Subtitles section Play video Print subtitles Shall we begin now? So as usual, and as promised, I will tell you what we have done so far, and that's all you really need to know, to follow what's going to happen next. The first thing we learned is if you're studying a particle living in one dimension, that's all I'm going to do the whole time because it's mathematically easier and there is not many new things you gain by going to higher dimensions. There's a particle somewhere in this one-dimensional universe. Everything you need to know about that particle is contained in a function called the wave function, and is denoted by the symbol ψ. By the way, everything I'm doing now is called kinematics. In other words, kinematics is a study of how to describe a system completely at a given time. For example, in classical mechanics for a single particle the complete description of that particle is given by two things, where is it, and what its momentum is. Dynamics is the question of how this changes with time. If you knew all you could know about the particle now can you predict the future? By predict the future we mean can you tell me what x will be at a later time and p will be at a later time. That's Newtonian mechanics. So the kinematics is just how much do you need to know at a given time, just x and p. Once you've got x and p everything follows. As I mentioned to you, the kinetic energy, for example, which you write as ½mv^(2), is given just in terms of p, or in higher dimensions the angular momentum is some cross product of position and momentum, so you can get everything of interest just by giving the position and momentum. I claim now the equivalent of this pair of numbers in this quantum world is one function, ψ(x). So it's a lot more information than you had in classical physics. In classical physics two numbers tell you the whole story. Quantum theory says "give me a whole function", and we all know a function is really infinite amount of information because at every point x the function has a height and you've got to give me all that. Only then you have told me everything. That's the definition. And the point is ψ can be real, ψ can be complex, and sometimes ψ is complex. So we can ask, you've got this function, you say it tells me everything I can know, well, what can I find out from this function? The first thing is that if you took the absolute square of this function, that is the probability density to find it at the point x. By that I mean if you multiply both by some infinitesimal Δx that is the probability that the particle will be between x and x dx. That means that you take this ψ and you square it. So you will get something that will go to 0 here, go up, go to zero, do something like that. This is ψ^(2), and that's your probability density and what we mean by density is, if the function p(x) has an area with the x-axis like p(x)dx that's the actual probability that if you look for this guy you'll find him or her or it in this interval, okay? So we will make the requirement that the total probability to find it anywhere add up to 1. That is a convention because--well, in some sense. It's up to you how you want to define probability. You say, "What are the odds I will get through this course, 50/50?" That doesn't add up to 1 that adds up to 100, but it gives you the impression the relative odds are equal. So you can always give odds. Jimmy the Greek may tell you something, 7 is to 4 something's going to happen. They don't add up to 1 either. I mean, 7 divided by 11 is one thing and 4 divided by 11 is the absolute probability. So in quantum theory the wave function you're given need not necessarily have the property that its square integral is 1, but you can rescale it by a suitable number, I mean, if it's not 1, but if it's 100 then you divide it by 10 and that function will have a square integral of 1. That's the convention and it's a convenience, and I will generally assume that we have done that. And I also pointed out to you that the function ψ and the function 3 times ψ stand for the same situation in quantum mechanics. So this ψ is not like any other ψ. And if ψ is a water wave 3 inches versus 30 inches are not the same situation, they describe completely different things. But in quantum mechanics ψ and a multiple of ψ have the same physics because the same relative odds are contained in them. So given one ψ you're free to multiply it by any number, in fact, real or complex and that doesn't change any prediction, so normally you multiply it by that number which makes the square integral in all of space equal to 1. Such a function is said to be normalized. If it's normalized the advantage is the square directly gives you the absolute probability density and integral of that will give you 1. That's one thing we learned. You understand now? What are the possible functions I can ascribe with the particle? Whatever you like within reason; it's got to be a single value and it cannot have discontinuous jumps. Beyond that anything you write down