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PROFESSOR: Today we're
going to just continue
what Allan Adams was doing.
He's away on a trip
in Europe, so he asked
me to give their lecture today.
And we'll just follow
what he told me to do.
He was sort of sad to give me
this lecture, because it's one
of the most interesting ones.
This is the one where you get
to see the Schrodinger equation.
But anyway, had to be a
way, so we'll do it here.
He also told me to take off
my shoes, but I won't do that.
So let's go ahead.
So what do we have so far.
It's going to be a list
of items that we have.
And what have we learned?
We know that
particles, or systems,
are governed by wave functions,
described by wave functions.
Wave functions.
And those are psi
of x at this moment.
And these are complex numbers,
belong to the complex numbers.
And they're continuous
and normalizable.
Their derivatives need
not be continuous,
but the wave function
has to be continuous.
It should be also
normalizable, and those
are two important
properties of it,
continuous and normalizable.
Second there's a probability
associated with this thing.
And this probability
is described
by this special p of x.
And given that x is a continuous
variable, you can say well,
what is the probability that the
particle is just at this point
would be zero in general.
You have to ask, typically,
what's the probability that I
find it in a range.
It's any continuous
variable that
postulated to be given
by the square of the wave
function and the x.
Third there's superposition
of allowed states.
So particles can be
in superpositions.
So a wave function
that depends of x
may quickly or generally be
given as a superposition of two
wave functions.
And this is seen in many ways.
You have these boxes.
A particle was a superposition.
A top, and side,
and hard, and soft.
And photon superposition
of linearly
polarized here or that way.
That's always
possible to explain.
Now in addition to this,
to motivate the next one
we talk about relations
between operators.
There was an abstraction
going on in this course
in the previous
lectures in which
the idea of the
momentum of a particle
became that of an operator
on the wave function.
So as an aside,
operators momentum,
we have the momentum
of a particle
has been associated
with an operator h
bar over i [INAUDIBLE] x.
Now two things here.
My taste, I like to
put h bar over i.
Allan likes to put I h bar.
That's exactly the same
thing, but no minus sign
is a good thing in my opinion.
So I avoid them when possible.
Now there's is d dx here
is a partial derivative,
and there seems to be no need
for partial derivatives here.
Why partial derivatives?
I only see functions of x.
Anybody?
Why?
Yes.
AUDIENCE: The complete
wave function also
depends on time, doesn't it?
PROFESSOR: Complete
wave function
depends on time as well.
Yes, exactly.
That's where we're
going to get to today.
This is the description of
this particle at some instant.
So within [INAUDIBLE] time,
the time here is implicit.
It could be at some time,
now, later, some time,
but that's all we know.
So in physics you're allowed
to ask the question, well,
if I know the wave
function of this time
and that seems to be what I
need to describe the physics,
what will it be later?
And later time will come
in, so therefore we'll
stick to this
partial d dx in here.
All right, so how
do we use that?
We think of this operator as
acting on the wave functions
to give you roughly the
momentum of the particle.
And we've made it in
such a way that when
we talk about the expectation
value of the momentum,
the expected value of the
momentum of the particle,
we compute the
following quantity.
We compute the integral from
minus infinity to infinity dx.
We put the conjugate of
the wave function here.
And we put the operator,
h bar over i d dx acting
on the wave function here.
And that's supposed
to be sort of
like saying that this evaluates
the momentum of the wave
function.
Why is that so?
It is because if you're trying
to say, oh any wave function,
a general wave
function need not be
state in which the particle
has definite momentum.
So I kind of just say the
momentum of a particle
is the value of this
operator on the function.
Because if I act with this
operator on the function,
on the wave function,
it might not return me,
the wave function.
In fact in general, we've
seen that, for special wave
functions, wave functions
of the form psi, a number,
e to the ikx.
Then p, let's think p hat
as the operator on psi.
Would be h bar
over i d dx on psi.
Gives you what?
Well, this h over I,
the h remains there.
When you differentiate with
respect to x, the ik goes down,
and the i cancels, so you get
hk times the same wave function.
And for this, we think that
this wave function is a wave
function with momentum hk.
Because if you act with
the operator p on that wave
function, it returns for you hk.
So we think of this as
has p equal hk, h bar k.
So the thing that
we want to do now
is to make this a
little more general.
This is just talking
about momentum,
but in quantum
mechanics we're going
to have all kinds of operators.
So we need to be more general.
So this is going to be, as Allan
calls it, a math interlude.
Based on the following question,
what this is an operator?
And then the physics question,
what do measurable things
have to do with our operators?
So about operators aren't
measurable things, quantities.
Now your view of operators
is going to evolve.
It's going to evolve
in this course,
It's going to evolve in 805.
It probably will
continue to evolve.
So we need to think
of what operators are.
And there is a simple
way of thinking
of operators that is going
to be the basis of much
of the intuition.
It's a mathematical way
of thinking of operators,
and what we'll sometimes
use it as a crutch.
And the idea is that
basically operators
are things that act on
objects and scramble them.
So whenever you
have an operator,
you really have
to figure out what
are the objects
you're talking about.
And then the operator
is some instruction
on how to scramble it,
scramble this object.
So for example, you
have an operator.
You must see what it
does to any object here.
The operator acts on the object
and gives you another object.
So in a picture
you have all kinds
of objects sets,
a set of objects.
