The Biggest Ideas in the Universe | 12. Scale

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll I wanted to express my appreciation for those of you who have stuck with it this far I know that the last three videos in particular a series building up some fundamental ideas from quantum field theory they can get pretty abstract they can get pretty challenging not because they're intrinsically hard but because they're very different from what we're usually used to either in our everyday lives or even in our first elementary physics courses okay so I thought we would get a little break today a little bit of a palate cleanser there's some big ideas in here absolutely but I wanted something a little bit more tangible that we could sort of relate to things we're familiar with so today's big idea is scale this is a very big idea indeed but it is no less profound as we will see than the other ideas scale is just the size of things and it particularly you might ask how different sizes of things depend on other variables if you change one thing how do other things scale with them is a question that we often ask we've already talked about this of course a little bit especially in the last video and we normalization we talked about the fact that you could put an ultraviolet cutoff in your quantum field theory and that sort of sets a scale an energy or a wavelength scale beyond which you say we don't know what's going on we don't know what the laws of physics are and Ken Wilson and others figured out ways to nevertheless do physics below those energies without knowing physics above those energies and this raises the fact that in particle physics in particular and in quantum field theory there are a set of scales characterized by nature or maybe the other way around nature is characterized by a set of scales so I want to talk about that a little bit today but of course because this is what we do here I want to dig in a little bit more to where some of these skills come from and how they relate to each other it turns out that the whole idea of where scales come from is enormous and difficult and we're not going to get to all the different scales and some that we don't know some of them we do know we'll have to wait a little bit to figure out where they come from so when I talk about the size of things what is that supposed to mean well it depends on what units you're thinking about right size could mean length of course could just mean distance but it also mean any combination of lengths so area is length squared volume of course but then of course you also have time right space-time we know is one thing so if you can measure lengths you can measure time and then we have mass we have energy we have a whole bunch of different ways in which you could talk about the size of different things the size of the energy scale the size the mass scale so we are highly-trained particle physicists so we will use natural units and that will make our lives much easier and I'm not sure why it was the particle physicist who got to come up with the name natural units there's a whole bunch of unit systems out there we can use so we set h-bar plunks constant the reduced Planck's constant with the two pi in it equals one and the speed of light C also equals one and then we need to pick a unit right we need to pick one unit to actually stick with so we know we went over before is that the units of energy in natural units are the same as the units of mass and that's because you know that there's an equation e equals MC squared and if the speed of light is equal to one then energy just equals mass you could equally well to make that argument use the equation for kinetic energy right equals MC squared is the rest energy the kinetic energy in nonrelativistic physics is e equals one-half MV squared and the half is dimensionless M still has a dimensions of mass and if the speed of light is equal to one that means every velocity is dimensionless because every velocity is measured in the fractions of the speed of light so e equals one-half MV squared also tells you that in natural units energy and mass have the same units and we're gonna measure them using electron volts evey okay I will explain a little bit I gave the technical definition of electron volt you have a vault of electrical potential it requires one electron volt of oomph of energy to move it across that one volt of potential let's say between the ends of a one volt battery I guess that's completely meaningless who does that in their everyday life moving electrons across one volt batteries but will give you a feeling for how much an electron volt is during this video bye-bye sort of comparing it to other things that we actually have feelings for sometimes we use GeV G 4 Giga this is billion electron volts okay so 10 to the 9 electron volts in fact particle physicists are shameless about using electron volts but also GeV 4 billion electron volts but also MeV with a capital M 4 mega electron volts 10 to the 6 ke V kill electron volts 10 to the 3 MeV with a lowercase M means milla electron volts 10 to the minus 3 TV tera electron volts etc I'm gonna try to mostly use electron volts sometimes I will use GeV because I'm so used to using GeV for certain things that that's what I just understand the units in the different quantities but I'm gonna try it like if you don't know it I'm gonna try to make your life easier by just sticking with evey most of the time and then you have you know space and time so length and time have units of 1 over energy so we will measure those in units of 1 over electron volts so if you wanna know how tall somebody is you can give their height in inverse electron volts okay that's one of the lessons of this lecture and the conversions in in natural units just to relate to more natural things one gram interestingly you know there's not a unit of energy that everyone is familiar with like probably the unit of energy most people are familiar with is the kilowatt hour how many kilowatts you use in an hour sorry say the way around kilowatts times a number of hours so kilowatt is a rate at which you're using energy so if you go to your electrical meter outside it measures the rate which you're using energy and then it integrates it over time to get kilowatts times hours and that's even to total energy but also completely bizarre unit of energy in CGS units centimeters gram seconds energies measured in eggs in MKS meters kilogram seconds its measured in joules but very few people have a feeling for either what an ERG jool actually is so I'm not gonna tell tell you how to convert but there it is electron volt so you can just look it up I will tell you how to convert grams gram one gram is approximately ten to the thirty three electron volts and just to reiterate that I'm very much a theoretical physicist here we are going to use one significant figure in all our calculations so of course there's a much more precise number relating grams to electron volts but since we only really care about the basic way that things relate to each other we're not gonna be plugging in and doing building any machines based on this I'm just gonna give you the order of magnitude here okay so one gram ten to the thirty three electron volts so as a mass one electron volt is really really tiny okay one electron volt is ten to the minus 33 grams and grams are pretty tiny a gram is the mass of one cubic centimeter of water and a cubic centimeter is about a fifth of a teaspoon for you Americans out there so a gram is a very tiny amount of mass electron volts are much tinier than that one centimeter is approximately what do I have here you can write it two different ways ten to the five inverse electron volts right as I said the units of length and natural units are inverse electron volts but sometimes it's more convenient to write this as the quantity ten to the minus five electron volts to the minus one power okay so if you're using units of inverse evie one centimeter is ten