Heisenberg’s Uncertainty Principle

All physicsof the 19th century and earlier is called classical physics. Examples areNewtonian mechanics, which we dealtwith this whole term, and electricity and magnetism, which you will encounterthe next term.

In the early partof this century, when we learnedabout the composition of atoms, it became clear thatclassical physics did not work on the very small scaleof the atoms. The size of an atom is onlyten to the minus ten meters. If you take 250 million of themand you line them up, that’s only one inch.

In 1911, the English physicistRutherford demonstrated that almost all the massof an atom is concentrated in an extreme small volumeat the center of the atom.

We call that the nucleus,it’s positively charged. And there are electronswhich are negatively charged, which are in orbitsaround the nucleus, and the typical distances fromthe nucleus to the electrons is about 100,000 times larger than the sizeof the nucleus itself. As early as 1920,Rutherford named the proton, and Chadwick discoveredthe neutron in 1932, for which he receivedthe Nobel Prize.

Now, let us imagine thatthis lecture hall is an atom. And the size of an atomis defined by the orbits, the outer orbitsof the electrons. If I scale it properly, now,in this ratio 100,000 to 1, then the sizeof the nucleus would be even smallerthan a grain of sand. And it just so happens that yesterdayI went to Plum Island, I walked for three hourson the beach and I ended upwith some sand in my pockets. And so I will donateto you one proton; make sure you hold onto it… Ooh, this is two protons,that’s too generous. So keep it there–this is one proton. And there would bean electron, then, anywhere there, near the walls,going around like mad in orbit and that would then bea hydrogen atom. Just think aboutwhat an atom is. An atom is all vacuum. You and I are all vacuum. You think of yourself as beingsomething, but we are nothing. You can ask yourselfthe question, If you are all vacuum, why is it, then,that I can move my hand not through the other hand, likea ghost can walk through a wall?

That’s not so easy to answer, and in fact, you cannot answerit with classical physics and I will not returnto that today. But you are all vacuum. According to Maxwell’sequations, Maxwell’s law of electricityand magnetism, an electron, because of theattractive force of the proton, would spiral into the proton in a minute fractionof a second, and so atoms could not exist. Now, we know that’s not true. We know that atoms do exist. And so that createda problem for physics and it was the Danish physicistNiels Bohr who in 1913 postulated that electrons move around thenucleus in well-defined orbits which are distinctly separatedfrom each other, and that the spiraling-in ofthe electrons into the nucleus does not occur, for the reasonthat an electron cannot exist in between these allowed orbits. It can jumpfrom one orbit to another, but it cannot exist in between. Now, Bohr’s suggestionwas earth-shaking, because it would also imply that a planetthat goes around the sun cannot orbit the sunjust at any distance.

You couldn’t move itjust a trifle in or a trifle farther out. It would also requirediscrete orbits. It would also mean thatif you had a tennis ball and you would bouncethe tennis ball up and down, that the tennis ballcould not reach just any level above the ground, but it would only bediscrete levels, and that is very muchagainst our intuition. We’d like to think thatwhen you bounce a tennis ball, that it can reachany level that you want to. You give itjust a little bit more energy and it will go a little higher. That, accordingto quantum mechanics, would not be possible.

Now, all this seemsrather bizarre, as it goesagainst our daily experiences, but before we dismissthe idea of quantization– see, the quantization comes in when you talkabout discrete orbits– you have to realizethat the differences in the allowed heightsof the tennis ball and the differences between the allowed orbitsof the planets around the sun would be soinfinitesimally small that we may never be ableto measure it.

In other words, quantummechanics really plays no role in our macroscopic world. Now, atoms are very, very smallcompared to tennis balls, and the quantization effectsare much larger in the sub-microscopic worldof electrons and atoms than in our familiar world of baseballs,pots and pans, and planets. So before we continue,I would like to repeat to you one of the cornerstonesof quantum mechanics. And it says that the electronsin atoms can only exist at well-defined energy levels–think of them as being orbits– around the nucleus, andthey cannot exist in between. Now, when I heat a substance,the electrons in the atoms can jump from inner orbitsto allowed outer orbits, and when they do so, they can leave a hole,an opening, an empty spacein the inner orbits.