And the operators are things.
You can have a
list of operators.
And they come here and
move those objects,
scramble them, do
something to them.
And we should distinguish them,
because the objects are not
operators, and the operators
are not the objects.
So what is the simplest example
in mathematics of this thing
is vectors and matrices.
Simplest example.
Objects are vectors.
Operators are matrices.
And how does that work?
Well, you have a two
by two, the case where
you have a set in which you have
vectors with two components.
So example, a vector that
has two components v1 v2.
And the matrices,
this is the object.
And the operator is a matrix.
a 11, a 12, a 21, a
22 as an operator.
An m on a vector is a vector.
If you are with a matrix
on a vector, this 2
by 2 matrix on this common
vector, you get another vector.
So that's this simplest example
of operators acting on objects.
In our case, we're going to
talk about a more-- we're
going to have to begin,
in quantum mechanics
we're required to begin with a
more sophisticated one in which
the objects are
going to be-- objects
are going to be functions.
In fact, I will write
complex functions of x.
So let's see it, the
list of operators.
And what do the operators do?
The operators act
on the functions.
So what is an operator?
It's a rule on how
to take any function,
and you must give a rule
on how to obtain from that
function another function.
So let's start with an examples.
It's probably the
easiest thing to do.
So an operator acts
on the functions.
So an operator for
any function f of x
will give you another function.
Function of x.
And here's operator o, we put
a hat sometimes for operators.
So the simplest operator,
the operator one.
We always, mathematicians, love
to begin with trivial examples.
Illustrate anything
almost, and just kind of
confuse you many times.
But actually it's good
to get them of the way.
So what is the operator one?
One possibility it
takes any function
and gives you the number one.
Do you think that's it?
Who thinks that's it?
Nobody?
Very good.
that definitely is
not a good thing
to do to give you
the number one.
So this operator
does the following.
I will write it like that.
The operator one takes f
of x and gives you what?
AUDIENCE: f of x.
PROFESSOR: f of x.
Correct.
Good.
So it's a very simple
operator, but it's an operator.
It's like what matrix?
The identity matrix.
Very good.
There could be a zero operator
that gives you nothing
and would be the zero matrix.
So let's write the more
interesting operator.
The operator would d dx.
That's interesting.
The derivative can be
thought of as an operator
because if you
start with f of x,
it gives you another
function, d dx of f of x.
And that's a rule to get
from one function to another.
Therefore that's an operator,
qualifies as an operator.
Another operator that
typically can confuse you
is the operator x.
x an operator?
What does that mean?
Well, you just
have to define it.
At this moment, it
could mean many things.
But you will see that
[INAUDIBLE] is the only thing
that probably makes some sense.
So what is the operator x?
Well, it's the operator
that, given f of x,
gives you the function
x times f of x.
That's a reasonable thing to do.
It's multiplying by x.
It changes the function.
You could define the operator
x squared that multiplies
by x squared.
And the only reasonable
thing is to multiply it by.
You could divide
by it, and you may
need to divide by it as well.
And you could define
the operator 1
over x gives you the
function times 1 over x.
We will need that
sometime, but not now.
Let's see another set of
operators where we give a name.
It doesn't have a
name because it's not
all that's useful in fact.
But it's good to
illustrate things.
They operator s
sub q for squared.
S q for the first two
letters of the word square.
That takes f of x
into f of x squared.
That's another function.
You could define more
functions like that.
The operator p 42.
That's another silly operator.
Well certainly a lot
more silly than this one.
That's not too bad.
But the p 42 takes f of x And
gives you the number 42 times
the constant function.
So that's a function of x.
It's trivial function of x.
Now enough examples.
So you get the idea.
Operators act on functions
and give you functions.
And we just need to
define them, and then we
know what we're talking about.
Yes?
AUDIENCE: Is the Dirac
delta and operator?
PROFESSOR: The Dirac
delta, well, you
can think of it as an operator.
So it all depends how
you define things.
So how could I do find
the Dirac delta function
to be an operator?
So delta of x minus a.
Can I call it the
operator delta hat of a?
Well, I would have to tell
you what it does in functions.
And probably I
would say delta had
on a on a function of x is equal
to delta of x minus a times
the function of x.
And I'd say it's an operator.
Now the question is, really,
is it a useful operator?
And sometimes it will
be useful in fact.
This is a more general
case of another operator
that maybe I could define.
o sub h of x is
the operator that
takes f of x into h
of x times f of x.
So that would be
another operator.
Now there are operators
that are particularly nice,
and there are the
so-called linear operators.
So what is a linear operator?
It's one that respects
superposition.
So linear operator
respects superposition.
So o hat is linear.
o hat is a linear operator.
If o hat on a times f
of x plus b times g of x
does what you would
imagine it should do,
it that's on the first, and
then it acts on the second.
Acting on the first,
the number goes out
and doesn't do anything,
say, on the number.
It's linear.
It's part of that idea.
And it gives you o on f of
x plus b times o on g of x.
So your operator may be linear,
or it may not be linear.
And we have to just guess them.
And you would imagine
that we can decide that,
of the list of
operators that we have,
let's see, one d dx-- how much?
Which one?
Sq, p 42, and o sub h of x.
Let's see.
Let's vote on each one
whether it's linear or not.
A shouting match whether I
hear a stronger yes or no.