is fine. That's like saying what are the allowed positions, or allowed momentum for a particle in classical mechanics? Anything, there are no restrictions except x should be real and p should be real. You can do what you want. Similarly all possible functions describe possible quantum states. It's called a quantum state. It's this crazy situation where you don't know where it is and you give the odds by squaring ψ. That's called a quantum state and it's given by a function ψ. All right, now I also said there is one case where I know what's going on. So let me give you one other case. Maybe I will ask you to give me one case. The particle is known to be very close to x = 5 because I just saw it there and ε later I know it's still got to be there because I just saw it. Now what function will describe that situation? You guys know this. Want to guess? What will ψ look like so that the particle is almost certainly near x = 5? Student: > Prof: Yeah? Centered where? Student: Centered at 5. Prof: At 5, everybody agree with that? I mean the exact shape we don't know. Maybe that's why you're hesitating, but here is the possible function that describes a particle that's location isn't known to within some accuracy. So one look at it, it tells you, "Hey, this guy's close to 5." I agree that you can put a few wiggles on it, or you can make it taller or shorter if you change the shape a bit, but roughly speaking here is what functions, describing particles of reasonably well know location, look like. They're centered at the point which is the well-known location. On the other hand, I'm going to call is ψ x = 5. That is a function, and the subscript you put on the function is a name you give the function. We don't go to a party and say, "Hi, I am human." You say, "I'm so and so," because that tells you a little more than whatever species you belong to. Similarly these are all normalizable wave functions, but x = 5 is one member of the family, which means I'm peaked at x = 5. Another function I mentioned is the function ψ_p(x). That's the function that describes a particle of momentum p. We sort of inferred that by doing the double slit experiment. That function looks like this. Some number A times e^(ipx/ℏ). Now you can no longer tell me you have no feeling for these exponentials because it's going to be all about the exponential. I've been warning you the whole term. Get used to those complex exponentials. It's got a real part. It's got an imaginary part, but more natural to think of a complex number as having a modulus and a phase, and I'm telling you it's a constant modulus. I don't know what it is. But the phase factor should look like ipx/ℏ. So if I wrote a function e to the i times 96x/ℏ, and I said, "What's going on?" Well, that's a particle whose momentum is 96. So the momentum is hidden in the function right in the exponential. It's everything x of the i, the x and the ℏ. Whatever is sitting there that's the momentum. I am going to study such states pretty much all of today. So let's say someone says, "Look, I produced a particle in a state of momentum p. Here it is. Let's normalize this guy." To normalize the guy you've got to take ψ^(2) and you've got to take the dx and you've got to get it equal to 1. If you take the absolute square of this, A absolute square is some fixed number. I hope you all know the absolute value of that is 1 because that times its conjugate, which is e^(−ipx/â„ ) will just give you e^(0) which is 1. I want 1 times dx over all of space to be equal to 1. That's a hopeless task because you cannot pick an A to make that happen, because all of space, the integral of dx over all of space is the length of the universe you're living in, and if that's infinite no finite A will do it. So that poses a mathematical challenge, and people circumvent it in many ways. One is to say, "Let's pretend our universe is large and finite." It may even be the case because we don't know. And I'm doing quantum mechanics in which I'm fooling around in a tiny region like atoms and molecules, and it really doesn't matter if the universe even goes beyond this room. It goes beyond this room. It goes beyond the planet. It goes beyond the solar system. I grant you all that, but I say allow me to believe that if it goes sufficiently far enough it's a closed universe. So a closed universe is like this. A closed one-dimensional universe is a circle. In that universe if you throw a rock it'll come back and hit you from behind. In fact you can see the back of your head in this universe because everything goes around in a circle. All right, so that's the world you take. Now that looks kind of artificial for the real world, which we all agree seems to be miles and miles long, but I don't care for this purpose what L is as long as it's finite. If L is finite, any number you like, then the ψ of p of x will look like 1/√L e^(ipx/â„ ). Do you agree? If you take the absolute square of this you'll get a 1/L. This'll become 1. The integral of 1 of L over the length of space is just