to the five of them 100,000 of them but what you want is sort of some sort of physical relationship right if someone is quoting you a length in electron volts you want to know like okay what is the length that one electron volt corresponds to or vice versa what are the number of electron volts that one centimeter corresponds to so one centimeter corresponds to ten to the minus five electron volts to the inverse powers as well see when we dig in a little bit you'll get a little bit more of a physical feeling for what that actually means and one second traditional unit of time is about 10 to the 15 inverse electron volts so in other words 10 to minus 15 electron volts to the minus one power so time and space relate to each other by C equals one it should not be a surprise to you that one centimeter corresponds to a much smaller number of inverse electron volts than one second because in one second the speed of light says that light is gonna move a lot more than one centimeter right it's gonna be a very big distance so one second in energy units corresponds to a much much much tinier energy than one centimeter does okay so that now with those in hand we can do the traditional thing this has been done many times the powers of ten kind of diagram and I don't want to belabor it too much but I know that some people been asking in the videos you know like the Planck scale what is that or you know how what is an electron compared to a proton things like that so we can actually just go over these very very quickly if I can get them right I think we'll be in good shape here so I think I can draw this this way let's see okay it's a that's close enough to a straight line and I'm gonna cheat not cheat a little bit but I'm gonna ask your indulgence a little bit you know what I want to talk about thirty orders of magnitude of difference between energies at the low end I'm gonna say that this is 10 to the minus 3 electron volts and I want to go all the way up to 10 to the 27 electron volts okay so that's why I've divided into three because then by powers of 10 this is 10 to the 7 electron volts and this is 10 to the 17 electron volts okay sadly 10 is an awkward number we only pick it because we have 10 fingers so what I'm gonna do is draw one two little subdivisions and they're gonna go by 3 and then by 4 right does this make sense so this is I'm at the right smaller I suppose 10 to the well this is just one one electron volt this is 10 to the 3 electron volts then so that's 3 between there and there another 3 between there and 3 between there and there 3 between there and therefore between there and there that's what I'm trying to say here then same thing here so this is 10 to the and Evi 10 to the 13 Evy same thing here this is ten to the 20 Evy 10 to the 23 Evy okay this is just the way that I thought that would be most convenient to actually do this and then want to tell you where the different masses of particle physics are so for example an electron has a mass of about 500,000 electron volts so I'm tempted to just call that a million electron volts but that factor of two kind of really does matter so it's a little bit less than a million electron volts so let's put it here this is the mass of the electron so what I'm gonna do here to start is convert masses into energies and just tell you where the mass scales are and then we'll talk about some real energies later so this is 5 times 10 to the 5 electron volts so a capital M evie is a million electron volts mega electron volts you will often see the electron mass written is 0.5 MeV okay whereas protons and neutrons are about 1 GeV about 1 billion electron volts so that's here write it like there okay so this is protons neutrons and that's 10 to the 9 electron volts 1 GeV 1 Giga electron volt and then things like the Higgs boson which is 125 GeV the top quark is 174 so again ignoring anything other than the order of magnitude that is 10 to the 2 times the GeV so it's 10 to the 11 electron volts okay so that's here this is Higgs top quark is about the same 10 to the 11 electron volts okay and then everything that we know in the standard model of particle physics is mostly in between right things like muons and Tau's are in between electron and Higgs all the quarks are in between here so this is basically the range that's it and you might ask well why do we have to go all the way from 10 to the minus 3 to 10 to 20 710 and 27 electron volts is of course the Planck scale this is what we call the reduced Planck scale there's a factor of four pi or something like that that goes in there but 10 to the 18 GeV or ten to twenty seven electron volts is about what we mean so you can see the huge range you know this it covers a lot of sins the fact that we put 30 orders of magnitude into this picture this is an enormous difference between the mass of the Higgs and something like the Planck scale okay it's a really big difference but there it is and it's important once gravity comes into the game and you can instantly see sort of why gravity is not important in particle physics the Planck scale the Planck energy is basically the energy at which you would have to do some particle collisions before gravity became noticeable compared to the other forces of nature and here we are doing experiments that let's say the Large Hadron Collider it is roughly here you know ten or a hundred orders of magnitude sorry ten or a hundred times bigger what is the LHC something roughly speaking at this order magnitude level ten T V so 10,000 GeV so that is 10 T V is 10 to 12 10 to 13 yeah there we go 10 to the 13 Evie so let's say this is where we'll put the LHC I really should sort of rehearse and playing these lectures a little bit better okay and then of course you have things like atomic nuclei right like an atomic nucleus is typically a collection of protons and neutrons it's always a collection of protons and neutrons so nuclei sort of span some range here so as all the physics that we know is in this little range except there is of course at the neutrino right neutrinos there's three of them electron neutrino muon neutrino tau neutrino neutrinos are remarkably complicated and subtle actually even though I say there's three of them there's also three different masses for neutrinos the lightest one the middle one and the heaviest one but there's something called neutrino mixing that says that the lightest neutrino is not the electron neutrino it's a combination of the electron the muon and the Tau neutrino and likewise for the middle one and likewise for the heaviest one there's not a one-to-one line up between what we call the flavor of the neutrino are they electoral honor they muon or they tell what that means is are they produced along with those particles so when the muon decays it can turn into an electron but only at the cost of making a muon neutrino so the muon miss is conserved and then an electron antineutrino so electron this is not created and those flavor states of neutrinos do not line up with the mass States and this is why neutrinos are complicated but we don't even know what the masses are we think that they're somewhere in between 10 to the minus 3 and 1 electron volts that's why I need to go down all the way here and also interestingly there's another parameter that is down there which is remember we talked about the cosmological constant the vacuum energy that pushes apart the universe so we said that there is an energy density Rho lambda equals the energy density and empty space the amount of energy in any one cubic centimeter or something like that and so that has units of energy to the fourth power so we can write it as some energy scale Evac to the fourth power and you run through the numbers you compare to the observations that we've made and Evac turns out to be about 10 to the minus 3 electron volts so that's not the mass of any particle okay that's not the mass of anything that is the scale that is the energy scale that you raised to the fourth power