The Biggest Ideas in the Universe | 7. Quantum Mechanics

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll today's big idea quantum mechanics you knew this is gonna happen this is one of the biggest ideas we have like a previous idea time quantum mechanics is something about which I have written a book so you can buy the book something deeply hidden quantum worlds and the emergence of space-time wherever you can buy books and to be honest quantum mechanics is such a big topic that it's gonna be more than one idea but in this particular video what we'll be talking about is sort of the black box the recipe what we teach our undergraduate students in physics classes about how to use quantum mechanics okay then there's deeper mysteries about how quantum mechanics really is what the ultimate description of reality according to the quantum viewpoint should be thought of as and that will come in a subsequent video but right now how to use quantum mechanics oh by the time this hour is over you'll be at the level of a you know sophomore at Caltech or something like that - some mathematical details and I don't want to be too historical about it we've never been very historical in these lectures but we can be inspired by history so I'm gonna move into quantum mechanics in a sort of pseudo historical way and the reason for doing this is because quantum mechanics is so radical is so different you know it's it's such a big break from the previous way of doing physics that it really is important to motivate it not just to state what is going on unlike special relativity where I just sort of told you the answer in quantum mechanics I would really like to see why that is what the answer apparently is and then you know maybe you'll be empowered to think about other ways that it could be however I'm not going to stick to the actual history of quantum mechanics too much it's too rich too interesting too complicated they were all these brilliant minds arguing with each other we're gonna do a simplified version of quantum mechanics and you know the reason for all this throat-clearing is quantum mechanics is special in the whole world of physics in the whole set of physical theories ever invented you know everything you've been talking about up till now has been in the context of this classical paradigm for doing physics handed down to us by Isaac and others quantum mechanics is a replacement for classical mechanics no one before the 20th century thought the classical mechanics would need to be replaced it was just working so well everyone thought it was just gonna be a matter of figuring out what is the specific version of classical mechanics that we would need so quantum mechanics comes along it tells us something completely different about the world and that thing is so completely different so alien to the ways that we think that even today we don't claim to really know the final answer for what quantum mechanics is all about so that's why we need to break up the discussion into sort of the accepted parts the used way that we approach quantum mechanics and then the deeper parts that are still controversial we don't actually know many many answers to questions that are very easy to ask about quantum mechanics so that's why it's so important quantum mechanics is simultaneously absolutely central to all of modern physics and yet something that we don't claim to understand so to get there the way I like to do it is to actually imagine a fake history physics that didn't actually happen so this would would be if quantum mechanics hadn't come along what we where were we going you know what is the sort of idea of what physics is that people might very reasonably have had near the end of the 19th century beginning of the 20th century but things didn't work out that way because quantum mechanics got in the way so the thing that we would have thought well you know classical physics right and again as I've said before we usually illustrate classical physics with particles you know doing things with positions and velocities but it works equally well for things like the electromagnetic field in the middle of the 1800s Maxwell and Faraday and others worked out the theory of electromagnetism which is a field theory it doesn't fit into Newton's idea of absolute space and time electromagnetism was the inspiration for Einstein and others to invent special relativity but the basic paradigm of things with physicians or configurations and also with momentum that you could solve equations for predict the future deterministically the clockwork universe all that was perfectly compatible with everything that electromagnetism said even statistical mechanics which came along at the same time in the 1870s again Maxwell but also people like Boltzmann and Thompson and Gibbs and so forth it was a slightly different paradigm because in statistical mechanics you imagine you don't know for sure what the positions and velocities of everything are but you still think that there are such things as positions and velocities so it was still basically the classical paradigm so that's the direction of which people were going and they thought that you could say that matter in the universe the you know the stuff of which we are made was made of particles so we had by the end of 1800s we already knew about the electron we were soon gonna figure out something called the Rutherford atom which we'll get to in a second but we didn't know about protons and neutrons yet but those were coming very soon but we knew about atoms and atoms had been thought of as particles in in some way so the point of a particle is this is gonna seem like a little bit too much detail but it'll it'll be useful later if this is space a particle is something that has a location there it is X and I'm gonna put a little vector sign because it's in three-dimensional space at any one moment in time a particle is here it is not anywhere else and along with the matter equaling particles again physics that didn't happen you had the idea that forces were given by fields we already of course it had seen the gravitational field Laplace had figured out that you could do Newtonian gravity in terms of a gravitational potential field Faraday and Maxwell and so forth show that electromagnetism fit into the same paradigm and unlike a particle a field is sort of the converse of that fields are everywhere so sort of if this is space the field exists at every point in space okay in particular at every point in space the field has a certain value and what what matters for the field is not where it is because it's everywhere what matters is what the value is and how the value changes from place to place so at each point in space like that point right there you might have for example an electric field which is itself a vector and it's located at that point in space at that moment in time so you have particles with locations you have fields that are spread all over and roughly the picture that was arising was particles move around being pushed by the forces generated by the fields so for example if you have a proton which is a positively charged particle you have an electron which is a negatively charged particle there is some pole between them which is a force due to electromagnetism they pull together they also pull together due to gravity whereas if you have an electron and another electron electromagnetism electromagnetism pushes them apart gravity pulls them together and so you had this complicated but really very rich picture where particles interacted through different forces which could compete in different