But later on, they can fall backto fill that opening. They can occupythat place again. And when I keep heatingthis substance, there is some kind ofa musical chair game going on. The electrons will goto outer orbits, they may spend there some time and then they may fall tolower orbits, to inner orbits. You see here a vase,a very precious vase, and when I pick up this vase,I have to do work. I bring it further awayfrom the center of the Earth. Now, is that energy lost? No. I could drop the vase, and itwould pick up kinetic energy. I will get that energy back.

Gravitationalpotential energy will be convertedto kinetic energy. It will crash to pieces,and it will generate some heat. In fact, the breaking itselfof this vase would take some energy. In a similar way, the energythat you put into electrons when you bring themto outer orbits is retrievedwhen the electrons fall back. So there is a parallel– dropping this vase and gettingyour work back that I put in. It wouldn’t be a nice thingto do to this 500-year-old vase, but as far as I’m concerned, perfectly reasonableto do it with Ohanian, so we can let that go, and the energy will come outin the form of heat and also in the formof, perhaps, some noise. When electrons fall from an outer orbitback to an inner orbit, it’s not kinetic energythat is released, but it comes out oftenin the form of light, electromagnetic radiation. Light has energy.

Einstein formulatedthat a light photon, the energy of a light photon,is h times the frequency, and h is Planck’s constant–named after Max Planck– and h is about 6.6 times 10to the minus 34 joule-seconds. Now we’ve also seen in 8.01 that lambda,the wavelength of light, equals the speed of lightdivided by the frequency. And so if I eliminate thefrequency, I also can write that the energyof a light photon equals hc divided by lambda. And so you see, the more energythere is available, the smaller the wavelength. And the less energythere is available, the longer the wavelength. And so if the jump from an outer orbitto an inner orbit is very high, then the wavelengthwill be shorter than when the jumpis relatively small.

I can make you some kind of anenergy diagram of these jumps. And these are energy levels, soenergy goes in this direction, but if you want to,you can think of these as the position of how far the electrons are awayfrom the nucleus, if you like that,if it helps you, so this will be the electron that will be the closestto the nucleus. So these would be allowedenergy levels, allowed orbits. And if this electron had jumpedall the way here, then it could fall backat a later moment in time and the energy could be so much that you couldn’t even seethe light.

It could be ultraviolet, and this jump maystill be ultraviolet, but now this jump, which isa little less energy, that may be in the blue partof our spectrum. So we may see thisas blue light. And this one, which isa little less than this, this energy may generate, this jump may generategreen light. And the jump from here to here,which is even less, may generate red light. And a jump from here to here,which iseven less, may again be invisible,so this may be infrared. And so as the electrons fall from outer orbitsto inner orbits, you expect very discreteenergies to come out, very discrete wavelengths, and these wavelengths that youwould see correspond, then, to these allowed transitionsbetween these energy levels.

So if we could look at thatlight and sort it out by color, we would, in a way, seethese energy levels. Now, you havein your little envelope a piece of plastic,which we call a grating, and the grating has the ability to decompose the lightin colors, which we call a spectrum, and we’re going to shortly usethat grating to look at light from heliumand light from neon. But before we do that,I’d like to hand out– as a souvenir to a few people,randomly picked– somethingthat they can also use.

It’s not as goodas your grating, though, but it’s also nice. You will see a more spectacularresult, but not as clean. It’s not as clean. All right, one for you,one for you, one for you and one for you. And you want one– I can tellthat– and you want one. And here, for you, for you. Oh, no, this side hasn’thad anything. I’ve got to walkall the way over now. So this is really for children’sparties, which I’m handing out. Oh, George Costa, you want one,of course. Professor Costa wants one–I couldn’t bypass him. And you want one, okay,and you want one.

So, by all means,use your grating, but then, at the very end, you can always usethese little spectacles, which don’t work nearly as well,but, uh… this kind of thing. I’m going to light here this bulb, this light,which has helium in it, and what you’re going to seewith your grating, if you holdyour grating properly– you may have to rotateit 90 degrees; you will see how that workswhen you try it– you’re going to seevery, very sharp, narrow lines at various colors.

