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  • 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