OK?
One is a linear operator, yes?
AUDIENCE: Yes.
PROFESSOR: No?
Yes.
All right.
d dx linear.
Yes?
AUDIENCE: Yes.
PROFESSOR: Good.
That's strong enough.
Don't need to hear
the other one.
x hat.
Linear operator?
Yes or no?
AUDIENCE: Yes.
PROFESSOR: Yes, good.
Squaring, linear operator?
AUDIENCE: No.
PROFESSOR: No.
No way it could be
a linear operator.
It just doesn't happen.
If you have sq on f plus g,
it would be f plus g squared,
which is f squared plus g
squared plus, unfortunately
too, fg.
And this thing ruins it, because
this is sq of f plus sq of g.
It's even worse than that.
You put sq on af,
by linearity it
should be a times the operator.
But when you square a times f,
you get a squared f squared.
So you don't even
need two functions
to see that it's not real.
So definitely no.
How about p 42?
AUDIENCE: No.
PROFESSOR: No, of course not.
Because if you
add two functions,
it still gives you 42.
It doesn't get you 84, so no.
How about oh of x?
AUDIENCE: Yes.
PROFESSOR: Yes, it does that.
If you act with this operator on
a sum of functions distributive
law, it works.
So this is linear.
Good.
Linear operators
are important to us
because we have
some superposition
of allowed states.
So if this is a state
and this is a state,
this is also good state.
So if we want superposition
to work well with our theory,
we want linear operator.
So that's good.
So we have those
linear operators.
And now operators have
another thing that
makes them something special.
It is the idea that there's
simpler object they can act on.
We don't assume you've studied
linear algebra in this course,
so whatever I'm
going to say, take it
as motivation to learn some
linear algebra at some stage.
You will be a little more
linear algebra in 805.
But at this moment,
just basic idea.
So whenever you have matrices,
one thing that people do
is to see if there
are special vectors.
Any arbitrary vector, when
you act within the matrix,
is going to just jump and
go somewhere else, point
in another direction.
But there are some
special vectors
that do act-- if you
have a given matrix m,
there are some funny vectors
sometimes that acted by n
remain the same direction.
They may grow a little
or become smaller,
but they remain
the same direction.
These are called eigenvectors.
And that constant
of proportionality,
proportional to the action of
the operator on the vector,
is called the eigenvalue.
So these things
have generalizations
for our operators.
So operators can have special
functions, eigenfunctions.
What are these eigenfunctions?
So let's consider
that operator a hat.
It's some operator.
I don't know which
one of these, be we're
going to talk about
linear operator.
So linear operators
have eigenfunctions.
A hat.
So a hat.
There may be functions that,
when you act with the operator,
you sort of get the
same function up
to possibly a constant a.
So you may get a
times the function.
And that's a pretty
unusual function,
because, on most functions,
any given operator
is going to make a mess
out of the function.
But sometimes it does that.
So to label them better
with respect to operator,
I would put a subscript a,
which means that there's
some special function
that has a parameter a,
for which this
operator gives you
a times that special function.
And that special
function is called--
this is the eigenfunction
and this is the eigenvalue.
And that the
eigenvalue is a number.
It's a complex number
c there over there.
So these are special things.
They don't necessarily
happen all the time to exist,
but sometimes they do, and
then they're pretty useful.
And we have one example of
them that is quite nice.
For the operator a equal
p, we have eigenfunctions
e to the ikx with eigenvalue hk.
So this is the connection
to this whole thing.
We wanted to make clear for
you that what you saw here,
that this operator
acting on this function
gives you something times this
function is a general fact
about the operators.
Operators have eigenfunctions.
So eigenfunction e of x with
eigenvalue hk, because indeed
p hat on this e to ikx,
as you see this h bar
k times e to the ikx.
So here you have p hat is the a.
This is the function
labeled a would be like k.
Here is something like k again.
And here is this thing.
But the main thing operator
on the function number
times the function
is an eigenfunction.
Yes?
AUDIENCE: For a given operator,
is the eigenvalue [INAUDIBLE]?
PROFESSOR: Well, for a given
operator good question.
a is a list of values.
So there may be many,
many, many eigenfunctions.
Many cases infinitely
many eigenfunctions.
In fact, here I can put
for k any number I want,
and I get a different function.
So a belongs to c and may take
many, or even infinite, values.
If you remember for nice
matrices, n by n matrix
may be a nice n by n matrix
because n eigenvectors
and eigenvalues are
sometimes hard to generate,
sometimes eigenvalues have
the same numbers and things
like that.
OK.
Linearity is this some of two
eigenvectors and eigenvector.
Yes?
No?
AUDIENCE: No.
PROFESSOR: No, no.
Correct.
That's not necessarily true.
If you have two
eigenvectors, they
have different eigenvalues.
So things don't work
out well necessarily.
So an eigenvector plus
another eigenvector
is not an eigenvector.
So you have here, for
example, A f1 equals a1f1.
And A f2 equal a2f2,
then a on f1 plus f2
would be a1f1 plus
a2f2, and that's
not equal to something
times f1 plus f2.
It would have to be
something times f1 plus f2
to be an eigenvector.
So this is not necessarily
an eigenvector.
And it doesn't help to put
a constant in front of here.
Nothing helps.
