiply heads by three or heads by a complex number it doesn't mean anything you just have two points heads and tails and that's all you have uh you don't have a thing that you would call three times heads or or minus three times heads or two plus four times i times heads so in saying that state vector that states of a system are a vector space this is something new and radical and weird you should not understand this at this point unless of course you've done quantum mechanics before this should not make any sense at all but for the moment we're doing some abstract mathematics to get some definitions and then i'll show you how those definitions uh apply so first of all you can multiply a vector an abstract vector by any complex number and you get a new vector that's an operation that you can do and the next operation is that you can add any two vectors now of course if you can add any two vectors and you can also multiply them by complex numbers then you can also then you can also multiply the first vector by a complex number the second vector by a complex number and that gives you some new i won't call it c i'll call it c prime so what you can do with the vector space is you can take any pair of vectors multiply both of them by arbitrary complex numbers add them and you get a new vector so the notion of addition is well-defined of course for ordinary pointers the notion of addition is also defined and it's defined by the operation of drawing the parallelogram ordinary vectors satisfy both of these rules that you can multiply a vector by any real number i'm sorry not complex numbers ordinary pointers as i said are vector spaces over the real numbers and that means that you can multiply any of them by a real number and you can add any pair of any pair of pointers to get a new pointer so ordinary pointers in space are a special case of a vector space over the real numbers let me give you a couple of other examples in particular vector spaces over the complex numbers this was the reason hilbert originally invented the notion of hilbert space to describe functions functions of x let's say functions of one variable for simplicity that's the simplest example take the class of functions of one variable x let's take x to be the variable but complex functions of one variable functions psi of x which are the sum of two terms let's call it a real part psi real of x where psi real of x always takes on real values plus i i being the square root of minus 1 times the imaginary part of psi every complex number is a sum of a real part plus an imaginary part and every complex function now this is a function of only one variable it's not a function of a complex variable it's a function of an ordinary variable but the function itself can be complex all right so this means that for every point x there is a complex number psi of x or equivalently two real functions now let's take any complex function let's take any complex function like this we can multiply it by a complex number any complex number defines a new function you can multiply the complex number psi plus psi real plus i psi imaginary by air complex number alpha in other words complex numbers can be multiplied and therefore you can take any complex number and multiply it by any function so first of all functions of x satisfy the first rule the collection of functions the collection of functions of x a collection of complex functions of x form a vector space that's the that's the assertion we're going to check that first of all you can multiply any function by a complex number and you get a new function all right so it satisfies rule number one what about rule number two you can take any pair of functions and add them together and you get a new function so if you have two functions psi and phi let's call them that's another function a perfectly good one so if psi and phi are complex functions their sum is a complex function that's enough to tell you that complex functions are a vector space over the complex numbers real functions are a vector space over the real numbers but they're not a vector space over the complex numbers why not because you can't multiply a real function by a complex number and get back a real function you'll get back a complex function you don't get back a real function so complex functions form a complex vector space and that's the that actually is this with the vector space that that hilbert was first interested in that's an example of a hilbert space it's an important example in quantum mechanics maybe the most important example but for now i'm just illustrating the the abstract mathematical definitions complex functions of x form a vector space over the complex numbers let me give you another one it's uh very similar let's let's invent a thing called a column vector a column vector is just a collection of numbers which have and we can decide how many we fix the number of numbers uh one two three any number we like but let's fix it right and we simply arrange them in a column let's call those numbers let's give them let's call them a 1 a 2 a 3 and a 4. this would be a vector space of dimension 4. we could also think of vector spaces of dimension 2 a1 and a2 just pairs of numbers pairs of complex numbers here's quadruples of complex numbers we could make quintuples of complex numbers sex tuples of complex numbers and we could even imagine infinite columns of complex numbers or just one complex number just one complex number one unit long a one of x all of these each one separately not all together but each one separately four dimensions or two dimensions or three dimensions each of these form the collection of such objects form a complex vector space vector space over the complex numbers so let's take this particular complex vector space here a 1 a 2 a 3 a 4 where the a's are arbitrary complex numbers if these are arbitrary complex numbers and we display them in a column like this then i assert that there are rules which will allow this to be a vector space so what are the rules i'm going to invent now a rule of addition notice so far there's been no rule of multiplication of vectors only addition of vectors so the rule for adding two vectors here's two vectors one i'll call a the other one i'll call b b1 b2 b3 b4 just add the entries this is the definition of adding of column vectors a1 plus b1 a2 plus b2 a3 plus b3 a4 plus b4 obviously with this definition if you take any two vectors and you add them you get another vector the a's and b's are arbitrary complex numbers but each specification of four complex numbers defines a vector there are two vectors one labeled a one labeled b here's a third one that you can make out of it all right so i started uh for some reason i started with the addition rule here but it's also true that we can invent the idea of multiplication by a complex number if we want to take a vector let's call it a 1 a 2 a 3 and a 4 and multiply it by the complex number alpha we just do that by multiplying every