let's think about thewaves for the electron. so here i've written outa wave, which is just the same form as the wavesi had for the photon. now let's think a little bitabout the negative energy solutions. if i let the energygo to minus energy, so i turn a positive solutioninto a negative solution. then you see that if ialso take the time to minus
the time, then, thisway it doesn't change. this changes sign, this changessign, nothing changes all together. so what this tellsus, straight away, is that these negative energysolutions are the same thing as if you have an electron. normally, electronstravel forward in time, for the positiveenergy solutions. but here, for thenegative energy solution,
t goes to minus t. so a negative energy solutionis an electron travelling backwards in time. that sounds a little bitlike science fiction, but i'll hope to convinceyou by the end of the lecture that actually, it's not. this is actuallywhat really happens. this is real fact. let's put anexclamation mark there,
because it really is kind of asurprising statement to make. now let's try to write down awave equation for the electron, like we did for the photon. so an equation thatwill give those waves, and give the correctrelation between the energy and the momentum. now, for the electron,we want to be clear about which is thepositive energy piece, and which is the negativeenergy piece, because they're
separate. and so to do that,our wave equation can only have singlederivatives in it. because as soon as wehave two derivatives, then we square everythingand we lose the sign. so if we write down a waveequation for the electron, you might think that you candifferentiate with respect to x. you can differentiatewith respect to time.
if we're only going tohave single derivatives, then it should looksomething like that. then, that will then act onthe electron wave function, since this is the electron,this is presumably the positive energy solution,so i'll put a little plus on it. and let's see what happensif that is equal to zero. well, you remember a d by dxgives you a factor of momentum. d by dt gives youa factor of energy. so this equation tells youthat p is equal to e over c.
so this is indeed thepositive energy solution. and we can write down a negativeenergy one, just the same way, by changing thesign in the middle. that changes thesign of the energy. so this is now thenegative energy solution, and this corresponds top equals minus e over c, because i changed the sign here. now that's still not goodenough for our electron, because the electron has a mass.
and so the tricky thing,is how you get the mass into this equation, sothat everything works out. and the answer, is that -which you find by staring at it for a while - is that youhave to couple these two equations togetherusing the mass. so the equation forthe positive solution, when coupled to thenegative solution, is a factor of the mass (timesa factor of c over hbar, as it turns out) - sothat multiplies psi minus.
and then you do the same thing,couple the negative energy solution to the positiveenergy solution. now, why does this work? well, we have to do alittle bit of algebra. so, to solve this, let'stake this derivative that's acting on the negativesolution and act it on the equation withthe positive solution. so to write that out,i'm going to have d by dx plus 1 over c, d bydt, acting on the whole
of the left-hand side. now, if i do that, if thisacts on the left-hand side, then to keep theequation true, i have to also have this acton the right-hand side. but if this acts onthe right-hand side, this, then that's justwhat we have here, in the second equation. so this, acting onthis, is the same thing (and we have this extrafactor of m c over hbar)
- so if i write thism c over hbar here, then this is actuallyjust equal to this. and since i put an extram c over hbar here, i've got to put one here,to keep everything true. now you see, if wecompare this with this, i have a equationjust for psi plus. now if we look atwhat that equation is, then let's just multiply outthese derivatives on the top. you get a d2 by dx squaredhere, a d2 by dt squared here,
and because thesign is different, the cross terms will cancel. so we wind up with d2by dx squared, minus 1 over c squared,d2 by dt squared, acting on psi plusis equal to the thing that we have here, whichis m c over h-bar squared, acting on psi plus. and actually, if you didthis the other way around, so you started withthe minus equation,
and worked throughthe plus equation, then you would get exactly thesame equation for the minus 1. so this actually workswhether you have psi plus, or whether you havepsi minus there. now, this equation we can solve. the solutions arejust waves, just like they were for the photons. and you rememberfor the photons, when you differentiatewith respect to x,
then you get a factor p squared. when you differentiate thewaves with respect to t, you get a factor of e, sotwice you get e squared. so we get p squared minusone on c squared e squared. and then on the right-hand side,we have m squared, c squared. and that's preciselythe relationship between energy andmomentum that we want for a massiverelativistic particle, so that we wantfor our electrons.
so you can see thatthe whole thing works. the equations that istarted from, here, these linearequations, those then, there are equations for ourelectron wave functions, the positive energy one, andthe negative energy ones. and these equations werefirst discovered by dirac, in 1928 in a very famous paper. so these things are calledthe dirac equations. so, what does all thisactually mean, physically?