There's no way to
construct an eigenvector
from two eigenvectors by
adding or subtracting.
The size of the eigenvector
is not fixed either.
If f is an eigenvector, then 3
times f is also an eigenvector.
And we call it the
same eigenvector.
Nobody would call it a
different eigenvector.
It's really the same.
OK, so how does that
relate to physics?
Well, we've seen
it here already.
that one operator that
we've learned to work with
is the momentum operator.
It has those eigenfunctions.
So back to physics.
We have other operators.
Therefore we have
the P operator.
That's good.
We have the X operator.
That's nice.
It's multiplication by x.
And why do we use it?
Because sometimes you
have the energy operator.
And what is the energy operator?
The energy operator
is just the energy
that you've always known, but
think of it as an operator.
So how do we do that?
Well, what is the energy
of a particle we've written
p squared over 2m plus v of x.
Well, that was the
energy of a particle,
the momentum squared
over 2m plus v of x.
So the energy operator
is hat here, hat there.
And now it becomes an
interesting object.
This energy operator
will be called E hat.
It acts and functions.
It is not a number.
The energy is a number,
but the energy operator
is not a number.
Far from a number in fact.
The energy operator is
minus h squared over 2m.
d second the x squared.
Why that?
Well, because p was h bar
over i d dx as an operator.
So this sort of arrow here,
it sort of the introduction.
But after a while you just say
P hat is h bar over a i d dx.
End of story.
It's not like double arrow.
It's just what it is.
That operator.
That's what we call it.
So when we square it, the i
squares, the minus h squares,
and d dx and d dx applied
twice is the second derivative.
And here we get
v of X hat, which
is your good potential, whatever
potential you're interested in,
in which, whenever you see
an x, you put an X hat.
And now this is an operator.
So you see this is not a
number, not the function.
It's just an operator.
The operator has this
sort of operator v of x.
Now what is this v of
x here as an operator?
This is v of x as an operator
is just multiplication
by the function v of x.
You see, you have here that the
operator x is f of x like that.
I could have written the
operator X hat to the n.
What would it be?
Well, if I add to
the function, this
is a lot of X hats
acting on the function.
Well, let the first one out.
You let x times f of x.
The second, that's
another x, another x.
So this is just x to
the n times f of x.
So lots of X hats.
X hats To the
100th on a function
is just X to the 100th
times a function.
So v of x on a
function is just v
of the number x on a function.
It's just like this
operator, the O in which you
multiply by a function.
So please I hope this
is completely clear what
this means as an operator.
You take the wave function,
take two derivatives,
and add the product of the wave
function times v of plane x.
So I'll write it here maybe.
So important.
E hat and psi of x
is therefore minus h
squared over 2m the [INAUDIBLE]
the x squared of psi of x
plus just plain v
of x times psi of x.
That's what it does.
That's an operator.
And for these
operators in general.
Math interlude, is it over?
Not quite.
Wow.
No, yes.
Allan said at this
moment it's over,
when you introduce it here.
I'll say something more here,
but it's going to be over now.
Our three continues
here then with four.
Four, to each observable we
have an associated operator.
So for momentum, we
have the operator P hat.
And for position we
have the operator X hat.
And for energy we have
the operator E hat.
And these are
examples of operators.
Example operator A hat
could be any of those.
And there may be more
observables depending
on the system you're working.
If you have particles
in a line, there's
not too many more
observables at this moment.
If you have a
particle in general,
you can have angular momentum.
That's an interesting
observable.
It can be others.
So for any of those,
our definition
is just like with
it for momentum.
The expectation
value of the operator
is computed by doing what
you did for momentum.
You act with the operator
on the wave function
here and multiply by the
compass conjugate function.
And integrate just like
you did for momentum.
This is going to be the
value that you expect.
After many trials on
this wave function,
you would expect the measured
value of this exhibit
a distribution which its
expectation value, the mean,
is given by this.
Now there are other definitions.
One definition that
already has been mentioned
is that the uncertainty of
the operator on the state
psi, the uncertainty,
is computed
by taking the square
root of the expectation
value of A squared minus
the expectation value of A,
as a number, squared.
Now the expectation value
of A squared, just simply
here instead of A
you put A squared,
so you've got A squared here.
That unless the function is very
special, it's very different
whole is bigger than the
expectation value of A squared.
So this is a number, and
it's called the uncertainty.
And that's the uncertainty
of the uncertainty principle.
So for operators, we need to
have another observation that
comes from matrices that
is going to be crucial
for us is the observation that
operators don't necessarily
commute.
And we'll do the most
important example of that.
So we'll try to see in
the operators associated
with momentum and
position commute.
And what we mean by commute
or don't communicate?
Whether the order of
multiplication matters.
Now we talked about
matrices at the beginning,
and we said matrices act on
vectors to give you vectors.
So do they commute?
Well, matrices don't commute.
The order matters for
matrices multiplication.
So these operators we're
inventing here for physics,
the order does matter as well.
So commutation.
So let's try to see if
we compute the operator p
and x hat.
Is it equal to the
operator x hat times p?
This is a very good question.
These are two operators
that we've defined.
And we just want to know
if the order matters
or if it doesn't matter.
So how can I check it?
I cannot just
check it like this,
because operators are only clear
what they do is when they act
on functions.
So the only thing
that I can do is test
if this thing acting on
functions give the same.