entry by alpha alpha a1 alpha a2 alpha a whoops alpha a3 and alpha a4 we multiply every entry by the same number that defines multiplication by a complex number and all i've done is multiply complex numbers here obviously given any vector in any complex number i can make another vector by multiplication not multiplication of vectors but multiplication of numbers by vectors and given any two vectors you can add them so the collection of objects arranged in a column this way also form a vector space now if i were to take the collection that consists of vectors of length 2 length 3 length 4 and combine them all together that's not a vector space because i haven't given you any rule for adding a vector with with length 2 to a vector of length 3. i haven't told you how to do that and i don't see any obvious rule for that so it's uh with this set of rules the collection of two-dimensional vectors three for the whole shebang does not form a vector space but for a given length fix the length it forms a vector space and the terminology is that it forms a vector space in this case of dimension four now of course this has an analog for ordinary pointers for pointers we can just think of these as the components of the pointer along the different axes pick some axes x y and z and pointers have components so we can also specify an ordinary pointer by specifying a collection of components or collection of components along different axes and that that also defines a uh a pointer so it's natural to call these numbers the components of a vector and we'll come back to the uniqueness of components you can if you like think of such a column as a function of the index variable here well let's let's get that's too abstract let's let's drop that for the moment we'll come back to it can be thought of as a function but uh we don't need to now of course i've told you nothing about in what possible sense what possible sense can it make to identify vectors with states of systems we're not going to do that tonight tonight we're just going to talk about about the abstract notion of a vector tonight and a bit of next week we'll talk about the abstract notion of a vector and then we'll talk about how vector spaces define uh the states of a system and then go on and work out some examples of uh of uh quantum mechanical systems described by such vector spaces at the moment this should be sort of um mumbo jumbo right success success what success in in making mumbo jumbo no i think i've given you rather accurate definitions that you can understand of what vectors are and how to manipulate them how to how to manipulate them how to do the abstract operations with them these abstract operations are in many ways analogous to the abstract operations that you do with set theoretic logic with boolean logic these abstract these abstractions for how you manipulate vector spaces are the generalizations if you like of how boolean set theory allows you to ask uh less allows you to combine concepts together and or all these ideas have some meaning in uh set theory what is and and is you take two sets and you combine them and make the union of them or you take the intersection or is it the other way it's the other way isn't it it's the other way sorry and is the intersection or is the is the union of two sets the analogous logic in quantum mechanics doesn't operate on sets it acts on vector spaces so it's best that we get rid that we go through vector spaces once and for all it's the basic underlying mathematics and once we get familiar with it it won't seem so uh so much mumbo-jumbo but at the moment we're doing abstract mathematics any other question okay so let's continue with the abstractions now there's a notion of every for every such vector space well let's let's leave this here there's a notion of a dual vector space the dual vector space is in one to one correspondence with the original vector space well let's go back a step let's just talk about complex numbers for complex numbers incidentally complex numbers are themselves a vector space you can multiply complex numbers by a complex number you can add complex numbers they're just the case of one dimension one-dimensional vector spaces over the complex numbers are just the complex numbers now there's an operation that you can do on complex numbers which is called complex conjugation it's a very important operation and all it is if you think about the cartesian plane if you think about the complex plane the real part of a number being on the horizontal axis and the imaginary part being on the vertical axis then the number then is described by a real part the real part is the horizontal component and the the imaginary part of the vertical component and we will just write that this number usually called z is equal to x plus iy the complex conjugate number is just the number reflected about the horizontal if this is a number called z x plus i y then x minus i y is called z star x minus i y is the complex conjugate of x plus i y or z star is the complex conjugate of z let's uh let's manipulate something let's let's consider the product of z times z star z times z star what is it it's equal to x plus iy times x minus iy that's x squared now we get i x y and then we get minus i x y from the cross terms they cancel and then plus y squared why did i write plus y squared because i times minus i is plus one i times i is minus one so i times minus i is plus 1. z star z is x squared plus y squared what is x squared plus y squared x squared plus y squared is the square of the length of this hypotenuse here so z star z is the square of the length or the square of the magnitude of the number or its distance from the origin so complex conjugation is a convenient uh operation among other things the complex conjugate times a number is uh the mag square of the magnitude of the number um and if you're not most of you of course are quite familiar with complex conjugation i want you to think of it as a very very basic operation but more than that i want you to think of it as a mapping of the numbers back into the numbers given any number any complex number you can map it to its complex conjugates what about vector spaces i think we should probably quit now i think it's time to quit now next time we'll go through a little more about vector spaces vector spaces operators uh hermitian operators and eigenvalues and then we'll begin applying it to quantum mechanical problems the preceding program is copyrighted by stanford university please visit us at stanford.edu

Channel: Stanford

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Keywords: Physics, math, calculus, geometry, algebra, statistics, quantum, mechanics, infinite, precision, randomness, photons, law, of, physics, state, system, two, slit, theory, interference, detector, uncertainty, principle, momentum, set, states, vector, space, complex, function, number

Id: JzhlfbWBuQ8

Length: 111min 10sec (6670 seconds)

Published: Thu Apr 10 2008