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# PHYS 201: Fundamentals of Physics II

## Lecture 20

## - Quantum Mechanics II

### Overview

Lecture begins with a detailed review of the double slit experiment with electrons. The fate of an electron traversing the double slit is determined by a wave putting an end to Newtonian mechanics. The momentum and position of an electron cannot both be totally known simultaneously. The wave function is used to describe a probability density function for an electron. Heuristic arguments are given for the wave function describing a particle of definite momentum.

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html## Fundamentals of Physics II## PHYS 201 - Lecture 20 - Quantum Mechanics II## Chapter 1: Review of Double Slit Experiment Using Electrons [00:00:00]
But I know that it makes sense to me, because I’ve seen it, and I don’t know how it sounds to you. I have no clue. You know that and so you have to speak up. You can ask any question you want, and I will try to answer you, if it’s within the realm of possibility. Okay, so what have I said so far? So let me summarize. Even if you never came to last lecture, here is what you should know about the last lecture, okay? Here’s what I said. First thing I said is, everything is really particles, all things, electrons, photons, protons, neutrons. They are all particles, so let there be no doubt about that. By that, I mean if one of them hits your face, like an electron, you will feel it in only one tiny region, one spot. Electron dumps all its charge, all its momentum, all its energy to one little part of your face. So there’s nothing wavelike about that. It’s not like getting hit by a boxing glove, which can hit your whole face. An electron hits one dot, or if it’s an electron-detecting screen, only 1 pixel is hit by the electron. And into that pixel is given all the charge, all the momentum, all the energy of that electron. That’s exactly what particles do. So when you encounter an electron, it is simply a particle. So where does the problem come in? Where does the quantum mechanics come in? It comes in when you do the famous double slit experiment. That’s the key. The entire quantum mystery is in the double slit. Part of the resolution is in the double slit, but the rest are a little more difficult, and I’ll try to tell you. First I want to tell you what goes wrong with Newtonian mechanics. After all, if everything is a particle, what’s the big deal, what’s the problem? The double slit experiment is a problem. That’s what puts the nail on the coffin for Newtonian physics, and here it is in the basic version. You’ve got two slits. By the way, I’m going to call the particle the electron. They’re all doing the same thing, so what applies to one, applies to all of them. There is a source, like an electron gun, that emits electrons. In the old days, televisions had the electron gun. And the gun emits the electrons, they go and hit the screen, they make a little dot, and then the dot moves around, and you see your favorite show. Okay, this is the electron gun, and the electron gun has been engineered to send electrons off a definite momentum. That you can get by accelerating the electrons over a definite potential, and the gain of so many electron volts will turn into kinetic energy. As for direction, if this gun is really far away to the left, in principle 1 mile, then the only way electrons are going to go 1 mile and hit the screen is they’re all basically moving in the horizontal direction. Then you put a row of detectors in the back, which will detect electrons. Then this is slit 1 and this is slit 2. You block slit 1. In fact, let me say the following thing: what do we really know when we do the experiment? Once in a while this gun will emit an electron, and we know it’s emitted the electron, because it will recoil one way, just like a gun, rifle. It will recoil. That’s when we know the electron left. Then we don’t know anything, and suddenly, one of these guys says click. That means electron’s arrived here. This is what we really know. Everything else you say about the electron is conjecture at this point. You know it was here, you know it was there. The question is, what was it doing in between? Now if you say, “Look, things cannot go from here to there, except by following some path, I don’t know what path it is.” Maybe if it’s an ordinary particle, like a Newtonian particle, it will take some straight line, hit that slit, or go through that slit and arrive here. So you might say, “I don’t know the trajectory, but it’s got to be some trajectory, maybe like that, or maybe like that.” So the electron takes some path and you can label the path as either through slit 1 or through slit 2. Okay, now here is the problem. Suppose I do the experiment with slit 2 blocked, so you cannot even get through this one, and I sit at a certain location for a certain amount of time, maybe 1 hour and I see how many electrons