to get this number the energy density of the vacuum but that's okay this wonderful thing about natural units is we can compare energies and other scales and masses all to each other at the same time now you might say hmm it's interesting that the energy density of the vacuum is at the same energy scale as the mass of the neutrinos just I'd have anything to do with anything else no one knows because no one knows what the explanation for the vacuum energy actually is but you would not be the first to try to relate them if you went out and made a theory that had them related to each other okay what else can we talk about well we don't know what the dark matter is okay we're not or what the dark matter is but there are ranges of possibilities for the dark matter so one range one particle that could be the dark matter is the acción accion's could be maybe 10 to minus 4 electron volts very very light particles some of you we haven't gotten into cosmology in detail yet but some of you might know that one of the things we know about the dark matter is that it's cold and moves slowly and you might say well I would think that light particles should be moving fast because they're light so even a little bit of energy would make them move very rapidly why our accion's qualifying as cold dark matter the answer is that they're made with zero momentum from the start when the accion's are born they're born at rest and this is a complicated story about accion's which makes them very interesting to cosmologists the other possibility is that the dark matters up here sort of LHC type scales ok so if you have weakly interacting massive particles wimps I should put question marks here because we don't know if these particles exist but wimps could be at the scale where the Large Hadron Collider is looking maybe a little bit less than that actually to be to be honest because the hope is that you have more than enough energy to make them so let me be a little bit more honest there and put the wimps up here ok they might be comparable to the Higgs a bit less than the things we don't know there could be as much as a proton or something like that and then there's this idea called grand unification there's this idea that we don't know if it's true it became very popular 1970s because there's four forces of nature that we can easily recognize gravity electromagnetism the weak nuclear force and the strong nuclear force in some sense you can qualify electro and magnetism is two different forces but Maxwell unified those back in the 1800s so we call them one force in the 1960s Steven Weinberg played this trick where he showed how to unify the electromagnetic force with the weak nuclear force to create what we now called the electroweak theory and the energy scale of the electroweak theory is about these energies we've been talking about like somewhere around the Higgs boson the LHC etc okay a hundred GeV or ten to the 11 electron volts that's a typical scale for the electroweak force now gravity has always been hard gravity is a hard thing to put into the quantum mechanical framework but the strong force is perfectly quantum field theory so an obvious next thing to do is to do what Weinberg did but do it again rather than just unifying electromagnetism in the weak forest unify it with a strong force as well and that attempt was called grand unification so grand unification doesn't even include gravity it's just supposed to be the electromagnetic weak and strong forces and what you can do is predict using physics etc that there's a typical energy scale the grand unified theory energy scale again question mark times we don't even know if it's true but it's something two orders of magnitude maybe three orders of magnitude below the Planck scale 10 to the 10 to the 23 24 25 electron volts again we don't know so there's some some wriggle room there so this would be the scale at which all the different forces differentiated or at least the strong force differentiated from the weak from the electroweak force then electricity magnetism differentiate from the weak force down here near where the Higgs boson is so this is something that again is hard to probe experimentally it's very hard to build a particle accelerator that reaches those energies the hope of grand unification was that because it unified everything it gave you a way for protons and neutrons to decay into particles that were not strongly interacting at all now neutrons decay all the time but they decay into protons protons according to every experiment we've ever done don't decay at all but according to grand unified theories they should decay so people started building proton decay experiments and they were very disappointed that the proton is not observed to decay this was sort of this is you know since the 80s we've been looking for this and haven't found it this is the first hint that our attempt to go beyond the electroweak theory haven't been paying off yet and that's still true you know forty years later so what you get is in between here and here in these theories you get what is called a desert sometimes called class shows desert after Sheldon Glashow who was one of the people along with Howard Jehu was an originator of the grand unification idea the idea that maybe down here near ten to the ten evey ten to the twelve evey there's a whole bunch of particles and there's nothing new going on until you reach the grand unification maybe that is true maybe it is not we don't know we haven't looked yet but there is this big gap in scales that is one thing that I absolutely wanted to emphasize to you so there's another thing okay so this is basically this is the powers of 10-story at the particle physics level okay it would be nice to relate these to some more tangible things most of us have never seen proton with our naked eye most of us have never seen the Planck scale with our naked eye or anything like that how do you relate these like it's just a lot of energy or a little bit of energy okay so let me give you two different scales you can compare to let's make another color here what if you clap your hands like that I mean not like that because that's very slow what if you clap your hands as fast as you can right you know do your best to move your hands as fast as you can I don't know if this is if I got this right but this is the homework I did for you folks I sat down and I calculated the kinetic energy involved in clapping your hands together don't trust me on this I'm not very good at those kinds of back-of-the-envelope calculations but the answer I got was that a clap is around here this is the kinetic energy you get by clapping your hands together the energy released and sound and heat when you clap your hands together it's about 10 to the 18 electron volts okay and if you have a sort of medium sized jet airliner or a very big freight train loaded and they're moving at you know 100 miles an hour or whatever they move at those will have an energy that is basically the Planck scale either I think of Boeing 767 was what I read trains of course depends on how much you've load them how fast they go cetera but the kinetic energy but you pack into a moving train or plane is about the Planck scale so on the one hand the Planck scale is a lot of energy like you don't want to get hit by moving freight train but on the other hand you can clap your hands together and you create more energy than any particle accelerated by the Large Hadron Collider oh you might be wondering like why do we spend 9 billion dollars to accelerate particles to these energies when I can create much more energy just by clapping my hands the answer of course is that the amazing thing about the LHC is not that the energies are high but that they are packed into a single particle likewise for the Planck scale up here the amazing thing is not that the energy is just enormous ly big but that it is perhaps packed into a single particle or at least packed into a single energetic event whatever is