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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).

• 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