I want you to realizethat the reason why you seevery sharp, narrow lines is only because my light sourceare very sharp, narrow line. If you use it on something that is nota very sharp, narrow line, then you’re not going to seethrough that grating very sharp, narrow lines. So don’t confuse the linesthat are on the grating with the line sourcethat I have here. Now, when you look throughyour grating very shortly, you will see, on both sides,the wonderful lines. It’s a mirror image, and we will discuss itin a little bit more detail, but before you lookthrough your grating, I first want you to simply lookat it without the grating, because then it iseven more spectacular when you use the grating. Because you have no clue,when you don’t use the grating, what kind of colorsare hidden there. And the colorsthat you are going to see are these electron levels.

So I am going to make it dark. And I will turn this on. And this one, I believe,is helium. I have a grating here. So we have to rotate itso that you see vertical lines on either side. You may have to rotate it90 degrees, no more. And if you look closely– for instance, look onthe right side of the light– you’ll see a distinct blue line,a few blue lines, green, very nice bright yellow one,and you see red. And if you go furtherto the right, you see a repeat. It’s a little fainter,but you see a repeat of that.

That’s not important right now; I just want you to see that thislight, which you have no idea that it comes outin very discrete wavelengths, very discrete frequencies, and they correspond to thesejumps from allowed energy levels to other allowed energy levels,but there is nothing in between. And when you look on the leftside, you’ll see a mirror image of what you seeon the right side. Now, neon… excuse me, heliumhas only two electrons. I’m now going to put in the neonbulb and that makes it richer, for reasons that neonhas ten electrons, so you have many more orbits,so many more ways that the electrons can playmusical chair.

A lot of lines in the red–I’m not blocking you, I hope– a lot of lines in the red, and some beautiful linesin the yellow. I see some in the green, I don’t see much in the blue…a little bit in the blue. But the key thing is,I want you to see that these lines are discrete. It is not just any wavelengththat can be generated; it’s only the allowed orbits,the musical game when the electrons jumpfrom one orbit to another, and that gives youthis unique discrete spectrum. Now, these light spectrawere known long before Bohr camewith his daring ideas, but before quantum mechanics, these lines werea great mystery, but they no longer are.

I suggest you use this grating and use itwhen you are outside at night; look at some streetlights, particularly sodium lampsand mercury lamps. And, of course, the neon lampsare quite spectacular, but keep in mind, you will notsee very nice straight lines unless your light source itself is a very nice straight,narrow light source. Now, quantum mechanics tooka big leap in the ’20s, and it would beimpossible for me in the available amount of time to do justiceto all the basic concepts.

However, I will discusssome consequences that are rather nonintuitive. Prior to quantum mechanics, there was a long-standing battlebetween physicists whether light consistsof particles or whether they are waves. Newton believed stronglythat they’re particles, and the Dutchman Huygensbelieved that they were waves. And it seemed like, in 1801, that a conclusive experimentwas done by Young, which demonstrated unambiguouslythat light was waves; Huygens was right. But as time went on,discomfort was growing, as there were also experiments that showed rather conclusivelythat light really was particles. And it was one of the greatvictories of quantum mechanics that it showedthat light is both.

At times it behaves like waves and at other times,it behaves like particles; it all depends onhow you do your experiment. In 1923, Louis de Broglie madethe daring suggestion that a particle can behavelike a wave, and he specified, he was veryspecific, that the wavelength– which nowadays is calledde Broglie wavelength– is h, Max Planck’s constant, divided by the momentumof that particle and the momentum is the mass ofthe particle times the velocity, as we have seen in 8.01. If the momentum is higher,then the wavelength is shorter. A baseball will havea very high momentum, with a ridiculously low…short wavelength. Now, one ofthe startling consequences is that protons and electrons, which everyone of that timeconsidered particles, can then also be consideredas being waves. And in 1926, the Austrianphysicist Schrodinger drove the nail in the coffinwith his famous equation– Schrodinger’s equation,it’s called now– which is the ground pillarof quantum mechanics and it unifies the wave and theparticle character of matter.

Returning to my baseball, take a mass of the baseballof, say, half a kilogram and give it a speedof 100 miles per hour. Calculate the wavelengththat you would find, according to quantum mechanics. That wavelengthis so absurdly small, it is 20 ordersof magnitude smaller than the radius of an electron,so it is completely meaningless. So quantum mechanics playsno role in our macroscopic world of pots and pans and baseballs. But now take an electron. You take the massof the electron, 10 to the minus 30 kilograms. And you give the electron a speed of, say,1,000 meters per second. Now you get a wavelength whichis comparable to the wavelength of visible light, red light.