So I'm going act with this
on the function f of x.
And I'm going to have act with
this on the function f of x.
Now what do I mean by
acting with p times
x hat on the function f of x.
This is by definition
you act first
with the operator that
is next to the f and then
with the other.
So this is p hat on the
function x hat times f of x.
So here I would have, this is
x hat on the function p hat
f of x.
See, if you have a series of
matrices, m1, m2, m3 acting
on a vector, what do you mean?
Act with this on the vector,
then with this on the vector,
then with this.
That's multiplication.
So we're doing that.
So let's evaluate.
What is x operator on f of x?
This is p hat on x times f of x.
That's what the x operator
in the function is.
Here, what this x hat?
And now I have this, so I
have here h over i d dx of f.
Let's go one more step here.
This is h over i d ddx now
of this function, x f of x.
And here I have just the x
function times this function.
So h over i x df dx.
Well, are these the same?
No, because this d dx here is
not only acting on f like here.
It's acting on the x.
So this gives you two terms.
One extra term on the
d dx acts on the x.
And then one term
that is equal to this.
So you don't get the same.
So you get from here h over i
f of x, when you [INAUDIBLE]
the x plus h over i x the df dx.
So you don't get the same.
So when I subtract
them, so when I
do xp minus px acting on the
function f of x, what do I get?
Well, I put them in this
order, x before the p.
Doesn't matter
which one you take,
but many people like this.
Well, these terms cancel
and I get minus this thing.
So I get minus h over i f
of x, or i h bar f of x.
Wow.
You got something very strange.
The x times p minus p times
x gives you a number--
an imaginary number, even
worse-- times f of x.
So from this we
write the following.
We say look, operators
are defined by the action
and function, but for any
function, the only effect of xp
minus px, which we call
the commutator of x with p.
This definition, the bracket
of two things, of A B.
Is defined to be A B minus B
A. It's called the commutator.
x p is an operator acting
on f of x, gives you
i h bar times f of x.
So our kind of silly
operator that does nothing
has appeared here.
Because I could now
say that x hat with p
is equal to i h bar
times the unit operator.
Apart from the
Schrodinger equation,
this is probably the
most important equation
in quantum mechanics.
It's the fact that x and b
are incompatible operators
as you will see later.
They don't commute.
Their order matters.
What's going to mean is
that when you measure one,
you have difficulties
measuring the other.
They interfere.
They cannot be measured
simultaneously.
All those things
are encapsulated
in a very lovely
mathematical formula,
which says that this is the
way these operators work.
Any questions?
Yes?
AUDIENCE: When x-- the
commutator of x and p
is itself an operator, right?
PROFESSOR: RIght.
AUDIENCE: So is that
what we're saying?
When we had operators
before, we can't simply
just cancel the f of x.
I mean we're not really
canceling it, but it just
because I h bar is the only
eigenvalue of the operator?
PROFESSOR: Well, basically what
we've shown by this calculation
is that this operator,
this combination
is really the same as
the identity operator.
That's all we've shown, that
some particular combination is
the identity operator.
Now this is very
deep, this equation.
In fact, that's
the way Heisenberg
invented quantum mechanics.
He called it the
matrix mechanics,
because he knew that operators
were related to matrices.
It's a beautiful story how
he came up with this idea.
It's very different from
what we're doing today
that we're going to
follow Schrodinger today.
But basically his analysis
led very quickly to this idea.
And this is deep.
Why is it deep?
Depends who you ask.
If you ask a mathematician,
they would probably tell you
this equation is not deep.
This is scary equation.
And why is it scary?
Because whenever a
mathematician see operators,
they want to write matrices.
So the mathematician, you
show him this equation,
will say OK, Let me
try to figure out which
matrices you're talking about.
And this mathematician will
start doing calculations
with two by two matrices,
and will say, no,
I can't find two by
two matrices that
behave like these operators.
I can't find three by
three matrices either.
And four by four.
And five by five.
And finds no matrix
really can do
that, except if the matrix
is infinite dimensional.
Infinite by infinite matrices.
So that's why it's very
hard for a mathematician.
This is the beginning
of quantum mechanics.
This looks like a
trivial equation,
and mathematicians
get scared by it.
You show them for physicists
there will be angular momentum.
The operators are like this,
and there's complicated
into the [INAUDIBLE].
The three components
of angular momentum
have this commutation relation.
And h bar here as well.
Complicated.
Three operators.
They mix with each other.
Show it to a mathematician,
he starts laughing at you.
He says that best, the
simplest case, this is easy.
This is complicated.
It's very strange.
But the reason this is
easier, the mathematician
goes and, after five minutes,
comes to you with three
by three matrices that
satisfies this relation.
And here there weren't.
And four by four that
satisfy, and five by five,
and two by two, and all of them.
We can calculate
all of them for you.
But this one it's infinite
dimensional matrices,
and it's very surprising, very
interesting, and very deep.
All right, so we move
on a little bit more
to the other observable.
So after this, we have
more general observable.
So let's talk a
little about them.
That's another postulate
of quantum mechanics
that continues with this
one, postulate number five.
So once you measure,
upon measuring
an observable A associated
with the operator A hat,
two things happen.
You measure this quantity
that could be momentum,
could be energy, could
be position, you name it.
The measured value
must be a number.
It's one of the
eigenvalues of A hat.