ways and therefore give rise to all the complexity of life around us was a pretty darn good picture and furthermore there was some unification of known ideas within this picture so let's say you have an electron an electron has an electric field the electric field goes around and actually points toward the electron because it's negatively charged if it were a proton a positively charged proton the electric field would go out electron it goes in and you can take that electron and you can move it right well imagine that you do move it imagine you shake the electron up and down so the electric field here is the electric field surrounding the electron if you shake the electron the electric field wants to be pointing toward it but if you move it it takes time for the electric field to realize oh the electron has moved I should shift where I'm pointing toward okay in Newtonian gravity if you move a planet the gravitational field changes instantaneously all throughout the universe but Maxwell and others showed that this change in the electric field and likewise the change in the magnetic field does not happen instantaneously throughout the universe it propagates outward at the speed of light and that's because if you take this electron and shake it up and down multiple times the electromagnetic field vibrates it pulsates and the disturbance that you have created in that field propagates outward in the form of a wave and it moves at the speed of light because that's what light is light as Maxwell showed is a wave of electromagnetism so the electric and the magnetic fields vibrate and we see that as light of different wavelengths of different colors for example so that's nice you can see how you know something that you didn't know whether it was a particle or a wave namely light you say huh now we know what it is it's a way you know Newton thought the light were particles light were what he called corpuscles he had evidence for it he said when you have a your hand near the ground it casts a shadow with sharp edges the light doesn't go around like waves do but it turns out that's just because the wavelength of light is very very short so light is a wave in this field electrons which are particles created it's a very nice picture that is coming together sadly around the year 1900 not so sadly but remarkably around the year 1900 this picture begins to fall apart and it falls apart in two different ways falling apart this picture and how can it possibly fall apart well we could question classical physics but we're not going to do that yet we could question the particle nature of matter or the field like nature of forces and light so it turns out we did both we showed that light has particle-like properties this is very big sounding but that's important that it's vague sounding because when we came up with the idea we didn't have the full theory it wasn't like you went from classical physics boom the quantum mechanics we sort of claw our way up very gradually and so we made all these vague sounding statements before the whole thing finally clicks together so what does this mean this is the phenomenon of blackbody radiation and the people who figured it out were Max Planck I'm very inconsistent about whether I pronounce German and generally foreign names correctly or not but it's plunk if you were if you were mr. plunk the professor plunk that's that he who pronounce it and of course Einstein also came along to clean things up as he so often did so the point is that a blackbody is just something with a temperature okay you know when you look at things in the world you see they have colors and textures and all that stuff but you're seeing light reflecting off of it there's also the glow that the objects give off intrinsically usually you don't see that because room-temperature objects glow in the infrared and your eyes are not attuned to the infrared but if you heat something up so the clothes in the visible light then you can see it glowing and that's blackbody radiation and the nice thing is it doesn't matter what the thing is made of blackbody raishin radiation is exactly the same for everything the problem was if you took this paradigm matter is particle so there's all these vibrating atoms in the object you're heating up forces or fields so light is a wave of electromagnetism you could calculate how much energy should come out in the form of blackbody radiation and furthermore you should you could calculate at what colors it comes out right we know that long wavelength light is for the red end of the spectrum short wavelength light is toward the blue end of the spectrum and we have ultraviolet which is even bluer than blue very short wavelengths invisible and we have infrared longer than red even lower wavelengths longer wavelengths lower frequencies so it turns out that there was what is called an ultraviolet catastrophe it was only called that later but that's okay by I think paul ehrenfest called it that the classical theory of black body radiation predicts so there should be an infinite amount of energy being given off at very very short wavelengths from a blackbody that is clearly false it is clearly not what actually happens and so it was plunk who came along and said you know what I'm just gonna guess not exactly sure what his thought process was but his guess was rather than being emitted in smooth continuous waves the kinds of light that was given off by a blackbody came in discrete chunks there was a certain amount of energy given off one at a time it was not smooth continuous it was quantized in some sense okay this is the beginning of quantum mechanics circa 1900 when plunk first made his suggestion and the suggestion was that light comes in chunks of energy and the energy depended on the frequency of the light e the energy of each little chunk of light was a new constant of nature called H times the frequency H is cleverly now called Planck's constant and it joins things like the speed of light and Newton's constant of gravity as a fundamental feature of the natural world a fundamental parameter of physics I will tell you a little secret not much of a secret but a little trick usually in physics we define the angular frequency Omega so F the frequency is literally how often things go up and down you multiply by 2pi to turn that into an angular frequency so Omega is 2 pi times F that's the angular frequency and then we define H bar to be H Planck's constant that he invented divided by 2 pi so E is H bar Omega and this is a very famous formula plunks little formula and what plunk is saying remember is that black bodies are giving off light giving off radiation not as a smooth wave but in chunks he didn't quite go so far as to say that that's because light is individual particles that was Einstein Einstein came along and said forget it not just blackbody radiation but in other contexts in the photoelectric effect for example light comes out in these quanta because light really is particles and so we now call these particles of light we call them photons another thing that the name nomenclature came along later ok so that's interesting but it was also disturbing because you know literally one of the greatest triumphs of 19th century physics have been understanding that light was a wave in the electrum I medic field here's Einstein a plunk saying well it has a light also has a certain particle like kind of behavior under certain circumstances and it involves this new constant of nature plunks constant and it has sort of a quantum feeling quantum just meaning a discrete unit of something okay so that was in