PROFESSOR: This
is very important.
This is the beginning of
the uncertainty principle,
the matrix formulation
of quantum mechanics,
and all those things.
I want to just tabulate the
information of matrices.
We have an analog,
so we have operators.
And we think of
them as matrices.
Then in addition to operators,
we have wave functions.
And we think of them as vectors.
The operators act on the
wave functions or functions,
and matrices act on vectors.
We have eigenstate
sometimes and eigenvectors.
So matrices do the same thing.
They don't necessarily commute.
There are very many
examples of that.
I might as well give
you a little example
that is famous in the
theory of spin, spin 1/2.
There is the Pauli matrices.
Sigma 1 is equal to 1, 1, 0, 0.
Sigma 2 is 0 minus i, i 0, and
sigma 3 is 1 minus 1, 0, 0.
And a preview of
things to come--
the spin operator is
actually h bar over 2 sigma.
And you have to think of sigma
as having three components.
That's where it is.
Spins will be like that.
We won't have to deal
with spins this semester.
But there it is, that spin 1/2.
Somehow these matrices
encode spin 1/2.
And you can do simple things,
like sigma 1 times sigma 2.
0, 1, 1, 0 times
0 minus i, i, 0.
Let's see if I can
get this right.
i, 0, 0 minus i.
And you can do sigma 2 sigma
1 0 minus i, i 0, 0, 1, 1,
0 equals minus i, 0, 0, i, i.
So I can go ahead here.
And therefore, sigma 1
commutator with sigma 2
is equal to sigma 1, sigma
2 minus sigma 2, sigma 1.
And you can see that they're
actually the same up to a sign,
so you get twice.
So you get 2 times
i 0, 0 minus i.
And this is 2i times
1 minus 1, 0, 0.
And that happens to
be the sigma 3 matrix.
So sigma 1 and sigma 2
is equal to 2i sigma 3.
These matrices
talk to each other.
And you would say, OK,
these matrices commute
to give you this matrix.
This thing commutes to give you
a number so that surely it's
a lot easier.
You couldn't be more wrong.
This is complicated,
extraordinarily complicated
to understand what this means.
This is very easy.
This is 2 by 2 matrices
that you check.
In fact, you can write
matrices for x and p.
This correspondence is
not just an analogy.
It's a concrete fact.
You will learn-- not too much
in this course, but in 805--
how to write matrices
for any operator.
They're called matrix
representations.
And therefore, you could ask
how does the matrix for x look.
How does the matrix for p look?
And the problem
is these matrices
have to be infinite dimensional.
It's impossible to
find two matrices whose
commutator gives you a number.
Something you can prove in
math is actually not difficult.
You will all prove it through
thinking a little bit.
There's no two matrices that
commute to give you a number.
On the other hand,
very easy to have
matrices that commute to
give you another matrix.
So this is very strange and
profound and interesting,
and this is much simpler.
Spin 1/2 is much simpler.
That's why people do
quantum computations.
They're working with
matrices and simple stuff,
and they go very far.
This is very difficult. x
and p is really complicated.
But that's OK.
The purpose of this
course is getting
familiar with those things.
So I want to now generalize
this a little bit more
to just give you the
complete Schrodinger
equation in three dimensions.
So how do we work
in three dimensions,
three-dimensional physics?
There's two ways
of teaching 804--
it's to just do everything
in one dimension, and then
one day, 2/3 of the way
through the course--
well, we live in
three dimensions,
and we're going to
add these things.
But I don't want to do that.
I want to, from the
beginning, show you
the three-dimensional
thing and have
you play with
three-dimensional things
and with one-dimensional
things so that you don't get
focused on just one dimension.
The emphasis will be in
one dimension for a while,
but I don't want you to
get too focused on that.
So what did we have
with this thing?
Well, we had p equal
h bar over i d dx.
But in three dimensions,
that should be the momentum
along the x direction.
We wrote waves like that with
momentum along the x direction.
And py should be
h bar over i d dy,
and pz should be h
bar over i d dz--
momentum in the x,
y, and z direction.
And this corresponds
to the idea that if you
have a wave, a de Broglie
wave in three dimensions,
you would write this--
e to the i kx minus
omega t, i omega t.