come, and I get 5 electrons, with only 1 slit open. And if I move that observation point, I get some pattern, pretty dull, looking like that, and I’m going to call it Now I want to open both and ask, what will I get? In Newtonian mechanics, there’s only one possible answer to that question, and that is 10, because we’ve got 5 this way and you’ve got 5 that way. And you open both, whoever is going this way will keep going that way; whoever is going this way will keep going this way. They will add up to give you 10. Now I told you, some people may say, “Well, maybe it’s not 10, because with both slits open, maybe someone from here can collide with someone from there. How do you know that will not happen?” So I’m saying, do the experiment with such a feeble beam of electrons, there’s only one electron at a time in the whole lab. It’s not going to collide with itself. Then you wait long enough, and you have to get 5 + 5 = 10. And what I’m telling you is that if you go to the location marked That is the end of Newtonian physics. And I told you that something like that never happens in your daily life. I gave an example with machine guns. This is a machine gun. This is a concrete wall with 2 holes in it, and there’s some target here, you. And then you see how many are coming through this and how many are coming through that. Then you go there and you wait. And both are open. Somehow, nothing comes. With the second hole in the wall, you are safe. With one hole in the wall, you’re not safe. That can never happen with bullets. So these electrons are not following any path, because the minute you commit yourself to saying it follows one path, either through this one or that, you cannot avoid the fact that with both of them open, the intensity with 1 + 2 has to = Therefore you abandon the notion that electrons have any trajectories. You don’t want to abandon it, but you have to, because that assumption, which is very reasonable, just doesn’t agree with experiment. Then you say, “Okay, what should I do? Newtonian mechanics is wrong. What’s going to take its place?” To find that, you have to move away from this But this is such a familiar pattern. If you’re a trained physicist, which you guys are, you will say, “Hey, this reminds me of this wave interference, with water waves or sound waves or any waves. ” Obviously there’s some wavelength. The minute you give me wavelength and a slit separation, I can calculate this pattern. So you can successfully reproduce this pattern, but what does it tell you about what’s going on? What good is that pattern? The pattern tells you that if you repeat the experiment with this electron gun a million times or a billion times and you plotted the histogram patiently, the histogram will eventually fill out and take this shape. So this wave is not the wave associated with a huge stream of electrons. A single electron in the lab is controlled by this wave. You need a whole wave for 1 electron, so it’s obviously not a wave of electrons. It’s not a wave of charge, like the wave of water. It’s a mathematical function and you are drawn to it, because the only way you know how to get this wiggly graph is to take something with definite wavelength and let it interfere. So you’re forced to think about this wave. And the intensity of the wave, the brightness if you like, the square of its height, gives you what? Gives you the graph you will get if you repeat the experiment many times. And what does it mean for the individual trial? What does it mean for the millionth + one electron? For the millionth + one electron, it gives you the odds of where it will land on that screen, okay? You can never tell exactly where it will be. You can tell what the odds are, and the only way to test the statistical theory is to do the experiment many times. And if you do it, it works, and it seems to work for everything, for electrons, for protons, for photons, whatever it is, the wavelength and momentum are connected in this fashion. So this wave is forced upon us, and it gives you the odds of finding the electron somewhere. And we say that the probability — I’ll be a little more precise in a minute on what I mean by the probability to find it at a location Once you tell me that the fate of a particle is controlled by a wave, you’re immediately led to some other conclusions, so I’m going to tell you what they are. First conclusion is this: if I make a single slit, let’s call this the x direction, no momentum in the y direction, therefore py was strictly 0, no uncertainty. Dpy is 0 and the fellow I catch here is moving horizontally with that momentum p whose _{0}y position is within d, and I can make the d as small as I like. This is Newtonian physics. But we have now learned that the fate of the particle is not in its own hands. It’s contained in this wave. So what should I do in this context to find out what it will do? Any idea what I should do to find out what will happen in this experiment, given what we learned? Yes?