the back of everyone's mind that is something that is going on that was fact number one light has particle-like properties fact number two that people began to discover matter like electrons and atoms has wave-like properties or field like properties if you want where did that come from well remember this picture that we had that we were putting together of classical physics the the world that could have been but it didn't actually happen featured a bunch of particles interacting with forces and they came together to make atoms so it was the Rutherford atom Nina Terence Rutherford Rutter hoops that is the famous picture you always see so the Rutherford atom has you know protons and neutrons in it there's proton to neutrons and there are electrons circling around in in orbits circles or ellipses or whatever it's kind of like a little solar system okay that's the Rutherford atom and he got this experimentally by shooting a little helium nuclei at gold foil and seeing that sometimes they recoil by a huge amount as if the helium nuclei had hit something small and dense and hard to move namely what we now know as the nucleus of the atom and this is a very successful picture this is so successful it's used basically as the symbol for physics even today right the Rutherford atom this famous picture the problem is it can't be right we knew instantly that this picture couldn't possibly be the right answer why because we learned we knew that if you take an electron and move it it gives off electromagnetic radiation that's what where light comes from literally all the light around you right now is coming from vibrating electrons and in this Rutherford atom you have electrons moving in circles to beat the band that counts as moving okay that counts as vibrating these electrons should be giving off light so if I move this over here this is the picture that Rutherford thought was the case but what actually should be the case is something more like nucleus at the center protons and neutrons and what the electron should be doing is spiraling into the center very very quickly why because as the electron is moving it should be giving off light it should be radiating that's a terrible looking photon I just drew there I can't really draw from right to left very well so I'm gonna draw it this way it's giving off photons it's giving off light at all times it's losing energy it's spiraling into the middle so every atom in the universe if Rutherford's model had been correct should be wildly unstable how long does it take for all the electrons to fall into the center of the nucleus about 10 to the minus 11 seconds so 1/100 of a billionth of a second so all the matter in the universe including you and the tables and earnst Rutherford etc should be dramatically unstable that seems to not be how things actually are so this was a challenge this was a challenge to this otherwise SuperDuper successful way of thinking about particles and forces and so forth what to do about it well there were other clues that were clues from how light is given off by gases spectral lines and and so forth that led Niels Bohr to suggest a modification of the Rutherford atom Niels Bohr actually served as basically postdoctoral researcher in Rutherford's lab for a little while and Bohr said that you have the nucleus just like Rutherford thought okay but rather than the electron being just a particle that can be anywhere there are certain allowed orbits that the electron can be in and other orbits is are just places where the electrons are not allowed okay so the electron can be there or it can be there but it can't be in between that's just the rules why because Niels Bohr said so this is circa 18:13 approximately and well you know you might want to ask why is that true and Bohr's answer was he didn't know yet he was just making a guess like plunk made for blackbody radiation you might also say well what what orbits are allowed okay the answer is if this is if this distance is R the distance from the nucleus to the electron then the angular momentum of the electrons turns out if you want to fit the data if you want to understand the light that is given off by these gases the angular momentum which is given by just the ordinary momentum P so you have some electron moving here with a velocity and so it has P equals MV momentum is mass times velocity you can special relativity is all this stuff if you want that's the newtonian version P times R is the angular momentum and board just says you know what this quantity the angular momentum of the electrons going in its orbit is quantized because I said so and what units is it quantized in it is n so that's just some integer some number 1 2 3 4 etc times H times Planck's constant so that's where you begin to think that something miraculous is happening right plunk came up with his constant by saying that light which is supposed to be a wave has particle-like properties and you need to introduce a new constant of nature to fit the data here Bohr is saying that electrons a completely different phenomenon have to be quantized in a different way only certain orbits are allowed but to fit his data and to fit the data that he cared about not the data he collected he was a theorist but to fit the data that he knew about the quantization condition the chunkiness of which these electrons had to participate was also characterized by the same number as what plunked him up with the same constant of nature Planck's constant by the way you might want to know you know what is the number plunks constant it's some you know some number of joules seconds whatever it is I have no idea what it is because every good physicist sets H bar equal to 1 just like we set the speed of light equal to one and then we get on with our lives because no one remembers how many joules or whatever per second h-bar is so we use much more sensible notation than that and therefore in fact it's almost impossible for me to remember where the H bars go in almost any equation so if I if I write down equations and I forget the H bars that's because in my brain they're set equal to one okay so that's evidence that something is going on I mean it seems like a little clucky right little ad hoc and it seems like you're doing violence to the fundamental principles of classical physics and that sounds bad but it also seems to fit the data and it was sort of nicely consonant with what plunk and Einstein had said so that sounds good but you still want to know why something like this is true and the answer came along from a couple people the answer is basically that like we said up here matter has wave-like properties so the answer is that rather than just thinking of the electron is a little particle moving in a circular orbit think of matter waves and the first person to say this out loud was Louis de Bruyne in his PhD thesis so you know those are the standards for the PhD thesis you're Louie de boy say electrons are waves not particles okay and what he says is you know very much like what plunk said and Einstein for photons for particles of light he said that you can associate a wavelength with an electron the wavelength lambda is the Devoy wavelength and guess what it is H the Planck's constant divided by P the momentum so a fast moving particle has a short debris wavelength a slow moving particle is a very long wavelength why why is this an interesting thing to say well if you go back up to Bohr's atom to Bohr's condition the angular momentum has to be a certain amount and you say well I'm gonna interpret that as saying that when the electron does an orbit there needs