PROFESSOR: That brings us to
claim number four, which is
perhaps the most important one.
I may have said it already.b
The eigenfunctions of Q
form a set of basis functions,
and then any reasonable psi
can be written as
a superposition
of Q eigenfunctions.
OK, so let's just
make sense of this.
Because not only, I
think we understand
what this means, but let's
write it out mathematically.
So the statement is any psi
of x, or this physical state,
can be written as a
superposition of all
these eigenfunctions So there
are numbers, alpha 1 psi
1 of x plus alpha 2 psi 2 of x.
Those are the expansion
coefficients with alphas.
And in summary, we say from
sum over i, alpha i psi i of x.
So the idea is that
those alpha i's exist
and you can write them.
So any wave function
that you have,
you can write it
in a superposition
of those eigenfunctions
of the Hermitian operator.
And there are two
things to say here.
One is that, how would you
calculate those alpha i's?
Well, actually, if you
assume this equation,
the calculation of
alpha i's is simple,
because of this property.
You're supposed to know
the eigenfunctions.
You must have done the work to
calculate the eigenfunctions.
So here is what you can do.
You can do the
following integral.
You can do this one, psi i psi.
Let's calculate this thing.
Remember what this is.
This is an integral,
dx, of psi i star.
That's psi.
And psi is the sum over
j of alpha j psi j.
You can use any letter.
I used i for the sum, but
since I put that psi i,
I would make a great
confusion if I used another i.
So I should use j there.
And what is this?
Well, you're integrating
the part of this.
That's a sum.
So the sum can go out.
It's the sum over j alpha j
integral of psi i star psi j d.
And what is this delta ij?
That is our nice orthonormality.
So this is sum over
j alpha j, delta i j.
Now, this is kind
of a simple sum.
You can always be done.
You should just think a second.
You're summing over
j, and i is fixed.
The only case when this
gives something is when j,
and you're summing
over, is equal to i,
which is a fixed number.
Therefore, the only
thing that survives
is j equals to i, so this is 1.
And therefore, this is alpha i.
So we did succeed
in calculating this,
and in fact, alpha i is equal
to this integral of psi i
with psi.
So how do you
compute it now for i?
You must do an integral.
Of what?
Of psi i star times
your wave function.
So in this common interval.
So the alpha i's are
given by these numbers.
This would prove.
The other thing
that you can check
is if the wave function
squared dx is equal to 1.
What does it imply
for the alpha i's?
You see, the wave
function is normalized,
but it's not a function of alpha
1, alpha 2, alpha 3, alpha 4,
all these things.
So I must calculate this.
And now let's do it,
quickly, but do it.
Sum over i, alpha i, psi i star,
sum over j, alpha j, psi j.
See, that's the integral
of these things squared dx.
I'm sorry.
I went wrong here.
The star is there.
The first psi, starred,
the second psi.
Now I got it right.
Now, I take out the sums i, sum
over j, alpha i star alpha j,
integral dx psi i star psi j.
This is delta i j, therefore
j becomes equal to i,
and you get sum over i
of alpha i star alpha
i, which is the sum over i
of, then alpha i squared.
OK.
So that's what it says.
Look.
This is something that should
be internalized as well.
The sum over i of the alpha
i squared is equal to 1.
Whenever you have a
superposition of wave
functions, and the whole
thing is normalized,
and your wave functions
are orthonormal,
then it's very simple.
The normalization is computed
by doing the sums of squares
of each coefficient.
The mixings don't exist
because there's no mixes here.
So everything is separate.
Everything is unmixed.
Everything is nice.
So there you go.
This is how you expand any
state in the collection
of eigenfunctions of
any Hermitian operator
that you are looking at.
OK.
So finally, we get it.
We've done all the
work necessary to state
the measurement possibility.
How do we find what we measure?
So here it is.
Measurement Postulate.
So here's the issue.
We want to measure.
I'm going to say
these things in words.
You want to measure the
operator, q, of your state.
The operator might be the
momentum, might be the energy,
might be the angular momentum,
could be kinetic energy,
could be potential energy.