happening at the Planck scale the first excited modes of superstrings would have energies near the Planck scale and that would have in a single particle the energy of a moving freight train okay so that's why it is so much energies it's a lot of energy in one little tiny package that's why it's hard to make I mean you can smash airplanes together all you want the total energy might be of the order the Planck scale but you're not concentrating them into one particle all the individual protons or neutrons that are hitting each other will have energies way way below the Planck scale so no interesting quantum gravitational effects are going to take place okay another way of comparing these energy scales and particle physics to everyday scales is to compare masses to energy right so so as far as masses are concerned let me get this out of the way because you already saw that and I need the space so to compare masses to energies remember that as far as mass is concerned one proton is about the same as one Neutron and that's about 1 GeV which is 10 to the 9 electron volts for comparison purposes an electron is about 118 hundredths of a proton in terms of mass okay so when you make up ordinary matter almost all of the matter almost all the mass comes from electrons comes from protons and neutrons comes from the nuclei electrons are important for the chemical and electrical properties of the matter but as far as the mass is concerned forget about the electrons it's all the actions in the protons and neutrons and conveniently protons and neutrons are about the same mass if you're a little bit more careful here I'm going to get the numbers wrong but I think that the proton is 0.93 eight of a GeV and the neutron is 0.93 nine of a GeV it's just a little tiny bit more than that but to our precision we're going to talk talk about it as just one GeV and then you know let me get this right okay a gram a gram is ten to the thirty three electron volts so one gram of mass ten to the thirty three electron volts so that's ten to the 33 - 924 proton masses so ten to the thirty three electron volts doesn't even fit on our plot right one gram one-fifth of a teaspoon of water has more energy in it by which we mean if you literally converted all of that rest energy all of the energy like we did a nuclear explosion but better than nuclear explosion if you had half of a gram of matter and half of a gram of antimatter and combine them together and they completely annihilated the energy released would be ten to the 33 electron volts so that little tiny bit of mass by every day scales has an enormous amount of energy by particle physics scales that's the point I'm trying to get to and this number ten to the thirty three electron volts might actually secretly be familiar to you because I've been hiding it by not doing the precision but there is something called Avogadro's number Avogadro's number is useful in chemistry just by fixing a certain amount of stuff but it's also useful as a rough for when things become macroscopic and everyday in typical things that you know about one centimeter across you might get roughly speaking of a God rose number of particles okay or of atoms in them and Avogadro's number is let me get it to one significant figure six times 10 to the 23 six times ten to the 23 when you round up is 10 to the 24 that's what Avogadro's number is it's roughly speaking the number of hydrogen atoms in one gram of hydrogen or if you have something like carbon which is actually used as the as the mass standard here Avogadro's number is the number of carbon atoms in 12 grams of carbon because the atomic weight of carbon is 12 okay chemists bless their hearts care about the real world where there are isotopes and so they care about the fact that you know hydrogen is not all atoms that have one proton and one electron sometimes there's deuterium with a proton and a neutron or even tritium with a proton and two neutrons we physicists can ignore that and imagine the spherical cow where all hydrogen atoms are one proton and one electron but Avogadro's number is roughly speaking the number of atoms or the number of particles in a gram or a few grams worth of stuff okay so of a God Rose number is the connection between microscopic numbers protons and neutrons and their masses and macroscopic numbers grams and cubic centimeters and things like that now what we could do is redo this powers of 10 plot by the way I should just for sake mention that this is a logarithmic plot that is to say the distance between the different tick marks is not a fixed amount of energy it's a fixed scaling in energy right every tick mark the energy goes up by 10 to the 3 as a multiplicative power and we'll get back to that a little bit later maybe depending on how long this video goes so you can do this again and you can do this for the powers of 10 thing for sizes for you know distances or for x and cosmology and stuff like that I'm not gonna do that because you can get that on the internet you can go to Wikipedia and get that stuff but I should just give you some numbers for comparison so I wrote some numbers down how many and so from now on so since we know that approach is about a GeV 10 to the 9 AV let's once we get to the macroscopic scale use that as our unit of measure how big is something means how many proton or neutron masses in it are in it which is the number of protons and neutrons roughly speaking so a bacterium it turns out has about the same number of protons and neutrons as a single human DNA strand I think this is true I mean I don't know I looked it up on Wikipedia so a bacterium or a human DNA strand bacteria have their own DNA of course but it's shorter than human is about 10 to the 12 protons nowhere near Avogadro's number which is a reflection of the fact that a bacterium or a DNA is much smaller than a gram that's not surprising a person has about 10 to the 28 protons most of us are more than one gram sadly the earth not really sadly that was a joke the earth is about 10 to the 51 protons again this is not the number of protons in the earth this is the mass of the earth in units of one proton mass okay most of the earth is neutrons not protons the Sun 10 to the 57 so I can read my handwriting here the Milky Way galaxy about 10 to the 76 it looks like mass at 76 or 70s your homework how heavy is the Milky Way and the observable universe is about 10 to the 82 again don't trust me any of these numbers because you know why I don't care it doesn't really matter what these exact numbers are of course it matters a lot if you're a biologist or a chemist or an astrophysicist or whatever all I'm trying to do is emphasize fact that once again the scaling differences are enormous there's an enormous hierarchy between the size of the universe the size of a galaxy the size of the solar system the size of a person the size of a bacterium the size of a proton that's interesting that's interesting all by itself right why are these hierarchies there you know part of the whole philosophy of scaling is if you have a certain mass scale out of which everything else is created if you didn't know anything else if you said okay the world is made of protons and neutrons and they have about the same mass you might expect with good reason that one or two or ten proton masses would be a typical scale of stuff in the universe right because that's what everything else is built out of now it's not true you know that everything else is just is simply built out of protons and neutrons this is part of the explanation electrons are important gravity is important and so forth but the point is that therefore you demand an explanation you want to know why isn't everything roughly speaking one proton mass where do these enormous differences in sizes come from I'm not gonna get to explain that all but will give us some hints of where these numbers come from along the way so this is the appropriate time to get back into that question of what is the physical