2pℏ/p. But if you want to know what will happen on the other side of this slit, I have to find the fate of that wave. Yes, you can put a screen, but what will I see on a screen? Will the wave just hit this region? You know it will spread out from diffraction. I’ve told you, the light will spread out. There are tiny wiggles we ignore, and this point, where you get most of the action, that angle, _{0}θ, satisfies dsinθ= l. This is just wave theory. That’s when you can pair up the points in the slit, in the hole, to things shifting, differing by half a wavelength, so for every one I can find a partner that cancels it, you will get 0 here. Beyond that, you may get a few more rises, but it’s pretty much dead outside this cone. That means you will observe this particle anywhere in this angular width.Now a particle cannot go from this slit to there unless it had a momentum, which had a component in the θ or by an angle θ, it is just p. Or, if you like, precisely, _{0}xθp times sin_{0}θ. But psin_{0}θ, sinθ is controlled by dsinθ = l, so this is l/d. But l is 2pℏ/p x d. You cancel that, you find D _{0}py X d = 2pℏ. That means D py D y is roughly — forget the 2p’s — of our ℏ. So you should understand this much completely without any doubt: if the future of the particle, the fate of the particle, is controlled by the wave, you try to narrow the location of the particle by making the hole smaller and smaller, the wave fans out more and more.## Chapter 2: Heisenberg’s Uncertainty Principle [00:20:28]That’s just wave theory. People knew this about waves hundreds of years back. What is novel is that this wave is going to tell you where the particle will end up. This wave is going to control the odds of where the particle will end up, and the odds are pretty much concentrated in this cone, not of 0 opening angle, but an angle
Okay, so what we learn is it’s when you combine waves and particles and go back and forth that you run into the situation. So you cannot make a state of perfect momentum. By the way, I said one thing, I thought about it, which is incorrect, which is, in the microscope, I said if you want to locate the position of an electron in a microscope, take a microscope with an opening, and electron is somewhere on this line. I said you’re shining light down here. It hits the electron, but it goes in through the slit by spreading out. So the photon that came in goes into the eyepiece with a certain uncertainty in its final angle. That means we know the incoming momentum, but we don’t know the outgoing momentum of the photon. The lens picks up everything inside that cone. That means we don’t know how much momentum it gave to the electron. It gives an indefinite amount of momentum to the electron. Therefore the Now what I don’t like about my experimental setup is that I had the incoming light also coming from inside the microscope, but that means incoming light, when it comes through this hole here, will itself spread. Then it will hit this guy, and that will go to the aperture. That will spread some more. This uncertainty in incoming momentum is unnecessary. We can do better than that, because in other words, when the light is picked up, it is picked up by this tiny hole. There’s no reason it should also come from the tiny hole. It can come from a source far away, say on the other side, so that it is a well defined direction, it’s not diffracting at all. So I want it to come in through a very broad hole, so it’s got well defined direction, so the light here has known momentum. It hits the electron and goes into the microscope. It is the final momentum of the photon I don’t know. And I cannot make it better. If I make it better, I’ve got to open this eyepiece a lot. If I open the eyepiece a lot, I don’t know where I caught this guy. So again the problem between taking a very tiny eyepiece, so that if I see a flicker, I know the electron was in front of it, but the light coming from the reflected electron fans out more and more. Okay, so anyway, this is the uncertainty principle and the uncertainty principle told us something very interesting. I asked you, what can be the function here that produces this interference pattern in the double slit? We know the wavelength. Wavelength was Do you understand that the experiment only showed you there’s a wavelength. It did not tell you what the actual function is? That’s very, very important. When Young did the experiment with the double slit, he found the oscillations and he could read off the wavelength. It’s just geometry. But he didn’t know what was oscillating. He didn’t know there’s an electric and magnetic field underneath all of that. But you can always read out the wavelength without knowing what’s going on. Likewise, we have the wavelength. We know it comes from a function with a well defined wavelength, so I make my first guess to be this function. But I told you what was wrong with this choice. You guys remember that? I said this function violates the uncertainty principle. The uncertainty principle says if you know the position to an accuracy D Y, due to cosine, of course does this. It prefers some locations to another, but you’re not supposed to have any preference for any ^{2}x, so we have a problem. How do I put in a wavelength into a function whose square is flat? That’s the problem