to be an integer number of debris wavelengths around the orbit okay then you get fours quantization condition you get it I'm not exactly sure I'm looking at my notes here I didn't actually write down these equations so I might get two PI's in the wrong place sorry about that but the point is that if you just take the idea that an electron has a wavelength and then you say the only orbits for the electron that are allowed are those that have an integer number of wavelengths then you naturally get that only certain orbits are allowed others are not that fits the data in exactly the way that Bohr had wanted to so Bohr had this phenomenological relationship right some certain electron energies and orbits are allowed certain ones are not Dubrow is trying to explain it by saying the reason why you get these energies is because light because electrons are waves so from both sides our classical intuition is being attacked we thought that matter was particles and we thought that forces were fields and we're realizing that matter has wave or field like properties and and waves have particle-like properties okay so it was all put together I'm gonna again cuz we're not being historical the first really full-blown quantum mechanical theory was called matrix mechanics it came from Heisenberg born and Jordan and physicists didn't like it like Einstein said you know this is just mathematical jiggery-pokery like I don't understand what he's what they're saying this is Einstein saying you know this to mathematically complicated for me but very soon after that picture came along Schrodinger comes along with his picture of quantum mechanics and it turns out to be equivalent Schrodinger and Dirac and others show that these two ways of doing quantum mechanics are equivalent but Schrodinger's way is just much more comfortable to physicists because it's a story of waves not of matrices matrices are mathematical objects their physical meaning was not very clear Schrodinger comes along with waves and physicists knew what to do with that they were very happy with the idea of waves so furniture says that we should have these things called wave functions and he didn't even as it turns out the way that he thought about it wasn't exactly the right way early on but still we we keep the idea and he had an equation the Schrodinger equation which tells us how the wave functions evolve so what is that what is going on what's the difference here I mean Deroy says electrons or waves but he didn't give a dynamical equation for how those waves evolve if you would ask you know how do two electrons bump into each other and scatter off he wouldn't have been able to tell you that okay the classical paradigm that we've been working with before is given juice by this idea that if you tell me the positions and velocities of everything in the world the Newton's laws or Hamilton's equations or the principle of least action or whatever will tell you how to move them forward in time how to evolve them in time that's the kind of equation you want to turn these ideas into a full-blown physical theory so that's what Schrodinger gave us and also what Heisenberg and born in Troodon gave us they gave us a way of taking initial conditions and being a quantum version of Laplace's demon of saying from those initial conditions what happens next so schrödinger's wave function capital sy the Greek letter saw is a typical way of writing it so for just one particle let's just take one electron we'll make the world more complicated later on so it's a function of where the electron could be and when so it looks very much like a field on space time but there's subtleties that we'll get to which is why we don't call it a field we call it a wave function that's what it is or the quantum state so aka also known as the quantum state and just as a mathematical fact at any one point the wave function is a complex number so this is a real number plus an imaginary number imaginary numbers are proportional to I which equals the square root of minus 1 so you're gonna see i's square root of minus 1 all over the place in quantum mechanics because the schrödinger's wave a wave function is a complex valued thing I'm not gonna get into any of those mathematical details but you should know that that's how it works ok and what Schrodinger says is I'm going to describe the electron as a wave this complex-valued wave function I will tell you what equation it obeys and to tell you that equation I need to remind you of the Hamiltonian remember the hamiltonian the Hamiltonian in classical mechanics H is a function of X and P so you give me the position and the velocity and for some physical system the Hamiltonian tells you how much energy it has right so in quantum mechanics you don't have X and P you have the wave function so remember X in this when I say here X that is any location in space that the particle could be subjunctive Li right and P is any momentum that it could have I can assign by the Hamiltonian and energy to that particular state in quantum mechanics the state is not a particle with a position and a velocity it's a wave function okay that is the state so this sigh of X and T replaces X and P here okay so we want to know what the Hamiltonian is of the wave function okay not of a position and a momentum so what sort of your figured out how to do that okay basically and we don't need to go into the details there's there's some math involved some derivatives some calculus as you might guess and fertig ER said in quantum mechanics you should treat the Hamiltonian as an operator so we put the little hat on it to indicate that it's an operator what that means is rather than just evaluating and getting a number you act it on the wave function to get a different function so you give the Hamiltonian sorry you give the wave function Phi of X all over space evolving through time but you just give it to me at one moment of time okay so sigh of X the hamiltonian says I'm going to act on it I'm going to operate on it and give you a new function okay and then Schrodinger gives you some Mathieu facts about how that happens and then he sets that equal to I square root of -1 h-bar plunks constant the reduced Punk's constant H over 2 pi times the derivative of sy with respect to time d sy DT and this is the funny DS of partial differentiation that we talked about before so this is Schrodinger's equation in its most general form a bunch of details in the letter capital H here in what the Hamiltonian is every physical system will have a different Hamiltonian just like in classical mechanics okay but the point is this is an equation that answers the question you give me the state namely sy the wavefunction this equation tells me how it evolves in time so this is the equation that is being solved by a quantum version of Laplace's daemon this is the equation that tells you how the trajectory of the wavefunction travels through the universe through time cool that sounds good now once you have an equation its full employment for physicists right you can just solve that equation you can give it as homework to your students boy does that happen a lot because there's a lot of very interesting situations you might want to say okay solve this equation for an electron in a hydrogen atom that's actually pretty easy to do solve it for two electrons in a helium atom turns out to be really hard because the two electrons are bumping into each other in some way but you can do it okay again full-employment you can in principle answer whatever question you have that's the usefulness of this equation what it doesn't