meaning of the conversion between energy and mass and length and time okay what does it mean when you say that a certain distance is a certain number of inverse electron volts and the answer comes in the form of this idea called the Compton wavelength of a particle the Compton wavelength came out of analysis of electromagnetic scattering by Arthur C Compton Arthur C Compton Compton anyway whatever his initials were but it actually can be explained best if we admit that individual particles are not individual particles but come out of quantum fields okay so the Compton wavelength has a very simple definition we're gonna use the letter lambda for it I try to minimize my use of Greek letters but lambda is just what we use for wavelength sorry about that it's the Greek version of L if you want that's why length is appropriate and the definition of the Compton wavelength of a particle if I put all the constants back in is h-bar over the mass of the particle times the speed of light we set H bar and C equal to 1 so it's just 1 over the mass of the particle ok so that's kind of not very helpful you just said okay I take the mass and I invert it and I get the wavelength so because distance or wavelength has units of 1 over energy and mass has units of energy this makes perfect sense units wise ok I mean basically the Compton wavelength is the mass except we're expressing it in different units now why why do we care about this particular number first let me tell you what the number is so the number for an electron the Compton wavelength is about I'm gonna give you one significant figure so thank me for this 2 times 10 to the minus 10 centimeters whereas for proton Compton wavelength is about 1 times ten to the minus thirteen centimeters now that makes sense remember I told you an electron and proton are different in mass by a factor of about 1800 so if I go to one significant bit you're a factor of two thousand and here you go there's the two thousand but what I want to emphasize is the Compton wavelength of a proton is shorter than the Compton wavelength of an electron that's obvious because it goes as 1 over the mass but in ordinary ways of thinking in ordinary world language you think of heavier things as being bigger ok you have more stuff in a heavy thing in order to put more mass in something given the same construction materials it will generally take more size so we have this intuitive feeling that heavier means not just more massive but larger in physical dimension but that's because we're imagining that something is made out of some pre-existing materials atoms arranged in certain configurations this is different the Compton wavelength I want to argue is what you should think of when you think of the size of an elementary particle and in quantum mechanics heavier particles are smaller in some very real sense so let me actually explain the sense in which the Compton wavelength should be thought of as a size so remember you have the uncertainty principle okay so you're relating the error were you to measure the location of a particle Delta x times Delta P the uncertainty were you to measure the position of the particle and we said that it's greater than h-bar over two but in my one order of magnitude precision one half equals one okay two equals one only a particle physicist could get away with that so you have to be careful when you set two equal to one you better be careful that it's really only two that is you're setting equal to 1 not 2 to the power 100 that would get you in trouble because the to the power hunter is very different than 1 to the power hundred but if there's just a single factor of 1/2 rolling around then we ignore it and of course h-bar equals 1 so Delta X Delta P is greater than 1 that's the Heisenberg uncertainty principle and we also have the relativistic energy formula relating energy to momentum we have the energy squared equals the momentum squared this is a vector of I'm gonna ignore the vector signs plus the mass squared so when the momentum is zero e squared equals M squared or equals M the C equals 1 and I've ignored all the other factors of C etc there's factors of C floating all around here if you try to convert from momentum to velocity for example okay so consider a particle with zero momentum let P equal 0 but imagine that you have Delta P the uncertainty in the momentum of order the mass okay so what I'm saying here is imagine you have a particle and it has a wave function so we're not just thinking classical particles within you a particle with a quantum wave function and the wave function might look like a wave packet so here is sine as a function of X and it's sort of flat and then it Wiggles and is flat again okay so you have a certain Delta X the size over which the wave packet Wiggles before it goes to 0 to the left or to the right and if you decomposed this wave packet two modes with different wavelengths you'll be able to figure out that there's an uncertainty momentum as well so I'm asking you to imagine that we have a particle sort of in a more or less confined region of space that we know it's momentum on average is zero but there's some uncertainty so there's some contributions from zero momentum some contributions from a small momentum some contributions from large momentum etc we're not exactly sure what answer we would get were we to measure the size of this particle the momentum of this particle and let's imagine that that momentum that uncertainty Delta P is of the same order as the mass of the particle okay we can do that one way of saying it is if you were to measure the energy right do you need to take into account the fact that the momentum is unknown so the energy that you might get were you to measure it is likely to be greater than two times M squared because you have M squared itself and then you have the contribution from the fact that you don't know what momentum the particle hands but what that means is once the change in energy the uncertainty in momentum becomes that large the uncertainty in energy becomes that large that you don't even know if you only have one particle there right once you get up to 4 M squared you have enough energy to make an extra particle if you know that what you're doing is really quantum field theory you're not quantum mechanics then you know that you your quantum field is a superposition of there are no particles there there's one particle there there are two particles there and so forth and here what you're saying is you're probing at the particle in such a way that you're not even sure that what you will see is a single particle anymore so the way to get around this is to say the one particle regime the part of the wave function of the quantum field that really describes really just one particle has the feature that Delta P is less than M yes that's correct and therefore you look over at the uncertainty principle so Delta x times Delta P needs to be greater than one but Delta P is less than M so therefore Delta my deltas are deteriorating sorry Delta X has to be greater than one over m otherwise you're not going to satisfy the uncertainty principle so in other words for the wave packet to have the shape and size that lets you have enough certainty about position about momentum that you know you're not making new particles the linear extent of the wave packet has to be large enough that it's larger than one over m which is of course the Compton wavelength that's what the Compton wavelength is I'm not really not sure that was the most elegant explanation of it ever given the Compton wavelength is not necessarily the wavelength of the actual wave function of a particle it's the smallest the wave function can be so that we can actually know with high confidence that there really is only just one particle there it's the smallest box we can squeeze a particle into and still stay within this one particle regime in quantum field theory if you try to squeeze the particle smaller than that or if you try to