And the momentum would be
equal to h bar k vector,
because that's how
the plane wave works.
That's what de
Broglie really said.
He didn't say it
in one dimension.
Now, it may be easier to write
this as p1 equal h bar over i d
dx1, p2 h bar over i d dx2,
and p3 h bar over i d dx3
so that you can say that all
these three things are Pi
equals h bar over i d dxi--
and maybe I should put pk,
because the i and the i
could get you confused--
with k running from 1 to 3.
So that's the momentum.
They're three momenta,
they're three coordinates.
In vector notation,
the momentum operator
will be h bar over i
times the gradient.
You know that the gradient is
a vector operator because d dx,
d dy, d dz.
So there you go.
The x component of the
momentum operators,
h bar over i d dx, or
d dx1, d dx2, d dx3.
So this is the
momentum operator.
And if you act on this wave
with the momentum operator,
you take the gradient,
you get this--
so p hat vector.
Now here's a problem.
Where do you put the arrow?
Before or after the hat?
I don't know.
It just doesn't look
very nice either way.
The type of notes I think
we'll use for vectors
is bold symbols so there will
be no proliferation of vectors
there.
So anyway, if you have this
thing being the gradient acting
on this wave function, e
to the i kx minus i omega
t, that would be h
over i, the gradient,
acting on a to the i kx
vector minus i omega t.
And the gradient acting on
this-- this is a vector--
actually gives you a vector.
So you can do
component by component,
but this gives you i k vector
times the same wave function.
So you get hk, which is the
vector momentum times the wave
function.
So the momentum operator
has become the gradient.
This is all nice.
So what about the
Schrodinger equation
and the rest of these things?
Well, it's not too complicated.
We'll say one more thing.
So the energy operator,
or the Hamiltonian,
will be equal to p vector
hat squared over 2m plus
a potential that depends on
all the coordinates x and t,
the three coordinates.
Even the potential is radial,
like the hydrogen atom,
is much simpler.
There are conservation laws.
Angular momentum works nice.
All kinds of beautiful
things happen.
If not, you just
leave it as x and p.
And now what is p hat squared?
Well, p vector hat squared
would be h bar over--
well, I'll write this-- p vector
hat dotted with p vector hat.
And this is h over i
gradient dotted with h over i
gradient, which is minus
h squared Laplacian.
So your Schrodinger
equation will
be ih bar d psi dt is equal to
the whole Hamiltonian, which
will be h squared over 2m.
Now Laplacian plus v
of x and t multiplied
by psi of x vector and t.
And this is the full
three-dimensional Schrodinger
equation.
So it's not a new invention.
If you invented the
one-dimensional one,
you could have invented the
three-dimensional one as well.
The only issue was recognizing
that the second dx squared now
turns into the full Laplacian,
which is a very sensible thing
to happen.
Now, the commutation relations
that we had here before--
we had x with p is
equal to ih bar.
Now, px and x failed to
commute, because d dx and x,
they interact.
But px will commute with y.
y doesn't care
about x derivative.
So the p's failed to commute.
They give you a number with
a corresponding coordinate.
So you have the i-th component
of the x operator and the j-th
component of the p operator--
these are the components--
give you ih bar delta
ij, where delta ij
is a symbol that gives
you 1 if i is equal to j
and gives you 0 if i
is different from j.
So here you go.
X and px is 1 and 1.
Delta 1, 1 is 1.
So you get ih bar.
But if you have x
with py or p2, you
would have delta
1, 2, and that's 0,
because the two indices
are not the same.
So this is a neat way of
writing nine equations.
Because in principle,
I should give you
the commutator of x
with px and py and pc,
y with px, py, pc,
and z with px, py, pc.
You're seeing that,
in fact, x just
talks to px, y talks
to py, z talks to pz.
So that's it for the
Schrodinger equation.
Our goal is going to be to
understand this equation.
So our next step is
to try to figure out
the interpretation of this psi.
We've done very nicely by
following these things.
We had a de Broglie wave.
We found an equation.
Which invented a free
Schrodinger equation.
We invented an interacting
Schrodinger equation.
But we still don't know what
the wave function means.