tell us is what is Sai you know how should we interpret it what - how should we think of this wave traveling through space after all didn't we have good reasons to think of electrons as particles in the first place now you can't just say well no they're not particles they're waves you have to explain to me why I should believe that and why they ever looked like particles in the first place so the first step is I mean after all let me also mention one interesting thing the whole business about the Bohr atom and the blackbody radiation and all that stuff seemed to be a story of taking smooth continuous things and making them discrete right chunking them up that's where the word quantum comes from in quantum mechanics and here in the Schrodinger equation you have a wave function that is smooth and continuous and you have an equation that says how it smoothly evolves through time there's nothing chunky or pixelated or discreet about this at all where does that come from the answer is I'm going to good a job I'm going to do with this but I'll try to explain what the answer is the answer is you solve this equation for what the wave function of electron is doing when the electron is sitting close to the nucleus of an atom okay if the electron is just out there in the middle of space then there's nothing quantum about it there's nothing quantized discrete pixelated you can have any energy move it any direction the wave function be spread out all over but basically when you put it in a box which you can think of the the nucleus as doing then the discreteness shows up imagine you literally have a box so this is just an analogy okay box analogy and you have some oops and you have I don't know what to say in here let's say you have a string attached to two different sides of the box so the box isn't doing much except holding together the two ends of the string now you can vibrate the string the string is analogous to the wave function okay the height of the string is Sai of X and you can pluck it like a violin string or a guitar string and it will vibrate but you know that there are sort of different vibrational modes allowed so there's one mode where it just goes up and down in a simple way like that okay but there's another mode where it sort of has twice the frequency and half the wavelength and there's another mode where it has three times the frequency and so forth okay so the point of these little pictures are that the solutions to the vibrational states of the string given that we have tied it down on both ends come in a discrete set there's the lowest wavelength that sorry the longest wavelength half that 1/3 of that etc the electron near the nucleus of the atom is like that it's not quite in a box but it's snuggled down there near the nucleus it's being held in there by the electromagnetic force its wave function tapers to zero very very far away and so the allowed wave functions that solve Schrodinger's equation for an electron near the nucleus of an atom are discrete there is a lowest energy one one of one higher energy one higher energy than that etc and you can go through and you can derive the Bohr atom you can derive these conditions that the momentum will only have the angular momentum only has these certain values that you can derive dubrow's wavelength versus momentum relationship all of that comes out of the Schrodinger equation but the quantumness of it is not because nature is fundamentally discrete or pixelated it's because when you solve this wave equation in a box or near the nucleus of an atom you get a discrete set of solutions okay that's where the quantum this comes from so Schrodinger with his wave equation is able to reproduce the successes of the Bohr atom with the discrete orbits and things like that what about the fact that electrons look like particles so we can do the following thing this is that was our analogy so now let's look at electrons so we don't need to just think about them sitting in atoms we can take a little electron gun cathode ray tube that is shooting out electrons and we can look at them this is like a cloud chamber cloud chamber is a physics experiment where you use sort of super saturate a gas and then you shoot charged particles through it and they make little bubbles or little clouds there's also bubble chambers which are slightly different but the same spirit that leave tracks and if you put in a magnetic field and the particles with different charges will be deflected in different directions etc and the point is you can go online you can look at pictures of particles in cloud chambers or bubble chambers or modern particle detectors at the Large Hadron Collider at CERN for example when you shoot electrons into a detector like this they leave tracks which doesn't sound surprising what else would you expect it to do well tracks are something you would expect a particle to leave but Schrodinger and deployer saying that electrons are really waves is that the actual prediction you would make for a way no the wave would spread out all over the place you'd see some big puffy cloud and it turns out that this is just one example but there's many more examples when you actually look at electrons you never see waves you never see these big puffy clouds you see particle-like behavior Schrodinger himself hoped that something like this would happen the hope the Schrodinger had was that if you started with an electron wave function so here's space and you had an electron wave function sigh like this that it would naturally evolve over time into something that was more concentrated in other words that the wave function electron in empty space was sort of naturally narrow and begin to look more and more point like and that would explain why electrons look like points just because they're waves but they're waves that are really concentrated near one point sadly this does not happen at all it's the opposite if you start with Schrodinger's equation with a little wave packet and just let it go in empty space it spreads out all over the place so we're a little bit stuck and this is why you need to think start thinking about interpretations of quantum mechanics it seems very very naively that the following thing is going on that when you're not looking at the electrons when the electrons are just sitting there in atoms and you want to explain their energy levels and stuff like that when you're not looking at them the electrons are behaving like waves and when you look at them they behave like particles that seems to be what's going on but that's crazy because clearly whether or not we're looking at something should not be part of the fundamental nature of reality right that shouldn't be part of our best theories of physics so the greatest minds in the history of physics came together and decided that the right way of thinking about this was to say that when we're not looking at the electron it's a wave and when we look at it it's a particle this is this is what we teach our students to this day and some people like Einstein and Freud anger never liked this idea but nevertheless this is this is what we say so this is the Copenhagen interpretation of quantum mechanics and I say that only with some trepidation because no one really agrees and what the Copenhagen interpretation is you know we attributed to Niels Bohr and friends but Heisenberg was just as influential and in fact like the first step in this direction probably could be credited to Max Born who's Heisenberg's collaborator but the idea is that we need an extra set of rules in physics so if you if you do classical physics you