I want to speak about the subject of statistical mixtures
in quantum mechanics.
This is something that is pretty useful,
and we haven't quite talked explicitly
about in this course.
So what is the question of statistical mixtures?
So suppose you have a spin one half particle.
Now, this applies to other things, of course,
but this is just an example.
Now, this spin one half particle can
be in an infinite number of states.
It's all in a two dimensional vector space,
but in a two-dimensional [INAUDIBLE] vector space,
there are infinitely many vectors.
So it may be in the state psi 1, or it
may be in the state psi 2, or it may be in the state psi 3,
and let's stop at psi k.
So these are many states, a sample of the states,
that the particle could be in.
Now, we've studied in quantum mechanics the case
where the particle is in some particular quantum state,
and then in that quantum state, that
may be a superposition of some basis states,
and you can do some measurements to try to determine
what this quantum state is.
You will need a lot of particles because you
can do just probabilistic analysis
and then see what is the problem with it,
whether you find it up, you find it down.
With many particles, you can do this,
but you are in this state.
But now, let's consider the possibility
that we have a slightly more interesting set up in which
you have this reservoir of states,
but they're not all in state psi 1.
There may be a fraction in state psi 1,
a fraction in state psi 2, and a fraction in all of them.
So how would we describe this?
We would say a fraction P1 of states
are in this-- of particles-- are in this state.
A fraction P2 are in this state.
A fraction P3 in this one, up to a fraction Pk in that one.
And since these are fractions, we
would require, of course, that the sum over i 1 to k of Pi
should be 1.
So this is a slightly different set up.
This set up could exist in many experiments.
You could say, well, half of my particles have spin up,
and half of my particles have spin down,
and I send them through a [INAUDIBLE] machine.
So this is a generalized version of a state in which, well, we
imagine the states, the particles, the set of particles
here involved in this fraction, P1, is in state psi 1.
This fraction P2 is in state psi 2, and so forth and so on.
So this is a new kind of state, a quantum mechanical state.
Maybe you could call it capitals to distinguish it.
So far, whenever you've talked about the state,
you've had a wave function, the ket of some definite state.
That didn't mean that the particle, since it could
be in a superposition of basis states,
there's still all these uncertainties.
But here, there's more uncertainty in the sense
that not all the particles are in this given quantum state,
and not all of them are this, this, and that.
So with this state, you can ask the same questions
you asked before about expectation values
or measurements.
So let's talk about expectation values.
Now, suppose you have an operator A, Hermitian operator
A, and you would like to find out
what is this expectation value.
So you would write expectation value of A
is equal-- suppose you were in the state psi 1.
So you would say, oh, easy.
Psi 1 A psi 1 is what I call the expectation
value of the operator in the state psi 1.
But now, in this new state, you're
in psi 1 only a fraction P1 of the time.
So you should multiply this by P!
and then consider the next one, P2, as a possibility
that there's 2A psi 2 all the way
until you have Pk psi k A psi k.
So this should be the expectation value
to take it as a definition.
And we can write it briefly as a sum from i equals 1 to k
Pi psi i k psi.
So that's the expectation value of the operator
in this new kind of state.
And it's very logical, very sensible
from your ideas of probability and statistics
that that's the way you should define the expectation value.
Now, whenever you have a particle say in this state,
we'll say you have a pure state.
Whenever you have an ensemble in which different fractions take
different quantum states, you have this capital state
is now a mixed state.
Now we can ask another question.
Suppose you want to do a measurement.
So we can try to find the probability.
What is the probability?
[INAUDIBLE] that the particle is in the basis state ej.
So let's call that the probability P sub j.
So what should that be?
Well, if you had before just a wave function,
you will put ej, a single state, you will do this,
and you will be done.
That is the probability that you find
this particle in this state.
But now, you have to do the same thing.
Well, it may be psi 1, psi 2, psi 3.
So let's put an i here.
And now we'll put the probabilities of i
and then the sum from i equals 1 to k.
And that's the probability that we will have
this particle in that state.
That's basically the content of what I wanted to say.
It is at first sight very simple,
but it's also very profound because it
leads to a concept that is the concept of a density
matrix that is subject of [? 806. ?]
But I can't resist saying a few words about it.
And it all begins by rewriting here of these two formulas
that we have obtained.
So we have obtained two formulas,
and now we're going to try to rewrite them.
And for that, we call that if you have something
like a vector u times a vector v,
you can write it as trace of the operator vu.
This is something you did in the homework.
So you can move the bra to the right so this is an operator,
and you can take the trace.
You derive this.
So we can do the same thing here,
and we're going to try to move this thing to the right.
So this would be sum over i Pi.
And now we'll have an A exposed psi i-- well,
I didn't [INAUDIBLE] the trace-- trace A psi i psi i.
So it basically did what this property
said by moving this bra to the right.
And now this is an operator because this is an operator.
An operator times an operator is another operator.
Now, the trace is linear, so the sum
of traces of different matrices is the trace of the sum.
So I can put the Pi in and even put the sum in.
So I'll go here.
This is equal to the trace of A times
the sum over i Pi psi psi i pi.
And this thing now takes a simple form,
which you see in that here's the operator
and here's all the data about the ensemble
of the statistical mixture.
So this whole object is a very important object.
It's called the density matrix.
Let's see if I have some space to write it down.
I'll write it down here.
Density matrix, we'll define it to be
a matrix sum over i Pi psi i psi i rho for density matrix.
And then this becomes nothing else than trace of A
and so on, a very beautiful thing.
In fact, this probability Pj we have here--
so this is a very nice formula-- this probability Pj can also
be written just in terms of the density matrix.
For that, I like you to do it-- think
of this inner paren squared as the bra-ket times the ket-bra,
and then manipulate it.
And you will see that, after a little work,
this Pj becomes nothing else than the expectation
value of the vector Pj, the density matrix.
So both of the observables we can have here
can be expressed in terms of the density matrix.
So the density matrix is a very nice construct,
and it summarizes everything about your statistical mixture.
Notice that the density matrix in the case of spin 1 half
particles is nothing else but a Hermitian operator.
So it's a two by two matrix, and it's characterized
by four real quantities.
Two by two Hermitian matrices are characterized
by four real quantities.
So all the information about this infinitely many states
you could put here in the mixture at the end of the day
is boiled down into four numbers,
which are the elements of the density matrix.
So here, you've taken a little peek
into something you will see in the future,
and something that is a very natural extension
of the concepts of quantum mechanics.