measure its position with a precision higher than that then you're not sure you're talking about just one particle that's what the Compton wavelength is there it is the Compton wavelength one over the mass is basically how small the particle this wave function can be so we're sure it really is just one particle so in other words if you try to make something out of particles you can't make things out of individual particles with sizes smaller than their Compton wavelength so particles individual particles that you would know and trust are really one particle at a time can't be smaller than the compliment wavelength I'll put a C under there just to make sure that you know is the Compton wavelength but if I'm talking about individual particles I will sometimes put the subscript for the name of the particle okay so that's what the Compton wavelength is it's sort of the minimum size that you can squeeze the wave function and you still call it a particle so the bad news is there for ant-man is not real you know in the movies not just ant-man but Fantastic Voyage many many movies that you've seen in different contexts there's the idea that you can take a macroscopic thing like a person and just shrink it right just make it smaller now usually those thought experiments do not tell you whether that happens because the thing that you're making out of has fewer and fewer atoms or because the individual atoms are smaller okay they're a little bit silent about that but ant-man goes down all the way to the size of an atom and therefore he's not just a single atom so that's not possible unless the individual atoms became smaller and if that were true they would have to become heavier if you're making the Compton wavelengths of these atoms tinier by the laws of physics they would have to get more and more massive so ant-man would actually be enormous ly massive to have that many atoms in his body squeezed down to that tiny little size but that's not what he is he's actually very light so that's not realistic according to laws of physics another way of spending the same conclusion is you might wonder you know there's human beings who live on planets that circle the solar system the Sun and then the Sun is part of that galaxy and maybe the universe is part of something bigger we seem to be in this nest of hierarchy of many different things and if you go small we're made of cells that are made of molecules that are made of atoms and you might casually ask yourself I wonder if it just keeps going my wonderful electrons are made of tinier things and quarks are made of tinier things I wonder if there could be like a whole civilization that rises and falls on the surface of an electron just like it does on the surface of the earth going around the Sun no it can't that's what this means that's what the Compton wavelength tells you if you want to stay in a regime where you're made of particles where you're not just high-energy wave functions then it's a minimum size that your particle could have given that it has a certain mass so this is why we know that electrons and other things are simple we have other ways of knowing also but there's no hidden amounts of variability in electrons or anything like that if there were little Wiggles and wavefunction electron or extra wave functions that also had really really tiny wavelengths they would need to have high energies and they would be more massive than the electron so you can't do it you got to keep your electrons simple so size for elementary particles is proportional to 1 over the mass that's a crucial lesson that we got here ok that's good for elementary particles crucially important here because we know this is not true for pickup trucks or anything like that right more massive things tend to be larger in the macroscopic world so how do we relate those two things you know where does that where does the big size of macroscopic things come from if the size of individual particles comes from the Compton wavelength remember again particles can't be smaller than the Compton wavelength but let's emphasize particles and by which we mean the wave function of a certain kind of particle can be larger than the Compton wavelength so the Compton wavelength is a minimum size for particle nasubi a sensible concept but they can be larger and in fact they are so let's see for example hydrogen you know you talked about hydrogen atoms either atoms just a single proton and electron going around it if someone told you that someone says okay here's physics here's a proton here's an electron they're bound together by electromagnetism how big should the hydrogen atom be well you have the Compton wavelength of the electron which we said 2 times 10 to minus 10 centimeters we have the Compton wavelength of the proton ten to the minus thirteen centimeters okay proton much smaller than the electrons you might say well the hydrogen atom should be roughly the size of the electron so it should be 2 times 10 to the minus 10 centimeters across now by dimensional analysis I should I should note that we can construct the number lambda P divided by lambda Compton wavelength the proton / the Compton wavelength electron this is about 10 to the minus 3 okay we're getting the factor of 2 right now so this is a dimensionless number and we could raise it to any power if we raise it to the power 10 then this is 10 to minus 30 so scaling arguments only take us so far when you have two masses in the game or two Compton wavelengths two lengths I'm gonna erase this because the silly thing but you can create whatever number you want in principle so you have to think a little bit in other words okay so let's do some thinking here if the Compton wavelength of the proton is tiny compared to the Compton wavelength the electron even if you don't say if you don't leap to the conclusion that the hydrogen atom should be the comfortably length of the electron there's no reason for the Compton wavelength of the proton to have anything to do with anything it's just a very tiny little number we would naturally expect if we even if we didn't work through and solve the Schrodinger equation that the proton was confined to a more or less tiny little spot and the electron was spread out over the atom which is exactly true so in some sense the thing that sets the scale for the size of the hydrogen atom is the Compton wavelength of the electron not the proton and that's true but the question is are there other things other considerations that come into the game well there is there is electromagnetism okay electromagnetism is what binds the electron to the proton an electromagnetism has a strength you can think of the potential V of R ok the electrostatic potential as a function of the distance between two objects one way of writing it and this is we're gonna gloss over some constants of nature and things like that is as minus alpha over R where alpha is the fine-structure constant 1 over 137 roughly okay so alpha is the number that tells you the strength of electromagnetism it tells you how important the electromagnetic force is when it comes to binding two things together and 137 roughly 100 okay so this is in an hour very sloppy way of thinking 10 to the minus 2 so there's a factor of 10 to the -2 that might come into the game when we figure out what the size of something is right that's a dimensionless number alpha doesn't have any units at all we could multiply it or we could divide it what are we supposed to do well so we can actually draw this potential very you know again we're being a little bit impressionistic here so here's our here's V of R and R is the distance from one object to another so by construction it is a positive number or zero it can't be a negative number I'm gonna draw it as if it's negatives that'll draw a three-dimensional thing and the potential looks like this 1 over R right again I'm a 1 over plus R down here so that the potential is always a negative number and the electron wants to be near zero but