So actually those eigenvalues,
remember the definition
of the eigenvalues.
It's there.
I said many, but
whenever you measure,
the only possibilities
that you get this number.
So you measure the momentum, you
must get this hk, for example.
So observables, we have
an associated operator,
and the measured values
are the eigenvalues.
Now these eigenvalues, in
order to be observable,
they should be a real numbers.
And we said oh,
they can be complex.
Well, we will limit
the kind of observables
to things that have
real eigenvalues,
and these are going to be called
later on permission operators.
At this moment, the notes,
they don't mention them.
You're going to
get them confused.
So anyway, special operators
that have real eigenvalues.
So we mentioned here they
will have to be a real.
Have to be real.
And then the second one, which
is an even stranger thing that
happens is something you've
already seen in examples.
After you measure, the
whole wave function
goes into the state which is the
eigenfunction of the operator.
So after measurement system
collapses into psi a.
The measure value is one
over the eigenvalues a of A.
And the system
collapses into psi a.
So psi a is such that
A hat psi a is a psi a.
So this is the eigenvector with
eigenvalue a that you measured.
So after you
measure the momentum
and you found that its h
bar k, the wave function
is the wave function
of momentum h bar k.
If at the beginning, it was
a superposition of many,
as Fourier told you,
then after measuring,
if you get one
component of momentum,
that's all that is left
of the wave function.
It collapses.
This collapse is a
very strange thing,
and is something about
quantum mechanics
that people are a little
uncomfortable with,
and try to understand better,
but surprisingly nobody
has understood it better after
60 years of thinking about it.
And it works very well.
It's a very strange thing.
Because for example, if you
have a wave function that
says your particle can be
anywhere, after you measure it
where it is, the
whole wave function
becomes a delta function at
the position that you measure.
So everything on
the wave function,
when you do a
measurement, basically
collapses as we'll see.
Now for example,
let's do an example.
Position.
So you have a wave
function psi of x.
You find measure and
find the particle at x0.
Measure and you find
the particle at x0.
So measure what?
I should be clear.
Measure position.
So we said two things.
The measured value is one
of the eigenvalues of a,
and after measurement,
the system
collapses to eigenfunctions.
Now here we really need a
little of your intuition.
Our position eigenstate is a
particle a localized at one
place.
What is the best
function associated
to a position eigenstate?
It's a delta function.
The function that says it's at
some point and nowhere else.
So eigenfunctions delta
of x minus x0, it's
a function as a function of x.
It peaks at x0, and
it's 0 everywhere else.
And this is, when you
find a particle at x0,
this is the wave function.
The wave function must be
proportional to this quantity.
Now you can't normalize
this wave function.
It's a small complication, but
we shouldn't worry about it
too much.
Basically you really can't
localize a particle perfectly,
so that's the little problem
with this wave function.
You've studied how you can
represent delta functions
as limits and probably
intuitively those limits
are the best things.
But this is the wave
function, so after you
measure the system, you go into
an eigenstate of the operator.
Is this an eigenstate
of the x operator?
What a strange question.
But it is.
Look, if you put the x operator
on delta of x minus x zero,
what is it supposed to do?
It's supposed to multiply by x.
So it's x times delta
of x minus x zero.
If you had a little experience
with delta functions,
you'd know that this
function is 0 everywhere,
except when x is equal to x0,
so this x can be turned into x0.
It just never at any other
place it contributes.
This x really can be turned into
x0 times delta of x minus x0.
Because delta functions are
really used to do integrals.
And if you do the
integral of this function,
you will see that it gives you
the same value as the integral
of this function.
So there you have it.
The operator acting
on the eigenfunction
is a number times this.
So these are indeed
eigenfunctions
of the x operator.
And what you measured was an
eigenvalue of the x operator.
Eigenvalue of x.
And this is an
eigenfunction of x.
So we can do the same
with the momentum.
Eigenvalues and eigenfunctions,
we've seen them more properly.
Now we'll go to the sixth
postulate, the last postulate
that we'll want to talk about
is the one of general operators,
and general eigenfunctions,
and what happens with them.
So let's take now our operator
A and its functions that
can be found.
So six, given an observable A
hat and its eigenfunctions phi
a of x.
So an a runs over many
values, many values.
OK so let's consider this case.
Now eigenfunctions
of an operator
are very interesting objects.
You see the eigenfunctions of
momentum were of this form.
And they allow you to
expand via the Fourier
any wave function as super
positions of these things.
Fourier told you you
can't expand any function
in eigenfunctions
of the momentum,
or the result is more general.
For observables
in general you can
expand functions,
arbitrary functions,
in terms of the eigenfunctions.
Now for that, remember,
an eigenfunction
is not determined up to scale.
You change multiplied by three,
it's an eigenfunction still.
So people like to normalize
them nicely, the eigenfunctions.
You construct them like this,
and you normalize them nicely.
So how do you normalize them?
Normalize them by saying
that the integral over x
of psi a star of x psi b
of x is going to be what?
OK, basically what you want
is that these eigenfunctions
be basically orthogonal.
Each one orthogonal to the next.
So you want this to
be 0 unless these two
different eigenfunctions
are different.
And when they are
the same, you want
them to be just
like wave functions,
that their total integral of
psi squared is equal to 1.
So what you put
here is delta ab.