do Newtonian physics whether it's fields or particles or whatever you have two ingredients you have the system which is described by a point in phase space its positions and velocities and then you had the dynamical equations at the equals MA or Hamilton's equations or whatever and that's it you stop in quantum mechanics in the Copenhagen interpretation you have two different sets of rules the first set of rules is you have a state in the state is the wave function for daggers functions Cybex and T and you have an equation telling you how that wave function evolves the Schrodinger equation it's exactly parallel to classical mechanics but then in the Copenhagen interpretation in what I'm calling the Copenhagen interpretation what you might want to call the textbook or or usual conventional interpretation of quantum mechanics there's a new set of rules and the new set of rules only has to do with what happens when you observe or measure a system the rules of quantum mechanics that we teach ourselves have a whole separate category for measurements and the outcomes of measurements unlike any other theory of physics and you know I'm not saying anything controversial here this is what you can find in everyone's quantum mechanics textbook the controversy comes in when you try to say well what's really going on when you do these measurements but for right now that's not where we are in this video that's for later contemplations right now I'm just telling you what the rules are okay there are basically three rules one rule is let me let me actually just say this is for when you measure or observe a quantum mechanical system rule one is you see definite outcomes and I'm trying to translate some very specific mathematical formalism into everyday language here the point is you can measure something you call the position of the electron you can ask the question where is the electron you can shoot it through a cloud chamber and look to see where it is and the first rule of quantum mechanical observations is you see it somewhere you never see it spread out all over the place okay you can think of this wave function Phi of X as attaching a different value to every possible place the electron could be seen and the wave function might be completely spread out but when you see it you see it somewhere you see some definite outcome for that measurement of course if you you never in the real world have perfect measuring precision right you never have a an infinitely good measuring apparatus so there's always some uncertainty about the measurement outcome but that uncertainty is not the famous uncertainty principle okay even in classical physics your measuring apparatus is never perfect that's not the point the point is if you had a perfect measuring apparatus you would see a definite place where the electron is that's the first extra rule the second extra rule is you cannot predict from what that outcome will be with perfect fidelity all you can do is say the probability of getting a different measurement outcome so the probability of a measurement outcome is given by a very simple formula remember the wave function psy is a complex number so we want to take its length we're going to square it and that means square the real part square the imaginary part and add them together the notation we use is we put bars around it so we take psy and we ask what is the value of the wave function associated with that measurement outcome okay so you see what I'm doing here I said that I was a function of position that's one way of talking about size one way of talking about the wavefunction there are other ways of talking about it which we'll get to in a second but the point is if we want to measure the position we say well okay what is the value of the wave function at every individual position and the probability of seeing it at that position is the wave function squared this is the famous born rule after Max Born Max born is not Niels Bohr those are two different people so it was born Max Born who said that's what schrödinger's wave function is okay it is a way of calculating the probability of seeing the electron doing different things it is not sort of something that you have direct access to observational E and this is the part that Schrodinger had not anticipated and didn't really like okay but it is this is what we teach our students and then after observation afterward by making this observation you have changed the wavefunction dramatically you get what is called wavefunction collapse after the new sy equals one at that outcome that you saw and zero for any other outcome this is called collapse of the wavefunction what does that mean so in pictures what this means is if before you've made a measurement you had position and you have some wave function like this sigh and then you measure it capital sine let's say you measure it and you see that it's here that's the outcome you actually get in your experiment then after you do your experiment the wave function is not that anymore here's what the wave function was here's what the weight and here's where you saw it and here's what the wave function is now by one I mean it is completely localized at that particular measurement outcome it's not really the number one yes the integral of the wave function has to give you the number one not not the actual value of the wave function but you get the point the wave function goes from being all spread out to collapsing on to the value that you saw and you can't predict ahead of time exactly what it was all you can predict is where the wave function is bigger there's a higher probability of getting that outcome where it's smaller there's a lower probability so these are the rules the extra rules dealing with measurement and observation in quantum mechanics okay this is why quantum mechanics is weird and hard and controversial because no other theories of physics have these new sets of rules for the outcome of measurements okay it seems that quantum mechanics is trying to say that there is a fundamental divide between what nature really is and what you see when you look at it it seems and I keep saying scenes cuz not everyone's gonna agree okay but the most straightforward way of interpreting this is the electron is a wave when you're not looking at it but you never see the way if you see a particle the things that you see the set of possible outcomes for measurements are fundamentally different than the reality behind the scenes when you're not looking at it okay that is the Copenhagen interpretation of quantum mechanics so this is this is complicated and and controversial for all sorts of reasons there's a very obvious reason let me mention two reasons why this might be problematic I'm not gonna answer these questions but here are the two problems problem one is the measurement problem by which I mean come on what do you mean when you measure something like what what is that supposed to tell me what counts as a measurement Duany can I do with my eyeballs do I need a microscope I eat a cloud chamber can my cats do measurements can like insects in the room do measurements can a video camera do a measurement what about an atom can that do a measurement what if I only look at it a little bit does the wavefunction still collapse you know you haven't told me how quickly measurements happen what kind of interactions counters measurements and if you ask if you're you know an undergraduate taking quantum mechanics and you ask these questions you're told to stop asking those questions that is not what the Copenhagen interpretation tells you now there are