And now it’s somethingthat becomes very meaningful, something that can be measured. Now, you may argue, “Gee,what difference does it make? “Who careswhether something is a wave or whether somethingis a particle?” Well, it makesa huge difference, because waves have crestsand they have valleys, and so if you take two sourcesof waves, either water waves– two sources, tapping up and downon the water– or you can taketwo sound sources, then there are certain locationson the surface of the water where the crest of one wave arrives at the same timeas the valley of the other, and so they canceleach other out.

There is nothing,there is no motion of the water. We call thatdestructive interference. Of course,there are other places where there isconstructive interference, where they support each other. Now, if particlescan do that, too… That is very hard to imagine– how can one particlewith another particle interfere and vanish, that thetwo particles no longer exist? So if, indeed,particles are waves, you should be ableto demonstrate that by having the interferencepattern of two particles, like the water waves, and make–at certain locations in space– those particles disappear,which turn out to be possible. But that’sa very nonintuitive idea.

So we think of ittoo classically when we say, “Well, two particlescannot disappear.” But in quantum mechanics, you can think in wavesif you want to, and then you have no problemswith the interference pattern and the destructive interferenceat certain locations. Now, there areother remarkable consequences of quantum mechanicsin classical mechanics. If you and I are clever enough, you think thatwe should be able to determine the position of an objectto any accuracy that we require, and at the same timedetermine also its momentum at any accuracy that we require.

It’s just a matterof how clever we are. Simultaneously,the object is right there and that is its massand that is its speed. However, the German physicistHeisenberg realized in 1927 that a consequenceof quantum mechanics is that this is not possible. Strange as it may sound to you,Heisenberg stated that the positionand the momentum of an object cannot be measured veryaccurately at the same time. And I will read to you Heisenberg’s uncertaintyprinciple, the way we know it. It says, “The very conceptof exact position of an object “and its exact momentum,together, have no meaning in nature.” It’s a profoundnonclassical idea, and it is hard for any oneof us– you and me included– to comprehend. But it is consistentwith all experiments that we can do to date.

I want to repeat it, because it’s going to beimportant of what follows. “The very concept of exactposition of an object “and its exact momentum,together, have no meaning in nature.” What does it mean? First, let me write down Heisenberg’suncertainty principle. Delta p, which isthe uncertainty in the momentum, multiplied by delta x, which is an uncertainty inthe position of that particle, is largeror approximately equal to Planck’s constantdivided by two pi– for which, in physics,we call that “h-bar”– and h-bar is approximately 10to the minus 34 joule-seconds. You see, h is 6.6 times 10to the minus 34.

If you divide that by two pi, you get about 10to the minus 34. What does this mean, now? What it means that if the position is knownto an accuracy delta x– we’ll give you some examples– that the momentum is ill-determined, is not determined, to the amount delta p, larger or equalthan h-bar divided by delta x. That’s what it means. And I’ll give you an example which I’ve chosenfrom a book of George Gamow. Gamow wrote a bookwhich he called Mr. Tompkins in Wonderland. It’s about dreams. Mr. Tompkins wantsto understand the quantum world, and there is a professor– you will see a pictureof the professor– who takes him, in his dreams, along the variousremarkable nonintuitive effects of quantum mechanics. And in one of these dreams, the professor suggeststhat we make h-bar one. And the professor takesa triangle in the pool table and he puts the triangleover one billiard ball, so the billiard ball isconstrained in its position and that delta x is roughly…say, 30 centimeters, 0.3 meters. That means that the momentumis not determined, not determined to an approximatevalue of one divided by 0.3, is about 3 kilogram-metersper second. Now, if we give the billiardball a mass of one kilogram, then delta p is m delta v,and so if m is one kilogram, then the speed of thatbilliard ball is undetermined, according to Heisenberg’suncertainty principle, by at least approximatelythree meters per second.

Three meters per second–that means seven miles per hour, and so that billiard ball will go around like crazyin that triangle, and that’s exactlywhat happens in the dream. And I will show you herea picture from that book. Mr. Tompkins isalways in pajamas, just to remind youthat it is a dream. And needless to say, the professor is a very old manand has a very nice beard; it adds to the prestige. And I will read youfrom this book. I will read you a very shortparagraph that deals with this.