what it means, isthat if you have massive relativisticparticles, like electrons - so, if the particles have mass,then besides the positive energy solutions, you must also havethe negative energy solutions. you can't get away fromthem, because they're coupled together like this. so, massive particles,e.g., electrons, imply, immediately,that there must be another sort of particlewhich has the same mass,
but is different. it's the negativeenergy particle. so if we haveelectrons, then we have to have what wecall anti-electrons. this is the negativeenergy solution. so what dirac actuallypredicted here is anti-matter. just by looking atthese wave equations, and looking at making themconsistent with relativity, they have to be wavesbecause of quantum mechanics,
then the only way you canfit those two things together if the particles have mass,is if there's antimatter. so the astonishing thing is thatdirac predicted anti-matter. now, i'd like to emphasise thatthis was a genuine prediction. sometimes when people saythat things are predicted, they really mean explained. so they're predicting somethingthat they already know about. but at the time when dirac didthis calculation, essentially the calculation thatwe've just done,
nobody had a clue that therewas anything like anti-matter. it seemed a fantastic thing,science fiction if you like - well it wasn'teven science fiction, no one had even thought of it. and so this was areal prediction. it was a prediction thatsomething strange should happen, that nobody hadever guessed would happen, and of course, at the time,nobody really believed it. now so far, we've onlydiscussed the dirac equation
in one spatial dimension. and that again, wasto make it easier. of course, thereal dirac equation is in three dimensions,spatial dimensions. and then those extrafeatures come in, just like with the photon. so if we do it inthree dimensions, then what we find is that,just like with the photon, there were two polarizations,a left-handed one,
and a right-handedone, and they carry angular momentumplus and minus hbar, you find a similarthing for the electron. so if you do all thisin three dimensions, then you find - and i'm afraidi'm not going to show you in any detail - that there aretwo different polarizations, we call them twospins, because they're intrinsic to the electron. and for the electron,there's something
rather curious, inthat the spins are not plus and minus hbar, likethey are for the photon, but they're plus orminus a half hbar. this is something that comesout of the dirac equation, when you do it in three dimensions. so for an electron, thereare actually four states. there's the positiveenergy state, left-handed. there's positive energystate, right-handed. there's the negative energystate left-handed, and also
the negative energystate right-handed. now let's think abouthow the field looks - or the waves if you like -look for the electron now, because of what theimplications are of the spin. so if you consider electronfields that go around in a circle, like we didfor the photon field, so we look at theangular momentum, then the fieldlooks like, again, a sine, the angletheta times the angular
momentum minus the energy timesthe time, divided by hbar. but now, the angular momentumis in units of h-bar over 2, not in units of hbar. now, you rememberfor the photon, it had to be inunits of h-bar, so that when we go round the circleand get back to the beginning, we get back to where we started. now, electrons dosomething really weird. for electrons, you have togo around the circle twice
to get back towhere you started. so for electrons,if you do theta, i goes to thetaplus 4 pi, then you can see this expressionwill stay unchanged - the half here cancels outthe - takes the 4 down to 2, and then you get backto where you started. if you only do thetagoes to theta plus 2 pi, so you only go aroundthe circle once, then the wave functiondoesn't go back to itself,
it goes to minus itself. so you get an extra minus sign. so if you think it's a littlebit strange that sometimes you have to go around twice in orderto get back where you started, then i have a little partytrick for you, which shows you that sometimes thisis actually necessary. so here we have a perfectlyordinary wine glass, i'll hold on the flatof my hand like that. now i'm going to turnthe wine glass around,
so you should watch carefullywhat happens to the wine glass. so first, i turn itaround once, like this. and you can see i'mnot not back to where i started, because itfeels very awkward. and then i carry on up,and turn it round again, and now i get backto where i started. now, this extra minus signthat we get for electrons is actually very important. you remember inthe last lecture,
when we were talking aboutthe double slit experiment, and we had two differentpossible things that could happen, andin order to work out the probabilityof that happening, we had to add up theamplitudes for those two different processes,and square it, and that gave usthe probability. so for photons,what we found then, was the probability - in thatcase of going through the two
slits, although it couldapply to just about any process where there aretwo ways of doing things - has to be the sum of thetwo amplitudes squared. now for electrons,i told you last time that it worked inexactly the same way. we take the two amplitudes,we put them together and square them. but, that wasn't quite true. you see, the two processeshere, because they're the same,
you get by swappingthe two photons around. now, when we turn - whenwe swap two photons around, it's a bit like takingthem round in the circle. so for the photons,we get a plus sign. but if you take twoelectrons round in a circle to swap them round, thenyou get this minus sign. you have to take themaround twice in order so for the electrons, there'sactually a minus sign here. so to sum up, we have - thisis another difference now,
between photons and electrons. for photons, we havea plus sign here. we have two differentpolarizations, which have angular momentum,plus or minus h-bar. and this is actually a genericsort of class of the way that things can happen. so objects that look likethis, you can call bosons. whereas for the electrons,we have a minus sign here. the minus sign is because thepolarisation states aren't
in units of h-bar, butin units of a half hbar. this puts them in a differentkind of class of object. these objects we call fermions. bosons are named after anindian physicist called bose, and fermions are namedafter an italian phycicist called fermi - that's wheretheir names come from. now, for fermions,like electrons, this has a veryimportant consequence. if you think about - ifyou have two electrons
in the same state, then, ifthey're in the same state, then these two amplitudesare equal to each other. and if the two amplitudes areequal, then they cancel out, so the probability is zero. so what this meansis that for fermions, then they can't bein the same state. there's no probabilityat all for them being in the same state. so they're alwaysin different states.
and this is something calledthe pauli exclusion principle. and it's because of thepauli exclusion principle that you can build up allsorts of different atoms. so that's why all thedifferent atoms exist, and why the worldaround us is so varied. because, if we have a proton,and an electron going around it, that would be hydrogen, say. if we want to addanother electron, it can't just siton top of this one,
it has to be somewhere else. so the atom looks different. because it can't sitin the same state. actually, it's notquite like that, because there's two spins. so actually, we can put twoelectrons in this orbit, one with spin up, onewith the spin down. that would give us helium. and then the nextone, lithium, has
to come in a different orbit. so that makes all the differentatoms look very different. and is one of themain things that explains the reason theworld's the way it is.