we have. When you think about it, you realize your trigonometric functions will not do the trick. If they have a wavelength, their square is not flat. The square is also oscillating.But then what came to the rescue is the following function, not a cosine but Y times Y*, which is A — I’m taking A to be real here, so A* is this, times e^{ipx/ℏ}, times e^{ −ipx/ℏ}.That cancels out, you just get So we are driven now to the very interesting result that the wave function for a particle of definite momentum So that’s roughly where I left you, and I want to remind you of a few other things, this further discussion of the result we have, okay? The discussion is, if the world is really this messed up at the microscopic level, why do I think it’s the world I see in the macroscopic level? Where are all these oscillations? Why is it that when there’s a concrete wall, making another hole is bad for me and not good? Why do all these things happen? Why do I think particles have definite momentum and position? Why do I think that if I make a hole in the wall and I send a beam, the beam will go on the other side of the wall with a shape precisely like the shape of the hole, no spreading out? It all has to do with the size of the object. The laws of physics are always quantum mechanical laws, but when you apply it to an elephant, you get one kind of answer; when you apply it to an electron, you get another kind of answer. You don’t have separate laws for big and small things. The real question is, how do these very same laws, when applied to big things — by big things, I mean things you see in daily life — give the impression that the world is Newtonian? So let’s look at the double slit experiment. Here’s a double slit and we are told, “Send something. See what happens on the other side.” And the prediction is that you get these oscillations, with the peculiar property that with two holes open, you don’t get anything somewhere. We don’t seem to see that in daily life, and you can ask, “Why is that so?” Well, you remember that the condition for the next minimum is like You find this number is 10 That’s how many oscillations you have, okay? You’ve got enough now? 3, 6, 9, 12, well, I don’t have enough time. That’s a lot of oscillations. You should check the numbers though, okay? I’m saying they typical angle will be 10 Another reason you won’t see it is that the particle should have a definite momentum. It’s got an indefinite momentum, it’s coming in with different momenta, then each will have its own interference pattern and they’ll get washed out. Finally, I told you, if you ever try to see which slit the particle took by putting a light beam here, the minute you catch the electron going through one slit or the other, this pattern is gone. It will do this “I’m not here and I’m not there” routine only if you never catch it being anywhere. That’s very interesting. The electron behaves like it does not go through any one particular slit, as long as you don’t catch it going through one slit. You put enough light to catch every electron, then you can add the numbers and you must get the sum of the two numbers. Now for the atomic world, it’s possible for the electron to go for a long time without encountering anything, and the interference effects come into play. In a macroscopic world, there is no way a macroscopic object can travel for any length of time without running into something. It will run into other air molecules. It will run into cosmic ray radiation. It can collide with black body radiation from the big bang, anything. The minute you have any contact with it, this funny thing will disappear. So that’s one reason you don’t see it. Now we can go on and on and give other numbers. I’ve given examples in my notes, which I will post later on. One of them is the uncertainty principle. Why does it look like the uncertainty principle is not important? Take again, this is 10 That’s all you don’t know about its location. That’s your So you see, these uncertainties are not important in real life. So everything that you think has a definite position and momentum actually has a slight uncertainty, but the uncertainties don’t lead to any measurable consequences over any distances that you can actually have. So what I’m trying to tell you is, there are these waves. They do all kinds of things, they do interference and everything, but the condition for them is really the microscopic world. The minute the masses become comparable to gram or kilogram and distances and slits and so on, or like a meter or a centimeter, these effects get washed out. But in the atomic scale, they are seen. ## Chapter 3: The Probability Density Function of an Electron [00:42:34]Now the final thing I want to mention before moving on to a completely new topic is the role of probability in quantum mechanics. We have seen that quantum mechanics makes probabilistic predictions. It says if you do the double slit experiment, I don’t know where this guy will land, but I’ll give you the odds. Okay, now that looks like something we have done many times in classical mechanics. For example, if you have a coin and you throw the coin, you flip it and you say, “Which way will it land?” well, it’s a very difficult calculation to do, but it can be done in principle, because a coin, once released