people who want to be more respectable and one of the sort of try to answer those questions but fundamentally the measurement problem is number one measurement plays a crucial role in the rules of quantum mechanics number two we haven't defined what we mean by measurement that's the measurement problem and that's an issue here's the other problem that I like to highlight that sort of gets swept under the rug sometimes the reality problem now I spoke in ways like I said I was I hesitated sometimes along the way here because not everyone agrees I spoke in a way that made it sound like the electron really was a wave when you're not looking at it that in other words the wavefunction sigh schrödinger's wave function really is the electron when you're not looking at it but you don't see it you never see sigh you only see different measurement outcomes like the position so is it really true that sigh is reality or is is schrödinger's wave function just a tool that we use for calculating the likelihood of certain measurement outcomes is side just a black box is it possible there are some other ways of characterizing quantum mechanical States which if we knew what they were would give us definite outcomes for quantum measurements that's what people like Einstein and also you know Dubrow and other people really hoped was going to be true other people say you know so there's there's the idea that there's hidden variables in some sense other variables that are not included in this discussion that would if only we knew them give us exactly what the measurement outcome was going to be that was sort of the Einsteinian de Broglie and point of view David Bohm invented a theory like that later on there's the Copenhagen point of view which says don't talk to me about reality don't worry about reality who cares reality we don't care about that we care about them observational outcomes we care about what we see okay that's a different philosophy there's you get a third philosophy that says actually let's face the music let's imagine that the wavefunction really is reality okay that's the one that I tend to like myself which is why I tend to talk that way but I want you to know that there are people who disagree so and let me let me drive home how important that is remember you know we were talking about these atoms and we said you know there was the Bohr model okay or the the Rutherford model the Bohr model where the electrons were in little orbits and electrons are really tiny little things so in this picture that we drew you can make the statement that atoms are mostly empty space right because there you go you know that you look at the size of the electron which is basically zero the size of the nucleus which is not quite zero but still pretty small and mostly it's empty space in between the problem with that is if you believe in quantum mechanics you shouldn't believe that the electron is a little dot moving in an orbit okay you should know better than that what you should do is imagine if anything if we know anything at all let's see if I can actually do this yeah we should imagine as the electron is sort of in a cloud maybe a little bit more likely to be nearby okay but the wave function of the electron is actually the best way to talk about the electron and that's spread out all over the place so if you believe that the wave function represents reality then you don't say that atoms are mostly empty space atoms are mostly the wave function of the electron the electron wave function takes up space in the atom that's why atoms have a size at all so even such a simple straightforward question like or atoms mostly empty space or not we don't know the answer to that or at least we don't agree on what the answer is some of us have ideas to what the answer is but people don't agree so you shouldn't make very definitive statements about it okay but let me just that's the basic story of quantum mechanics but let me kind of emphasize some of the features that come along with this story I talked about observational outcomes right so things like in in quantum mechanics things like position I don't need to write things like position momentum etc don't you're gonna some people gonna give me a hard time for this but they don't can you guess what the next word is exist what do I mean by that I mean you know of course you can measure the position of an electron I already told you you could do that but that's the point these things like position of momentum they are observational outcomes they're possible observational outcomes so in quantum mechanics there are things you can observe and things you can't ah I have not improved my ability to write and talk at the same time sorry about that things you observe things you can there are outcomes you can possibly get like the position of an electron you can ask what answer do I get when I measure the position of the electron so we say that position is an observable it's a thing that can be observed and you can have outcomes and the wave function gives you the probability of getting all the different observational outcomes but if you want to say well when I'm not looking at it okay when I the wave function is some spread out thing what is the position then and the best way of thinking about I think this should be true no matter what your favorite interpretation of quantum mechanics is the best way of thinking about is there is no such thing as the position or the momentum of the electron when you're not observing it when you are observing it you can ask the question where do I see the position of the electron under where do I measure how much do I measure the momentum to be but it's not something that has a definite existence when you're not looking at it in the general case so this is why quantum mechanics is hard because intuitively no matter what no matter how much formalism we've been taught etcetera we tend to treat things that we observe as real all right that's the world that we're seeing there it is and quantum mechanics is drawing this distinction between what we see and what is even things that are so tangible and obvious and intuitive like the positions of things our only observational outcomes they are not what really exists in this kind of way of thinking leads to a very profound and famous consequence called the Heisenberg uncertainty principle okay so you can have wave functions that are very close to or even exactly definite in some observational outcome so you can imagine a wave function psy as a function of X that looks like this as we drew it before something that is very very close to completely infinitely peaked around some definite outcome and then you could say well it's almost true we're pretty darn close to true that that particle has a location the X naught in this particular case but then you say okay well what is its momentum and you have no idea what that is why is that so what what is really going on well it's because this is a wave function of definite or pretty close to definite position you can also think about wave functions of definite or pretty close to definite momentum but they don't look anything like this the wave function is not separately a function of position and momentum right remember I said the wave function is a function of position you can figure out what the momentum is by telling me what the wave function is as a function of position so if I have a wave function that looks like a sine wave this is a wave function of definite momentum and that is completely compatible with this is because this is where louis de bois wavelength comes from right