because of the uncertainty principle it will not be exactly at zero and what what can we say about the wave function of the electron in the hydrogen atom so it's gonna look something like this right it's gonna be some spread out thing if it were squeezed in more that would cost energy from the change over space for squeezed out less if we were broader then it would be further away from the proton where it doesn't want to be so there is some give-and-take and what we can say is we can just sort of figure out how things should scale as alpha goes to zero so alpha never doesn't go to zero what we mean is if we were to imagine a fake world in which the fine-structure constant were very very very very very tiny how would that be different from the real world well what it would mean is the electron will be bound less strongly there would be less of an attraction between the electron and the proton to make a hydrogen atom and therefore what we would guess is that how do I write this the size of the atom would go to infinity right the electron wants to be spread out that there's micron in empty space there's an electron if there's an electron that is not bound to a proton it's one function just spreads out as much as it can it's much as it can that is the prediction of the Schrodinger equation so as the binding to the proton is artificially turned off that electron would puff up and become big so that means that the size of the hydrogen atom is inversely dependent on the fine-structure constant so you might guess well let's not gas let's just tell you the answer there's something called the Bohr radius named after Niels Bohr who figured it out long before quantum field theory it was known or even quantum mechanics was known they were inventing it at the time the Bohr radius which is traditionally called a knot which is a terrible thing to call it but there's other things that are called R and B and so forth so a knot turns out to be the Compton wavelength of the electron divided by alpha because as alpha goes to 0 the radius should get bigger so divide it by alpha so that's 137 times the Compton wavelength which works out to be about 10 to the minus 8 centimeters so had you guessed that the size of a hydrogen atom and the Bohr radius is roughly speaking the size of a hydrogen atom more specifically there's no such thing as a hard fast wall at the end of a hydrogen atom right if you look at the yellow curve here it tails off it's supposed to tail off gently it's not supposed to cross the axis but the play there's always some chance some probability you'll find the electron if you look for it anywhere so the Bohr radius is the radius that says there's a 50-50 chance you'll find the electron within the radius or outside the radius so it is one way to characterize the rough size of a hydrogen atom you might have guessed naively that the hydrogen atom should be the Compton wavelength of the electron it's proportional to that but the constant of proportionality is a factor of a hundred so it is bigger and that's because electromagnetism is weak because the fine-structure constant is small so in fact this is sometimes called one angstrom unit 10 to the minus 8 centimeters an angstrom is just a very convenient unit to use in atomic or molecular physics is it's a size of a hydrogen atom roughly speaking 10 to the minus 8 centimeters and as it turns out that's basically it it's not just hydrogen that this is the size of this is the Bohr radius is roughly the size of a hydrogen atom it's also rough the size of every other atom the atoms do become bigger as you add more protons and neutrons that nucleus becomes heavier that's completely irrelevant to the size because the nucleus is very tiny the number of electrons though also goes up that is relevant we'll talk later about how electrons need to take up different shells and so forth but still you get factors of a few or factors of 10 times bigger than the hydrogen atom it's not linear with the number of extra electrons so this is the number time in my say 10 meters one angstrom that sets the scale for atomic physics for the length scale for atomic physics the next thing you want to do is put together more than one atom right let's make a molecule and that's gonna be you know life I'm not gonna do all of molecules and chemistry and stuff like that but we should be able to figure out or at least say something about the energies that are associated with molecules so to do that let's figure out something about the energy associated with individual atoms right so we drew this potential here for the electron this is the potential energy of an electron surrounded by the in the vicinity of a single proton and you know remember we did the simple harmonic oscillator remember oops yeah remember what I wanna do here come on open up remember simple harmonic oscillator was a thing where we had a potential that was a parabola x squared and we said that the energy levels were equally spaced there were energy levels en that were basically proportional to n in fact n plus 1/2 if you really want to be sticky about it but they were equally spaced so you could have an infinite number of energy levels in the harmonic oscillator I'm not gonna draw the wave functions again but do you member I've drawn them before and there's an infinite number of them they go infinitely high same thing is true for the Coulomb potential as we call it Coulomb potential is the potential one over arm minus alpha over R so the potential itself looks like this oops that wasn't very good as a function of R but the energy levels have a slightly different behavior the energy levels e n are proportional to minus 1 over N squared I'm sorry that alpha is used for proportionality and for the fine-structure constant once again but basically there is a lowest energy level e 1 and this is works out to be thirteen point six electron volts so the difference between the energy of an electron that is outside the hydrogen atom not in it at all and the energy of a electron that is in its lowest energy state inside the hydrogen atom is thirteen point six electron volts but then that's a 1 and then e2 is closer to outside and then a 345 and they pile up there are also an infinite number of energy levels that an electron can have in a hydrogen atom but as the number gets higher is the number that we use to characterize which energy level were in they get closer and closer to zero difference between being in the atom and being outside what that means physically is any little perturbation will just kick the electron right out so an electron is lowest energy level is stuck inside the hydrogen atom as much as it can be stuck it takes this much energy thirteen point six electron volts to kick it out and set it free whereas the if an electron is up here in a higher energy level it's easier to kick it out is bound more loosely it is larger you know its wavefunction is more spread out and it's less tightly bound to the proton what will happen if you just have an electron all by itself in a high energy level is generally will give off a photon and go down to the lower energy level so electrons tend to settle down to lower energy levels now this number thirteen point six evie where did that come from I mean remember the mass of the electron is something like 5 times 10 to the 5 electron volts so there's another big hierarchy here between the mass of the electron and the energy the binding energy is thirteen point six EB is the binding energy or the ionization energy of an electron in a hydrogen atom where did that come from well we know that the energy of an electron the just the energy classically speaking is the potential energy and we know what that is it's minus alpha over R right the energy V of our potential energy is minus alpha over R and we know that the typical are the average value R is the Bohr radius a naught and a naught where is it lambda over alpha so the energy is roughly speaking alpha over R R is lambda over alpha