from your hand, can only land in one way. That’s the determinism of Newtonian mechanics. If you knew exactly how you released it with what angle or momentum, what’s the viscosity of air, whatever you want, if you give me all the numbers, I’ll tell you it’s head or tails. There’s no need to guess. In practice, no one can do the calculation. What you do in practice is, you throw the same coin 5,000 times and you find out the odds for head or tails and you say, “I predict that when you throw it next time, it will be .51 chance that it will be heads.” That’s how you give statistical predictions. Now you did not have to use statistics. You use it, because you cannot really in practice do the hard calculation. In principle, you can. Secondly, if you toss a coin and I hide it in my hand, I don’t show it to you, it’s either head or tails, and I say, “What do you think it is?” you’ll say, “It’s .1 chance that it’s heads.” And I look at it, it may be head or it may be tail. Suppose I got head. It means that it was head even before I opened my hand, right? The correct answer’s already inside my hand. I just didn’t know it. I’m using odds, but when I look at it, I get an answer. That’s the answer it had even before I looked. So I’ll give another analogy here. So this is a probability of locating me somewhere. This is my home town, Cheshire, this is Yale, and this is the infamous Route 10. So somebody has studied me for a long time and said, “If you look for this guy, here are the odds.” Either he’s working at home or he’s working at Yale, and sometimes he’s driving, okay? This is the probability. First thing to understand is the spread out probability does not mean I am myself spread out, okay? Unless I got into a terrible accident on Route 10, I’m in only one place. So probability’s being extended doesn’t mean the thing you’re looking at is extended. I am in some sense a particle which can be somewhere. These are the odds. Well, suppose you catch me here on one of your many trials. If you catch me only once, you don’t know if the prediction’s even good, so you repeat it. You study me over many times and you agree the person had got the right picture, because after observing me many, many days you in fact get the histogram that looks like this. The important thing is, every time you catch me somewhere, I was already there; you just happened to catch me there. I had a definite location. It was not known to you, but I had it. I had a definite location because in the macroscopic world I’m moving in, my location is being constantly measured. You didn’t ask or you didn’t find out, but I’m running through air molecules. I’ve slammed into them. They know that. I ran over this ant. That was the last thing the ant knew, okay? So I’m leaving behind a trail of destruction and they all keep track of where I am. My location is well known. You just happened to find out. But now let’s change this picture and say this is not me. This is an electron and it’s got two nuclei. This is nucleus 1 and this is nucleus 2. It can be either near this nucleus or that nucleus, and this is the We think of measurement as revealing a pre-existing property of the object. But in quantum theory, it’s not that you don’t know where the electron is. It does not know. It is not anywhere. It’s the act of measurement that confers a location or position on the electron. That state of being, where you can be either here or there, or simultaneously here or there, has no analog in the classical world. If anybody tries to give you an example, don’t believe it, because there are no examples in the macroscopic world that look like this. No analogies should satisfy you, because this has no analog in the real world, okay? So this is the interesting thing in quantum mechanics. If this is a possible wave function It describes an electron which upon measurement could be found here and could be found there. It’s not like finding me in Cheshire or finding me in New Haven, because in those cases, on a given day on a given measurement, you can only get one answer, depending on where I am. Right now if they look for me, they can only find me here. They cannot find me anywhere else. But in the case of the electron, the one and the same electron, on a given trial, at a given instant, is fully capable of being here or there. It’s like tossing a coin and it’s in my hand. We all know that when I reveal it to you, you can only get one answer, now that the toss has been done, it’s got one answer. If it’s a quantum mechanical coin, you don’t know, and it doesn’t have a value till you look. When you look, it has a definite value. Before you look, it doesn’t have a definite value. That’s exactly like saying, when you looked, it goes through a definite slit. When you don’t look, it’s wrong to assume it went through a definite slit. Yes?
Now till you find it, it’s not anywhere. It can be anywhere on this line at that instant. It’s only the act of measurement, or hitting a detector that tells you that’s my location. So you will have to get used to that. You’ll have to get used to the fact that things don’t have position, momentum, angular momentum, energy or anything, until you measure it. Okay, so I’ve got to tell you a little more now about just position. So let’s take — by the way, any questions so far? Yes?