Any Hermitian operator.
You want to measure
it in your state.
The first thing that
the postulate will say
is that you will, in general,
obtain just one number
each time you do a
measurement, but that number
is one of the eigenvalues
of this operator.
So the set of
possible measurements,
possible outcomes,
better say, is the set
of eigenvalues of the operator.
Those are the only
numbers you can get.
But you can get them with
different probabilities.
And for that, you
must use this plane.
And you must, in a
sense, rewrite your state
as a superposition of the
eigenfunctions, those alphas.
And the probability
to measure q1
is the probability
that you end up
of this part of
the superposition,
and it will be given by
alpha 1 squared, [INAUDIBLE].
The probability
to measure q will
be given by alpha 2 squared
and all of these numbers.
So, and finally, that
after the measurement,
another funny thing happens.
The state that was this whole
sum collapses to that state
that you obtained.
So if you obtained q1, well, the
whole thing collapses to psi 1.
After you've done
the measurement,
the state of the
system becomes psi 1.
So this is the spirit
of what happens.
Let me write it out.
If we measure Q
in the state psi,
the possible values
obtained are q1, q2.
The probability, p
i, to measure q i
is p i equals alpha i squared.
And remember what this
alpha i we calculated it.
This overlap of psi
i with psi squared.
And finally, after finding--
after, let's write
it, the outcome, q i,
the state of the
system becomes psi
of x is equal to psi i of x.
And this is a collapse
of the wave function.
And it also means that after
you've done the measurement
and you did obtain the value
of q i, you stay with psi i,
if you measure it again, you
would keep obtaining q i.
Why did it all become possible?
It all became possible
because Hermitian operators
are rich enough to
allow you to write
any state as a superposition.
And therefore, if you
want to measure momentum,
you must find all the
eigenfunctions of momentum
and rewrite your state as a
superposition of momentum.
You want to do energy?
Well, you must
rewrite your state
as a superposition of
energy eigenstates,
and then you can measure.
Want to measure
angular momentum?
Find the eigenstates
of angular momentum,
use the theorem to rewrite your
whole state in different ways.
And this is something we
said in the first lecture
of this course, that any
vector in a vector space
can be written in
infinitely many ways
as different
superpositions of vectors.
We wrote the arrow
and said, this vector
is the sum of this and this,
and this plus this plus this,
and this plus this plus this.
And yes, you need
all that flexibility.
For any measurement,
you rewrite the vector
as the sum of the
eigenvectors, and then you
can tell what are
your predictions.
You need that flexibility that
any vector in a vector space
can be written in
infinitely many ways
as different linear
superpositions.
So there's a couple
of things we can
do to add intuition to this.
I'll do, first, a
consistency check,
and maybe I'll do
an example as well.
And then we have to
define uncertainties,
those of that phase.
So any question about this
measurement postulate?
Is there something
unclear about it?
It's a very strange postulate.
You see, it divides quantum
mechanics into two realms.
There's the realm of the
Schrodinger equation,
your wave function
evolves in time.
And then there's a
realm of measurement.
The Schroedinger
equation doesn't tell you
what you're supposed
to do with measurement.
But consistency with a
Schroedinger equations
doesn't allow you many things.
And this is apparently
the only thing we can do.
And then we do a measurement,
but somehow, this psi of x
collapses and becomes one of
the results of your measurement.
People have wondered, if
the Schroedinger equation is
all there is in the world,
why doesn't the result
of the measurement come out
of the Schroedinger equation?
Well, people think
very hard about it,
and they come up with all
kinds of interesting things.
Nevertheless, nothing
that comes out
is sufficiently clear
and sufficiently useful
to merit a discussion
at this moment.
It's very interesting, and
it's subject of research,
but nobody has found a flaw
with this way of stating things.
And it's the simplest
way of stating things.
And therefore, the measurement
is an extra assumption,
an extra postulate.
That's how a measurement works.
And after you measure,
you leave the system,
the Schroedinger equation
takes over and keeps evolving.
You measure again,
something happens,
there's some answer
that gets realized.
Some answers are not
realized, and it so continues.
