PROFESSOR: The simplest
quantum system.
In order to decide what could
be the simplest quantum system
you could say a
particle in a box.
It's very simple, but in a
sense it's not all that simple.
It has infinitely many states.
All these functions on an
interval, and then the energy
is where infinitely many of
them, so not that simple.
OK, infinite bound says
something with one bound.
OK, a delta function potential
just one bound state,
but it has infinitely
many scattering states.
It's still complicated.
What could be simpler?
Suppose you have the
Schrodinger equation.
H psi.
And we work in general we
know that this thing has
energy eigenstates, and probably
we should focus on them.
So Psi equal E to the
minus I, Et over H bar.
Little psi, and then
have H Psi equal E psi.
That is quantum mechanics,
and you could say,
well it's up to me to decide
what the Hamiltonian is.
If I want to invent the simplest
quantum mechanical system.
On the other hand,
there are some things
that should be true.
These are complex
numbers, energies,
H must be an operator
that has units of energy.
And we also saw that if
we want probabilities
that are going to be
associated with PSI
squared to be conserved
we need H to be Hermitian.
There should be some
notion of inner product.
Some sort of operation
that gives us
numbers we used to defy
PSI that gives a number.
To complex numbers
in general, and
has the property of somewhat
conjugates this thing.
It has this, and integrates,
but maybe if you're
doing the simplest quantum
mechanical system in the world
it will be simpler
than an integral.
Integrals are complicated.
But anyway we have
something like that,
and we want H to be Hermitian.
Let me write this in
for any operator A,
this is equal to a dagger Psi.
And that's a
Hermitian conjugate.
That's a general definition,
and we want H to be Hermitian.
H dagger equal H.
OK, in some sense
you could say that's
quantum mechanics for you.
It's a Schrodinger equation, a
Hamiltonian, an inner product,
a notion of Hermitian
operators, and then you're
supposed to solve it.
And what we've done is solve
this for a whole semester,
and try to understand
some physics out of it.
But we started with the notion
that something simple would
be a particle living
in one dimension,
and that's a very
reasonable thought.
Motivated from
classical mechanics
that surely we have
particles that move,
and moving in three dimensions
is more complicated.
We waited towards the
end of the semester
to do three dimensions,
but moving in one dimension
is already kind of
interesting, and complicated.
We had Psi of X that represented
the fact that the particle
could be anywhere here.
How can I simplify this?
The key to simplifying
this is maybe
not to be too attached to
the physics for a while,
and try to visualize what could
you describe that was simpler.
Suppose the particle could only
live at two points X1, and X2.
The particle can
be here, or here.
Now we've re-aligned
down to just two points.
It can only be this
point, or that point.
And you say, that's
very in physical.
But let's wait a second,
and think of this.
What does that mean?
We used to have Psi effects
that could be anywhere,
and we wrote it as a function.
If I think of this
the simplest thing OK,
the simplest thing is a
particle is just at one point.
There is only one point.
The whole world for the particle
is one point, and it's there.
But that probably is
not too interesting
because the particle is there.
The probability defined
there is always one,
and what can you do with It?
But if you have two points
there's room for funny things
to happen.
We'll assume that the
particle can be in two points.
From F of this Psi effects will
go to a new Psi effects that
has two pieces of information.
The value of PSI at x1,
and the value of Psi at X2.
And those are two
numbers alpha, and beta.
Alpha squared would be the
probability to be at the X1.
Beta squared would be the
probability to be at X2.
And this may remind you
already of something we're
doing with interferometers.
In which the photon could be in
the upper branch, or the lower
branch, and you
have two numbers.
This is somewhat
analogous except that
the interferometer
you could eventually
put more beam splitters, and
maybe later three branches,
or four branches,
or things like that.
Here I want to consider
two things, particle there.
One thing that this could be
strictly that, but now let's
relax our assumptions.
It could also mean
for example, if you
have a box, and a partition.
And there's the left
side of the box,
and the right side of the box.
And the molecule can
either be on the left side,
or on the right side.
That's a fairly
physical question.
Here you could be
probability the amplitude
to be on the left, or
amplitude to be on the right.
Two component vector
just like that.
One would be the amplitude to
be in either one, and maybe
that amplitude changes in time.
Or it could be that
you have a particle,
and suddenly you discovered that
yeah, the particle is at rest.
It's not moving.
It's not doing anything.
It's one single
point, not two points.
But actually this particle has
maybe something called spin,
and the spin can be up,
or the spin can be down.
We it could invent something.
We could call it
spin, or a particle
could be in this state,
or in that state.
And if that's possible
for a particle
you could have here the
amplitude for up spin,
and the amplitude for down spin.
And those would be
the two numbers.
It's lots of possibilities
in the sense this
is a classic problem waiting
for a physical application
in quantum mechanics.
Let's push it a little more.
Now how would we
do inner products?
We decided OK, you need
to do inner products.
And what was the inner
product of two functions
phi and psi was the integral,
the X of phi star of X1 times--
phi star of X times psi of X.
And what you're
really doing is taking
the values of the first
wave function at one point.
Complex conjugating it, take
the value of the second wave
function at the same point
complex conjugating it.
If you would have two vectors
like this alpha, and beta
the first wave function.
Alpha one, beta one, and
the second wave function.
Alpha two, beta two.
The inner products
psi 1, psi 2 should
be the analog of
this thing which
is multiply things
at the same point.
You should do the
alpha one star.
That's alpha two plus beta
one star times beta two.
That would be the
nice way to do this.
You could think of this as
having transposed this alpha
one, and complex conjugated it.
Beta one, and then the matrix
product with alpha one,
beta one.
You transpose complex
conjugate the first,
and you do that with the second.
When you study a little more
quantum mechanics in 805
you will explore this
analogy even more
in that you will think of a wave
function as a column vector,
infinite one. psi at zero, psi
at epsilon, psi at two epsilon,
psi at minus epsilon.
So you've sliced the x-axis and
conserved an infinite vector.
And that's all wave function.
It's not so
unnatural to do this,
and this will be
our inner product.
How about H be in Hermitian.
That just means for
matrices that H transpose
complex conjugate that dagger,
Hermitian, is equal to H.
And you may have seen that
that's what dagger means.
You transpose a
complex conjugate.
If you haven't seen it you
could prove it now using
this rule for the inner product
because the inner product
will tell you how to construct
the dagger of any operator.
And you will find that indeed
the dagger what it does
is transposes, and
complex conjugates it.
And it sort of comes because
the inner product transposes,
and complex conjugates
the first object.