“So the professor says, “‘Look, here, I’m goingto put definite limits “‘on the position of this ball by putting itinside a wooden triangle.'” “As soon as the ballwas placed in the enclosure, “of the whole insideof the triangle “became filled upwith glittering of ivory. “‘You see,’ said the professor, “‘I defined the positionof the ball “‘to the extent of thedimensions of the triangle. “‘This results in considerableuncertainty in the velocity “‘and the ball is moving rapidlyinside the boundary. “‘Can’t you stop it?’asked Mr. Tompkins.

“‘No, it isphysically impossible. “‘Anybody in an enclosed spacepossesses a certain motion. “‘We physicists call itzero point motion, “‘such as, for example, the motion of electronsin any atom.'” So here you seequantum mechanics at work when h-bar is one. This isa very nonclassical idea, because you and I would think– and we’ve always dealtwith that in 8.01– that you can take an objectand place it at location “a,” and we say at time t zeroit is at “a” and it has no speedand we know the mass, so we know both the momentumand the position to an infinite accuracy. But according to quantummechanics, that’s not possible. So let’s now return to the realworld, where h-bar is not one, but where h-bar is10 to the minus 34, and let’s now put a billiardball inside this triangle. Now, delta x is the same, but since h-bar is10 to the minus 34, delta p is, of course,10 to the 34 times smaller, and so the velocity is 10to the 34 times smaller.

This undeterminedness… degree to which the velocityis now undetermined, is so ridiculously small– it is 3 times 10 to the minus34 meters per second– that if you allowed that ballto move with that speed, in 100 billion years, it would move only 1/100of a diameter of an electron, so it’s meaningless again. And so again, you see thatquantum mechanics plays no role in our daily macroscopic world of baseballs and basketballsand billiards and pots and pans. And therefore, it iscompletely okay for us to say, “I have a billiard ballwhich is at point ‘a,’ and its mass is one kilogramand it has no speed.” That is completely kosher,completely acceptable, and quantum mechanicshas no problems with that. Let’s now turn to an atom. Take a hydrogen atom. The diameter of a hydrogen atom is about 10to the minus 10 meters. So the electron is confined to a delta x of about 10to the minus 10 meters. That means the momentum of thatelectron becomes undetermined– according to Heisenberg’suncertainty principle– to about 10 to the minus 34,divided by 10 to the minus 10, is about 10 to the minus 24kilogram-meters per second. What is the mass of an electron? That’s about 10to the minus 30 kilograms.

So this, delta p,is also m delta v. So it means that delta v– that means the velocityof the electron– is undetermined, accordingto Heisenberg’s principle, by an amount which is at least10 to the minus 24, which is this delta p dividedby the mass of the electron, which is 10 to the minus 30. And that is about 10to the six meters per second– that is one-third of a percentof the speed of light. So the electron is moving only because of the factthat it is confined. That’s what quantum mechanicsis all about. The electron’s motionis dictated exclusively by quantum mechanics. I’m going to show youan experiment in which I want to convey to you how nonintuitive Heisenberg’suncertainty principle is. I have here a laser beam, and this laser beam is going tobe aimed through a narrow slit– I’ll make a drawing,I’ll turn this light off– and that slit,which is a vertical slit, can be made narrowand can be made wider.

Here is this light beam and hereis this opening, this slit. It’s only going to be confinedin this direction, not in this direction. And sothe light will come out here, and then, on a screen, whichis going to be that screen, at large distance capital L, we’re going to seethat light spot, due to the light beamgoing through the slit and this separation, capital L. I start offwith the slit all the way open and so you’re going to seethis light spot like this. And then I’m going to makethe slit narrower and narrower, and as I’m going to cutinto the light beam, what you’re going to seeis exactly what you expect. You expectthat this light disappears, and when I cut in further,you see exactly what you expect, that this light disappears. And so the light spotthere on that screen will become narrowerand narrower and narrower.

But then there comes a pointthat Heisenberg says, “Uh-uh, careful now, becauseyour delta x, your knowledge, “the accuracy in this directionwhere the light goes through “is now so highthat now I’m going to introduce “an uncertaintyin the momentum of that light. “The momentum of that light is now no longer determinedto infinite accuracy.” And what that means,if you start fooling around with the momentum of that lightin the x direction, it no longergoes through straight but it goes off at an angle,and I will make you a more quantitative calculationfor that. So let’s look at this slitfrom above. Here’s the slit, and the slithas an opening, delta x. And this delta x we’re goingto make smaller and smaller, and let us start with a delta xof about 1/10 of a millimeter, which is 10to the minus 4 meters. I have light, I knowthe wavelength of the light, and I know that lambdaequals h divided by p, according to De Broglie. I know the wavelength, I know h, and so I can calculatethe momentum of that light.