Now this is something
that you can always
do with eigenfunctions.
It's proven in
mathematics books.
It's not all that
simple to prove,
but this can always be done.
And when we need examples,
we'll do it ourselves.
So given an operator that
you have its eigenfunctions
like that, two things happen.
One can expand psi as psi of x.
Any arbitrary wave
function as the sum.
Or sometimes an integral.
So some people like to write
this and put them an integral
on top of that.
You can write it
whichever way you want.
It doesn't matter.
Of coefficients times
the eigenfunctions.
So just like any wave could be
a written, a Fourier coefficient
[INAUDIBLE] Fourier function.
Any state can be written a
superposition of these things.
So that's one.
And two, the probability of
measuring A hat and getting a.
So a one of the particular
values that you can get.
That probability is given by
the square of this coefficient.
Ca squared.
So this is P, the probability,
to measure in psi and to get a.
I think, actually,
let's put an a0 there.
So here we go.
So it's a very
interesting thing.
Basically you expand
the wave function
in terms of these
eigenfunctions,
and these coefficients
give you the probabilities
of measuring these numbers.
So we can illustrate that
again with the delta functions,
and we'll do it
quick, because we
get to the punchline of this
lecture with the Schrodinger
equation.
So what do we have?
Well, let me think of
the operator X example.
Operator X, the
eigenfunctions are
delta of x minus x0 for all x0.
These are the eigenfunctions.
And we'll write sort
of a trivial equation,
but it sort of illustrates
what's going on.
Psi of x as a superposition
over an integral over x0
of delta of x minus x0.
Psi of x0.
Delta of x minus x zero.
OK, first let's check
that this make sense.
Here we're integrating over x0.
x0 is the variable.
This thing shoots
and fires whenever
x is equal to x--
whenever x0 is equal to x.
Therefore the whole
result of that integral
is psi of x is a little
funny how it's written,
because you have x minus
0, which is the same
as delta of x0 minus x
is just the same thing.
And you integrate over x0,
and you get just psi of x.
But what have you achieved here?
You've achieved the analogue
of this equation in which these
are the psi a.
These are the coefficients Ca.
And this is the
sum, this integral.
So there you go.
Any wave function can be written
as the sum of coefficients
times the eigenfunctions
of the operator.
And what is the probability
to find the particle at x0?
Well, it's from here.
The coefficients, the a squared.
That's exactly
what we had before.
So this is getting
basically what we want.
So this brings us
to the final stage
of this lecture in which
we have to get the time
evolution finally.
So how does it happen?
Well it happens in a
very interesting way.
So maybe I'll call it
seven, Schrodinger equation.
So as with any fundamental
equation in physics,
there's experimental evidence
and suddenly, however, you
have to do a conceptual leap.
Experimental evidence doesn't
tell you the equation.
It suggests the equation.
And it tells you probably
what you're doing is right.
So what we're going to do now
is collect some of the evidence
we had and look at an
equation, and then just have
a flash of inspiration,
change something very little,
and suddenly that's the
Schrodinger equation.
Allan told me, in
fact, still sometimes
are disappointed that we
don't derive the Schrodinger
equation.
Now let's derive
it mathematically.
But you also don't derive
Newton's equations.
F equal ma.
You have an
inspiration, you get it.
Newton got it.
And then you use it and
you see it makes sense.
It has to be a
sensible equation,
and you can test very quickly
whether your equation is
sensible.
But you can't quite derive it.
In 805 we come a little closer
to deriving the Schrodinger
equation, which we say unitary
time evolution, something
that I haven't
explained what it is,
implies the
Schrodinger equation.
And that's a mathematical fact.
And you can begin unitary
time evolution, define it,
and you derive the
Schrodinger equation.
But that's just
saying that you've
substituted the Schrodinger
equation by saying there
is unitary time evolution.
The Schrodinger question
really comes from something
a little deeper than that.
Experimentally it comes
from something else.
So how does it come?
Well, you've studied some of
the history of this subject,
and you've seen that Planck
postulated quantized energies
in multiples of h bar omega.
And then came Einstein
and said look,
in fact, the energy of
a photon is h bar omega.
And the momentum of
the photon was h bar k.
So all these people, starting
with Planck and then Einstein,
understood what the photon is.
The quantum of
photons for photons,
you have E is equal h bar
omega, and the momentum
is equal to h bar k.
I write them as a vector,
because the momentum
is a vector, but we also write
them in this because p equal h
bar k, assuming you move
just in one direction.
And that's the way
it's been written.
So this is the result of much
work beginning by Planck,
Einstein, and Compton.
So you may recall
Einstein said in 1905
for that there seemed to be
this quantum of light that
carry energy h omega.
Planck didn't quite like that.
And people were not
all that convinced.
Experiments were done
by Millikan in 1915,
and people were still
not quite convinced.
And then came Compton and
did Compton scattering.
And then people said, yeah,
they seem to be particles.
No way out of that.
And they satisfy
such a relation.
Now there was
something about this
that was quite nice, that
these photons are associated
with waves, and that
was not too surprising,
because people understood that
electromagnetic waves are waves
that correspond to photons.
So you can also
see that this says
that E p is equal
to h bar omega k
as an equation between vectors.
You see the E is the first, and
the p is the second equation.
And this is actually a
relativistic equation.