That’s the question? Can a little macroscopic object simultaneously be in two places? Most of them seem to have a well defined location. Can you create a situation when it’s capable of being found here and found there? It’s very hard, because you have to isolate the particle from the outside world. That’s the first condition. That’s what ruins everything. A quantum computer, you must know, has got these bits called qbits and unlike the bits in your laptop, which are either 0 or 1, a qbit can also give you 0 or 1, but it can also be in a state where it can give either 0 or 1 on a given trial. The bits in your computer, the particular bit right now is either a 0 or a 1. Maybe you don’t know it, but it can only give you that answer because that’s what it is. Because that bit is in contact with the world and the world is constantly measuring its value. A qbit is a quantum system which can do one of two things, but it’s isolated and it’s neither this nor that. It’s like the electron going through both slits. So a quantum bit can explore many possibilities. If you build a computer with 10 qbits it can be doing 2 That’s why, as you know, one of the ways to securely send your credit card information is to use very large numbers, on the assumption that no one can factorize them. You can always multiply a 100 digit number by a 100 digit number on your computer instantaneously. But if I gave you the 200 digit number and told you to find the two factors you won’t find it. You won’t find it in the age of the universe. It’s amazing, but that simple problem of factorization cannot be done if the numbers are 100 digits long, and that’s the reason why people openly broadcast the product, they may broadcast one of the numbers, and only the other person knows the second number. Now there is something somebody called Shore, Peter Shore, showed that if you have a quantum computer, made up of these qbits, it can actually factor the number exponentially faster, namely, instead of taking 10 So why is it so hard to build a quantum computer? There are many, many quantum systems which can do one or two things, and can be the state, but they are both this and that. The problem is, they cannot be in contact with the outside world, because single contact with them is like a dream. Think about it, it’s gone. Same thing. Any measurement destroys it. Any unintended measurement also destroys it, so you’ve got to keep your system fully isolated. A system that is not talking to the outside world, unfortunately, is also not talking to you. So you cannot ask it any questions, and if it knows the answer, it cannot tell you. So sometimes you want it to talk. It’s like relationships. Sometimes you don’t want it to talk. So what do you do? You’ve got to build a system where sometimes, in a controlled way, you can make contact with your system, namely give it the problem. Then it does its quantum thing, then you’ve got to make a measurement to find out what the answer is. Then you want to be able to get into it again. So the challenge for quantum computers is how to keep them isolated long enough to do the calculation. That’s a challenge, how to keep it from — how to keep it in what’s called a quantum coherent state. A coherent state is really when it’s doing many things at the same time. All right, so I want to tell you now more formally how to do more quantum mechanics. So let’s take a simple example, a particle living on a line. That’s the function In quantum theory, you don’t even tell me where it is. For every possible The answer is, whatever you like. Anything I can draw, with no special effort, is a possible function for an electron. There are no restrictions. It’s like saying, what should the position of the particle be? ^{2}, is connected to the probability of finding it at x. That has to be in fact to be refined. That’s not precisely the story. I’ll tell you why. If a statistical event has got 6 possible answers, like you throw the die, you want to get any number from 1 to 6, you can give the probability for 1, probability for 2, 3, 4, 5 and 6. These are all the odds for getting any number from 1 to 6. Since there are only a finite number of things that can happen, I call them I = 1 to 6, there’s a probability for each I, and if you add all the probabilities, you should get 1. But if the set of things that can happen is a continuous variable like x, in other words, the location of the electron is not a discrete set of numbers. It’s any real number is a possible location. Then you cannot give a finite probability for any one x. If that was finite, since the number of points is infinite, you cannot make the total probability 1.So what we really mean by Y, let this be the height of ^{2}Y. Take a little sliver of width ^{2}dx. The area under the graph, P(x) dx, that is the probability of finding the electron, or whatever particle, between x and x + dx. You understand? It’s called a density. So you don’t give a finite probability