this is alpha squared over lambda or in other words alpha squared times the mass of the electron so there is this energy scale given to us by the mass of the electron but science inserts a factor of alpha squared which is 10 to the minus 4 in fact it's a little bit smaller than 10 to the minus 4 so that takes us down from 5 times 10 to the minus 5 times 10 to the 5 electron volts all the way down to just thirteen point six electron volts so it's two factors of the fine-structure constant that's where this particular hierarchy comes from and this is why again you know this is a pretty good energy this is this is as close as that one electron can get to the proton it has we will talk about later you know in chemistry in heavier elements there will be some electrons that are stuck there close to the nucleus and the nucleus is more charged in a heavier element like carbon or iron or whatever so the most bound electrons are more tightly bound in carbon or iron than they are in hydrogen but they are also irrelevant to chemistry okay because they're just staying there chemistry is all the excitement because of the electrons in the outer shells of the atom and they are less tightly bound then this one is so basically the energy scales of chemistry all of chemistry happens at energy scales of between 1 and 10 electron volts roughly speaking okay again very very roughly but and this is why because it's the mass of the electron times two factors of alpha times again some extra factors of N and N squared and things like that when you get to higher and higher energy levels so this is how hierarchies get built because there's a one number the fine-structure constant that is a little bit small ten to the minus two and it comes in multiple times that's what we're atoms get their energy from what about molecules hmm molecules it's a very similar story the only thing we have to play with are the masses of the nuclei the protons and the neutrons the masses the electrons and the fine-structure constant but they get hooked up together in more and more complicated ways happily the answer works out very easily I'm not gonna actually try to justify this answer both tell you what it is so you can see once again this hierarchy at work so you can have in molecules let's say that you have very simple molecule let's say you have one atom that is just bound to another atom well one thing is that these two atoms can vibrate right they can simply go back and forth you will not be surprised to learn that for small vibrations the energy of that kind of vibration looks like a simple harmonic oscillator because everything looks like a sub armonica oscillator when it is small vibrations small vibrations just look like harmonic oscillators so there's a lowest energy and next highest energy etc vibrational energies are typically something like 10 to the minus 2 electron volts okay things are a little bit farther apart larger distances means lower energies because distance goes inversely proportional to energy there are also rotations so let's imagine we have a more complicated molecule let's imagine a little molecule that has you know three different this is a heavy nucleus maybe this is methane or something like that right no ammonia sorry well this is a heavy nucleus and three hydrogen atoms and if we work in a frame of reference where we fix this one and don't move it the one at the bottom sorry now that the other two move over it's very much like when we said when you have a wave any arbitrary wave can be written as a superposition of plane waves with fixed wavelengths sine waves and cosines okay those that set of things you can add together to get any wave at all we called the modes okay here we have an interesting geometry but the same kind of analysis applies there's certain fundamental modes of oscillation from which you can build everything that actually happens so if you think about if you just fix your reference frame by fixing the bottom most atom and let the top most atoms move around what can they do they can oscillate in and out but they can also lastly in and out either in phase or in opposite phase to each other okay that's sort of elongating or shrinking I guess they can move apart from each other back and forth or they can sort of wiggle in concert and they can also have one go toward you and away and vice versa or they can go toward you in a way in concert also so there's a bunch of fundamental things that the molecule can do and they have a fundamental kind of energy so the rotation the whole thing can rotate around also or distortion of the molecules also have a typical energy scale maybe 10 to the minus 3 electron volts again like more moving parts means that it's slower means that it's larger so bigger spatial scales bigger temporal scales means lower energy scales so that's what you get I'm not deriving the number but I'm trying to justify that's basically where it comes from and from that of course the real world is built out of much much heavier complicated molecules than that so you get even lower and lower energies once you get the long hydrocarbon chains they can wiggle around they have all sorts of different modes so what I would like to do is talk about not just so these are individual molecules the next step would be to go from molecules to more complicated things alas I have a lot things to say about more complicated thing it's like living organisms and complicated things where you can see structure where you can see complexity your eyes and stuff like that I have enough things that I think that it's pretty clear that I should just do a separate video on them so I'm gonna stop this one here and this is the the discussion we've done on scaling I hope that some people aren't upset because there's a whole nother discussion to have on scaling that is just much more macroscopic from the start and you know how does the heartbeat of an animal scale with its body mass or things like that and I do want to talk about that because it is a big idea in the universe but for this one you know just a home you know there are two main points I wanted to get across one was this big set of hierarchies in fundamental particle physics which some of it we understand some of it we don't and all actually I didn't try very hard to explain this hierarchy you know I put an electron together with a proton to get a hydrogen atom but I didn't tell you why electrons are much lighter than the Higgs boson or why the proton is much is similar to the Higgs boson within a factor of 100 but we'll get there this is one thing I want to talk about and this is really the other thing I want to talk about the idea of the Compton wavelength and how it sets the scale for elementary particles and then by multiplying by effect factors of alpha in the correct way we go from that Compton wavelength to the size of atoms hydrogen but also other atoms it works for and then you just multiply by alpha again to get the energy of the different things that can happen inside an atom so all these different scales these hierarchies are built up by making more complicated things out of simple things and there are rules behind them sorry about ant-man not being real but there are other things equally cool in quantum mechanics in the universe so I hope that the reality is just as interesting as the Marvel movies

Channel: Sean Carroll

Views: 69,069

Rating: 4.947608 out of 5

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Id: cm22AogSULg

Length: 68min 19sec (4099 seconds)

Published: Tue Jun 09 2020

Reddit Comments

[u/QuakePhil]

I always found it interesting how scale seems to be a fundamental part of asymmetry, in our otherwise usually symmetric physics. Thank you for the post!

[u/[deleted]]

I do have to say this one stumbled a bit on units and prefixes. A quick overview of SI units and how logical they are, and then a similarly quick walk down the SI prefixes would show how logical and simple these are, would have got off on a better foot.