I have done that,take my word for it. It is about 10 to the minus 27kilogram-meters per second. That’s the momentum ofthe individual light photons. Think of them as particles, which you can do,according to de Broglie. So now I have a delta p, the degree to which the momentumis undetermined, according to Heisenberg, is going to be 10 to the minus34 divided by delta x– which is 10 to the minus 4, so that is 10to the minus 30, very small. But the momentum itselfis 10 to the minus 27, so it’s only one partin a thousand. So what will happen? If the light comesthrough here… And I now makea classical argument. I say, “This isthe momentum of the light as it comes straight in.” When it has to be squeezedthrough this narrow opening, Heisenberg’s uncertaintyprinciple demands that it is goingto be undetermined, the momentum in this direction by roughly 10 to the minus 30,or more.

Remember, it is alwayslarger or equal. In other words, if I introduce,for instance, in this directionor in this direction, delta p, then I would expect that some of that lightgoes off in this direction. It is this change in momentum, this undeterminednessin momentum, that makes it go off at anangle, only in the x direction. If I have the slit like this, don’t expect this to happenin this direction, because the uncertaintyin the y direction, that’s not the problem. Delta y is not very small,it’s delta x that is very small, so it’s this direction that’sgoing to give you trouble. It’s only in this direction that you know preciselywhere that light goes through. This direction is not the issue. So this angle theta can nowbe calculated very roughly. Theta is obviously delta pdivided by p, so theta is very roughly10 to the minus 3 radians, which is a fifteenthof a degree, and if you haveat a distance L– if this distance here is L– if you have here a screen, then the spot on this screen…if I call that x at location L, then x at location Lis obviously theta times L. And if theta is10 to the minus 3– and let’s assume this isabout 10 meters away from us, so L is about 10 meters– then you get 10to the minus 2 meters. That is one centimeter.

One centimeter in this direction and one centimeter in thatdirection– two centimeters. But when I make the slit width10 times smaller, if I make the slit widthonly 1/100 of a millimeter, then this becomes10 centimeters, because now I know delta x10 times better, and so delta p is 10 timesmore uncertain. So now I expect to see here at least a smearof 20 centimeters and at least a smearof 20 centimeters there. So the absurdity is that ateeny-weeny little light source which in the beginning youwill see as a very small spot… When I make this slitnarrower and narrower, indeed, you will seethat you will lose photons, and you will see this gettingnarrower and narrower, and then all of a sudden,it begins to spread out, and it begins to spread out, and by the time I’m closeto a tenth of a millimeter, the light spot will be yay big. Very nonintuitive. You make the slit smaller,and the photons spread out. And I want to showthat to you now. I have to make it very dark. And I need my flashlight,turn on the laser beam. There you see it. The slit is nowall the way open.

Yeah, it’s all the way open, and I’m going to closethe slit now slowly. And if you look closely,you will see that the… Let me also get my red laser,then I can point something out. You will see that the light will get squeezedin the horizontal direction. You can see alreadyat the left side, has a very sharpvertical cut-off, and the right side also. It’s getting narrower,it’s getting narrower. Getting narrower, but I’m nowhere nearlya tenth of a millimeter yet. It’s getting clearly narrower. You see, it’s getting narrower,it’s getting narrower. If I look here… oh, I’m not yet at thetenth of a millimeter, but I’m getting there. I’m going slowly, squeezing it. I’m squeezing those photons.

Those photons now are forced to go throughan extremely narrow opening and Heisenberg isvery shortly going to jump in and says, “You are goingto pay a price for that. “You know too well where thosephotons are in the x direction. “The price you pay– “that nature will now makethe momentum undetermined in the x direction.” And you begin to…you see it now. You really begin to see that the center portionis widening. Even photons appear. Here, you see some dark lines, which I will notfurther discuss today, but notice that the lightis spreading. Of course, when I squeeze thisslit, when I make it narrower, it’s obvious that I lose light, because the lightthat hits the side of the slit is not going through, so the light intensitywill go down. That’s just inevitable. I used fewer photons. But look at this. There are photons here,there are photons there. It’s at least 10 centimeters,this portion. From here to here isat least one foot. I squeeze more– this is morethan half a meter now. I squeeze more– this isabout one meter already.