to each point. For an infinitesimal region, you give it infinitesimal probability, which is the function P(x) dx. And the statement that the particle has to be somewhere, namely, all the probabilities add up to 0, is the statement that when you integrate this probability density from - to + infinity, namely, Ydx, from - to + infinity, that should be 1. This is called normalization. It’s a mathematical term. Norm is connected to length in some way, and these can be viewed as length squared. Anyway, this is called the normalization. Now if somebody gave you a Y, which did not obey this condition, here is a Y. This already tells you a nice story, right? It tells you the odds are pretty big here, pretty small there, 0 here. Now take a function that’s twice as tall. That gives you the same relative odds, you understand.^{2}So when you multiply Y by any number, you don’t change the basic predictive power of the theory. It is just that if your original Y had a square integral = 1, the new one may not have, but the information is the same. It’s really the relative height of the function. That’s another shocking thing in quantum mechanics. If Y stood for a string vibrating, 2 × Y (this is Y, this is 2 × Y) is a totally different configuration of the string. But in quantum mechanics, Y and any multiple of Y are physically equivalent, because what we extract from Therefore it’s like saying in 2 dimensions, that’s a vector, that’s a different vector. But suppose you only care about the direction of the vector. For some reason, you don’t care how long it is, you just want to know which way it points, then of course all of these are considered equal. And that’s really how it is for quantum mechanics. Every So let me do a concrete example, so you know what I’m talking about. So let’s take a function that looks like this. It is 0 everywhere, and it has a height A between +a and -a. That’s my
Y^{2}dx from - to + infinity, I want it to be 1. And I’ll pick A so that that is true. Well, we can do this integral in our head. What is this integral? This is just the square of A times the width of this region. That’s got to be 1.That tells me that Let me give you another example. There’s a very famous function, called the Gaussian function. It looks like this. The function p/a. That is just one of those tabulated integrals. So here’s a bell shaped function with this property. Now I want to make a quantum mechanical wave function that looks like the Y(x) = A e^{- x} squared over 2 D squared. That’s a possible wave function, right? Nothing funny about it, but what do you know about the wave function? It’s biggest at x = 0. It’s symmetric between + and -x. And it dies off very quickly, but how far should you go? You can easily guess that when x is much bigger than D, this function is gone, because x/D is going on the exponent. So if that number’s big, it’s e to the - big, which is very small. So roughly speaking the width of this graph is of order ^{2}D or 2D. I’m just going to call it D, just to give you an order of magnitude. So that’s an electron whose location is roughly known to an amount D. But this is not normalized, because if I take the square of this, I won’t get 1.So I will choose D^{2}dx. The 2 went away, because I squared the function Y, so don’t forget that. Now I look at the table of integrals and what is a? When I compare these 2, a is just 1/D. So it’s square root of p^{2}D^{2}. This is an easy thing, because I’m already giving you the integral you need to do, but I want you to get used to it. So this whole thing should be 1. That means A is 1/pD^{2} to the fourth root, the power ¼. Therefore the normalized wave function, Y normalized, looks like 1/pD^{2} to the ¼, e^{ −x(squared)}/2D^{2}.Normalization is just a discipline. You discipline yourself to take all functions and normalize them, because why do you normalize them? If you normalize them to 1, then Y is the relative probability density. It will still tell you the relative odds of this and that, but you cannot say this interval from here to here, the chances are 30 percent for catching it. You must take the region that you’re integrating, divide by the whole thing. But you don’t have to divide by the whole thing if you’ve normalized it to 1.^{2}Okay, this is just practice in normalization. So I’m going to give you a little hint on what is going to happen next, but I won’t do it now, so you guys don’t have to take down anything. Just ask the following question and we’ll come back to it on Wednesday. I’ve told you that in Newtonian mechanics, every particle has an P(x) dx is the probability of finding it between x and x + dx.Now we can say, “Okay, that’s enough about position. What about momentum?” I can measure the momentum of a particle. You talked about momentum on and off in the lecture. If I measure momentum, what answer will I get?” What are the odds for getting this or that answer? So given
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