It's a wonderful
relativistic equation,
because energy and momentum
form what is called
a relativity of four vector.
It's the four vector-- this is
a little aside on relativity--
four vector.
The index mew runs
from 0, 1, 2, 3.
Just like the x mews,
which are t and x.
Run from x0, which is t, x1,
which is x, x2 which is y,
three-- these are four vectors.
And this is a four vector.
This is a four vector.
This all seemed quite
pretty, and this
was associated to photons.
But then came De Broglie.
And De Broglie had
a very daring idea
that even though this
was written for photons,
it was true for particles
as well, for any particle.
De Broglie says good for
particles, all particles.
And these particles
are really waves.
So what if he write--
he wrote psi of x and t
is equal a wave associated
to a matter particle.
And it would be an e to
the i kx minus omega t.
That's a wave.
And you know that this wave
has momentum p equal h bar k.
If k is positive,
look at this sign.
If this sign is like
that, then k is positive.
This is a wave that is
moving to the right.
So p being hk.
If k is positive, p is positive,
is moving to the right,
this is a wave moving to the
right, and has this momentum.
So it should also
have an energy.
Compton said that this is
relativistic because this all
comes from photons.
So if the momentum is given
by that, and the energy
must also be given by
a similar relation.
In fact, he mostly
said, look, you
must have the energy being
equal to h bar omega.
The momentum, therefore,
would be equal to h bar k.
And I will sometimes
erase these things.
So what happens with this thing?
Well, momentum equal to hk.
We've already understood
this as momentum operator
being h bar over i d dx.
So this fact that these
two must go together and be
true for particles was
De Broglie's insight,
and the connection
to relativity.
Now here we have this.
So now we just have to try
to figure out what could we
do for the energy.
Could we have an
energy operator?
What would the energy
operator have to do?
Well, if the energy
operator is supposed
to give us h bar omega,
the only thing it could be
is that the energy
is i h bar d dt.
Why?
Because you go again
at the wave function.
And you think i h bar d
dt, and what do you get?
i h bar d dt on the wave
function is equal to i h bar.
You take the d dt,
you get minus i omega
times the whole wave function.
So this is equal h bar omega
times the wave function,
times the wave
function like that.
So here it is.
This is the operator
that realizes the energy,
just like this is the operator
that realizes the momentum.
You could say these are the
main relations that we have.
So if you have
this wave function,
it corresponds to a particle
with momentum hk and energy h
omega.
So now we write this.
So for this psi
that we have here,
h bar over i d dx
of psi of x and t
is equal the value of the
momentum times psi of x and t.
That is something we've seen.
But then there's a second one.
For this psi, we also that i
h bar d dt of psi of x and t
is equal to the energy of
that particle times x and t,
because the energy of that
particle is h bar omega.
And look, this is familiar.
And here the t plays no role,
but here the t plays a role.
And this is prescribing you
how a wave function of energy E
evolves in time.
So you're almost there.
You have something
very deep in here.
It's telling you if you
know the wave function
and it has energy E, this
is how it looks later.
You can take this derivative
and solve this differential
equation.
Now this differential
equation is kind of trivial
because E is a number here.
But if you know that you have
a particle with energy E,
that's how it evolves in time.
So came Schrodinger and
looked at this equation.
Psi of x and t equal
E psi of x and t.
This is true for any
particle that has energy E.
How can I make out of
this a full equation?
Because maybe I don't
know what is the energy E.
The energy E might be
anything in general.
What can I do?
Very simple.
One single replacement
in that equation.
Done.
It's over.
That's the Schrodinger equation.
It's the energy operator
that we introduced before.
Inspiration.
Change E to E hat.
This is the
Schrodinger equation.
Now what has really
happened here,
this equation that was trivial
for a wave function that
represented a particle
with energy E,
if this is the energy operator,
this is not so easy anymore.
Because remember, the energy
operator, for example,
was p squared over
2m plus v of x.
And this was minus h squared
over 2m d second dx squared
plus v of x acting
on wave functions.
So now you've got a really
interesting equation,
because you don't assume
that the energy is a number,
because you don't know it.
In general, if the
particle is moving
in a complicated
potential, you don't know
what are the possible energies.
But this is symbolically
what must be happening,
because if this particle
has a definite energy,
then this energy operator
gives you the energy acting
on the function,
and then you recover
what you know is true for a
particle of a given energy.
So in general, the
Schrodinger equation
is a complicated equation.
Let's write it now completely.
So this is the
Schrodinger equation.
And if we write
it completely, it
will read i h bar d psi dt is
equal to minus h bar squared
over 2m d second dx squared
of psi plus v of x times psi--
psi of x and t, psi of x and t.
So it's an equation, a
differential equation.
It's first order in time,
and second order in space.
So let me say three things
about this equation and finish.
First, it requires
complex numbers.
If psi would be real, everything
on the right hand side
would be real.
But with an i it would
spoil it, so complex numbers
have to be there.
Second, it's a linear equation.
It satisfies the proposition.
So if one wave function
satisfies the Schrodinger
equation, the sum
of wave functions,
and another wave function
does, the sum does.
Third, it's deterministic.
If you know psi at
x and time equals 0,
you can calculate psi
at any later time,
because this is a first order
differential equation in time.
This equation will
be the subject of all
what we'll do in this course.
So that's it for today.
Thank you.
[APPLAUSE]