I squeeze even more. I close the slit now,and I will open it slowly. I’m opening it very slowly, andat the moment that it opens… Look at this! You see this? You see this wonderful streak? It looks more like a comet. From here to here isat least a meter. That’s that center portionof the light. It has spread out,since the poor light was forced to go throughthis very narrow opening. Now I’m opening itmore and more.

I’m opening it more, and now, of course,the reverse is happening. Extremely nonintuitive. Now, not only have you seenquantum mechanics at work, in terms of electrons jumpingbetween orbits, but you now have also seen one othervery interesting consequence of quantum mechanics, which is Heisenberg’suncertainty principle. Now, the spreading of this lightcan very easily be explained without Heisenberg’suncertainty principle. In fact, it was known,even in the previous century, to a high degree of accuracy,why this happens, and the dark lines werevery accurately explained. All I wanted to show is that the spreading of thelight is entirely consistent with Heisenberg’s uncertaintyprinciple, and it better be, because it wouldnot be possible, it would be inconceivablethat you could do any experiment that would violate Heisenberg’suncertainty principle.

And if this light that you wouldsee on the screen there, if that light spot would getnarrower and narrower and narrower and narrowerall the time, as we would think classically,that would have been a violation of Heisenberg’suncertainty principle, and that is not possible. Now, there is no way in advance to predictwhich photons end up where. All you can dowith quantum mechanics is to do the experimentwith lots of photons and then you will geta certain distribution and the distribution will beexactly as you saw there. Quantum mechanicscan never predict, on an individual photon,where it will end up. We saw that bright spotin the center.

So if you did this experimentwith one photon per day– one photon per daygoing through this slit– and you hada photographic plate there, and you would keep it therefor months, and you would develop it, you would see the same patternthat you see there. This photon arrives today. Here arrives one tomorrow. Here arrives one the dayafter tomorrow. Here one the day after that,the day after that, the day after that,the day after that, the day after that, the dayafter that, and slowly are youbeginning to see that pattern that you saw. So don’t think that this interference patternthat you saw is the result of two photons going through the slitsimultaneously– not at all. You can do itwith one photon at a time and you would seeexactly the same thing. Now, this idea– that youcannot in advance predict what a particular photonwill do– is a very nonclassic idea, andit rubs us all the wrong way because our classical wayof thinking is– and you are no different frommy own feeling in this respect– that if you do an experiment a hundred timesin a controlled way, you should get a hundred timesexactly the same result. Not so, says quantum mechanics. All that quantum mechanicswill tell you is what the probability isthat something will happen. No guarantees, but it is very goodin predicting probabilities. Now, Einstein had great problems with this idea of not knowingprecisely what would happen, and he had endless discussionswith Bohr and others in whichhe tried to convince them that because you couldn’tpredict what happened, that something had to be wrongwith quantum mechanics, and Einstein’s famous wordswere, “God does not throw dice.”

This was the way,was his way of saying, “It is ridiculousthat the outcome of a well-controlledexperiment is uncertain.” Now, almost nine decadeshave gone by since the beginningof quantum mechanics, and we now know that God– ifthere is one– does throw dice. However, God is bound tothe rules of quantum mechanics and cannot violate Heisenberg’suncertainty principle. The light could not go straightthrough without spreading when I made the slit as narrowas I did.

So quantum mechanics isa bizarre world that we rarely experiencein our daily lives, because we are used to basketballs,baseballs, tennis balls. But yet it is the waythe world ticks, and atoms and moleculescan only exist because of quantum mechanics. That means you and Ican only exist because of quantum mechanics.

I hope that this will give yousomething to think about, but I warn you in advance, because if youstart thinking about this, it will give you headaches and it will give yousleepless nights.

And it has given mecountlesssleepless nights in the past, and even today, when I thinkabout the consequences– the bizarre consequencesof quantum mechanics– I still cannot comprehend it,I still cannot digest it and I still have headachesand sleepless nights. But it may be necessary to gothrough these sleepless nights if you want to eventually evolve as an independentthinking scientist, and I hopethat someday all of you will. Thank you. (class applauds )

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