lambda the de rooy wavelength was H over P the momentum so there's this inverse relationship between wavelength and momentum if you have a wave function that has a definite wavelength that is just oscillating with a fixed wavelength everywhere then it has a definite momentum but to have a fixed wavelength everywhere is the opposite of being localized in space it turns out this is a little bit deeper than you really need to go but if you if you want so let's think about when you turn on the radio okay you turn on the radio or not forgetting about the radio let's imagine you're listening to music or listening to sounds okay you can tell the difference when you listen between a high-pitched sound and a low pitch sound notes of different pitches are things that your ears can discriminate between right even though what is actually hitting your ears is just the density of air at any one moment in time what's happening is that the density of air is oscillating up and down okay you could plot the density of air near your ear and it would be going up and down and what your ear does is to separate out contributions in the sound waves from different pitches from different wavelengths from different frequencies of oscillation so there is a wonderful mathematical thing called Fourier analysis oops I missed a letter sorry mr. fool yang a lot of people named Fourier actually so Fourier analysis is the thing that says any function f of X can be written as a combination of functions with definite wavelengths so this is what your brain and ears are doing all the time they are taking a function what is the pressure of air at your ear as a function of time and they are writing it they are separating it out into a bunch of different notes a bunch of different wavelengths corresponding to different pitches of the music or the sound that you're listening to and this is possible because of Fourier analysis Fourier transforms they sometimes D called okay so in other words let's say you have some function f of X and it looks like you know it's small and then it sort of vibrates a little bit and does something like that there's X so you can write that as something with a long wavelength plus something with a slightly shorter wavelength plus something with an even shorter wavelength etc I'm not drawing these perfectly but all these are supposed to be perfect sine waves with absolutely definite wavelengths okay and this is called the Fourier transform and so this is you know well let's let's just say it we can take f of X in fact let's say let's just be very blatantly on point here you can take the wave function psy is a function at X right that's the schrödinger's wave function and you can Fourier transform it as we say to write it as psy as a function of P the momentum because these things of definite wavelengths are roughly speaking parts of the wave function with definite momentum now this point that we made up here is that if you are of definite position or pretty close you are not a definite wavelength and therefore not a definite momentum likewise if you're of definite momentum a smooth curve sine-wave then you are not a definite position so therefore there are no wave functions with the property that you know ahead of time when you measure the position what you will get and when you know ahead of time when you measure the momentum what what you will get you can do something that is pretty close you can have a wave function that is close to being localized in both position of momentum if it sort of looks like this right so it's zero to the left zero to the right Wiggles around a little bit and there is some sort of variation here that is not exactly located in position but there's a certain uncertainty if you want to call it that if you were to measure the position it would be within that band Delta X and if you did the Fourier transform to write this little oscillating wave packet as a sum of momenta it'll be pretty close to this P here this is approximately the wavelength okay of this particular wave but it's not exactly it's really a sum of different things and so Heisenberg comes along and says that he formalizes mathematically this fact that you cannot both be localized in position and momentum as Delta x times Delta P is always greater than equal to H bar over 2 there's details you have to worry about here but the reason why I'm doing this is because there's one emphasize to you that this has nothing to do with measurement all by itself okay Heisenberg's uncertainty principle is not the statement that you disturb a quantum system when you measure it that statement was already made that that savings in the collapse of the wavefunction before okay Heisenberg's statement is a statement about all the possible wave functions you can imagine the momentum in the position are not things that exist they are possible measurement outcomes okay and Heisenberg statement says that no matter what wavefunction you imagine it can't be one where you will know ahead of time both the position and the momentum that's what this is saying it says nothing - it's disturbing it but disturbing it is part of what quantum mechanics actually says ok I think we should I think we should call it there I have many more things to say but that's okay I've planned ahead of time this is only going to be one out of I'm gonna try to squeeze the rest of the fundamentals of quantum mechanics into another lecture into another video hopefully next week but this is the basic story and you can see why things are tricky okay this is this is why things are tricky right here because we need to talk about observations and measurements when we do quantum mechanics we don't know what that means that's a problem and we don't even know what the reality that we're actually describing is there's a sense in which generations of physicists have been more or less brainwashed into thinking that they shouldn't care about what reality is that all they should care about are the outcomes of measurements okay Steven Weinberg the famous physicist and one of my absolute heroes in physics you know among all the living physicists right now he's someone I admire as much as anybody and he wrote a textbook on general relativity where he makes this attitude very explicit he's talked about general relativity not quantum mechanics but it carries over he says all this talk about curved space-time and stuff like that who cares I don't care about that stuff what I care about is making predictions for where spectral lines will appear or you know photons will hit photographic plates and things like that and as much as I admire them I could not disagree more about that in my view physics is about understanding reality no you know twelve-year-old girl gets excited about physics because she says someday I will grow up to make predictions about observational outcomes what they get interested in is the cool stuff that we learned about the real universe and quantum mechanics is a place where we haven't yet figured out what the science is trying to teach us about the real universe what reality is actually is that what the reality actually is that we are describing with our formalism of quantum mechanics so we have to do better we're trying we're not there yet tune in next week maybe we'll make some progress

Channel: Sean Carroll

Views: 214,679

Rating: 4.9020667 out of 5

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Id: BJlkkOIFjx4

Length: 65min 27sec (3927 seconds)

Published: Tue May 05 2020

Reddit Comments

[u/Amida0616]

Is that Cthulhu behind him? Because non Euclidean geometry is a big idea.