We spoke about
superposition, and we showed how, when you have two
states that are superimposed, the resulting state
that is built up doesn't have properties that
are intermediate between the two states that you're
superimposing. But rather, when you
do a measurement, you obtain the result
that you would sometimes-- you sometimes obtain
the result that you would have with
one of the states, and some other times with
different probabilities, the result as if you
had the other state. So it's a strange kind of
way in which things are combined in quantum mechanics. So the next thing we have to
say is a physical assumption that is made here. And it is that if
you have a state and you superimpose
it to itself, you haven't done anything. So the superposition
of a state with itself has no physical import. So we can say this. A physical assumption
superimposing a state to itself does not change the physics. So if I have a state, this
is physically equivalent-- I'll write physically
equivalent with this symbol-- to the state a plus a,
which would be 2 times a. It's physically equivalent
to the state minus a. It's physically equivalent
to the state ia on anything. It's not equivalent
to 0a, because that would be the zero state. So it's physically
equivalent as long as you have a
non-zero coefficient. All these states are supposed
to be physically equivalent. And that will eventually
mean that we sometimes choose a particular one in
those collection of states that is one that is
convenient to work with. And that will be called a
normalized state, a state that satisfies other
properties having to do with the norm
squared of the state. That will come later. But it's important
that the number that is multiplying the physical
state of your system has no relevance. And you could say, well, why all
of the sudden you tell us this. Could this be shown
to be necessary? Or it's a physical
assumption, so can we test it? Does it make some sense? And we can make some sense of
this assumption at this level. And we do it with
states of light. So remember, we spoke about
photons hitting a polarizer. And we could speak of two
independent kind of photons-- photons polarized along the
x-axis and a photon polarized along the y-axis. And those are two quantum
mechanical states. Now suppose I decide
to superimpose those states to create the
most general photon state. I would have an alpha,
which is a number here, a complex number,
and a beta there. And I would say, OK, here is
my most general photon state. And how many parameters
does this state have? It has two complex parameters,
alpha and beta, and therefore, four real parameters. And then you think about
polarization states, how many parameters they have. And as we'll review in a second,
it's well known that photons-- their polarization
state can be expressed with just two real parameters. So some counting is not
going very well here. But here comes the help. If the overall coefficient
here doesn't matter-- if I can change it, I
can multiply everything by 1 over alpha, and therefore
get that the state is just the same, physically
equivalent to this state, beta over alpha photon y. So all the physics is contained
in this state as well. And if all the physics is
contained in that state, I must look how many
parameters it has. It still looks like
there's two numbers here, but only the ratio appears. So if you call beta over
alpha, the number gamma is just one complex parameter. And therefore, thanks
to this assumption, you now get that the most
general photon polarization state has just one
complex parameter, or just equivalently, two
real parameters. And that is the correct number. Indeed, if you have a
polarization, a wave that has some polarization, the
most general polarization state of a wave is an
elliptical polarization. You probably did study a lot
about circular polarizations, or maybe you also heard
about the elliptical one in which the electric field--
in a circular polarization, the electric field at any
point traces a circle. But if you have an
elliptical polarization, the electric field
traces an ellipse. And that ellipse has an
angle that is one parameter. And for an ellipse, the other-- the size doesn't matter. The size depends
just on the magnitude of the electric field. It's not a parameter of the
polarization of the wave. Since the size
doesn't matter, it's the shape of the
ellipse that matters. And that's characterized by the
eccentricity or by the ratio a over b of the semi-major axis,
so parameters, two parameters, and they are a over b and theta. So an elliptically
polarized wave, which is the most general state
of polarization of a wave, has two real parameters. And now, thanks to this physical
assumption, we get this right. And this is important
because that's something we're going
to use all the time, that the overall factor in a
wave function does not matter. So if we have
superpositions, I want to emphasize one more
thing about superpositions. And for that, I'm
going to use spins. So what is spin? Spin is a property of
elementary particles that says that actually,
even if they're not rotating around
some other particle, they have angular momentum. They have intrinsic
angular momentum, as if they would be made
of a tiny little ball that is spinning. I say as if because
nobody has ever constructed a model of
an elementary particle where you can really make
it spin and calculate how it works. Somehow, this
elementary particle has angular momentum is born. Even if it is a point particle,
it has angular momentum, and it's spin. And spin is very
quantum mechanical. And we can't quite
understand it without it. So what happens is that you can
measure the spin of a particle. And then if you
measure it, you have to decide, however, since
angular momentum is a vector, what direction you should use. And suppose you
use the z direction to measure the
spin of a particle. You may find that the
particle has either spin up or the particle has spin down. Spin. And the spin is the direction
of the angular momentum. And that's a funny thing
that happens with most matter particles. These are spin 1/2 particles. The spin can be up or it can
be down along the z direction that you measure. You measure it, and you never
find it's 0 or a little bit. It's just either up
with the full magnitude or down with the full magnitude. That is a spin 1/2 particle. And the state where it is
up, we sometimes denote it with an arrow up and call it
z because it's up along z. And this would be down,
an arrow down along z. If those are possible
quantum states, you could build a
new quantum state by superposition which would be
up along z plus down along z. Now, if I wish to
normalize it, I would put the factor
in front of this. I will not talk about
normalizations at this moment. They're not so important. If you are faced with
this quantum state-- so suppose you have an electron
that is not in this state nor in this state, but is
in this state, in a quantum superposition. So you go and you decide
to try to measure it. Now, since you cannot predict
what that electron is going to be doing-- we cannot predict things
in quantum mechanics with certainty-- we, since we're going
to do this experiment, avail ourselves of 1,000 copies
of this electron, all of them in this peculiar quantum state. So you have the 1,000 copies,
and you start measuring. And you decide to measure
the spin in the z direction. And now what do you get? Well, we mentioned last time
that you don't get an average, or since this is up and
this is down, you get 0. You measure the first
particle and you find it up. Measure the second, up, the
third, up, the fourth, down, five, down. And then you get a
series of measurements. At the end of the 1,000
particles, you find about 495 up and 505 down,
about half and half. And if you did it
with 10,000 particles, maybe it would be closer. Eventually, you'll
find 50% in this state and find 50% in this state. And if you think this is
strange, which you probably do, well, you could be justified. But here would
come Einstein along and would say all this
stuff of this superposition is not quite right. You had this 1,000 particles. But actually, those 1,000
particles, half of them were with a spin up and half
of them were with a spin down. So here you have
your 1,000 particles, your quantum state, this. But Einsten says, no, let's make
an ensemble of 1,000 particles, 500 up, 500 down, and
do the same experiment. And the result is
going to be the same. So how do you know you
really have this as opposed to somebody has given you 1,000
particles, 500 up, 500 down? How can you tell? And in fact, he
would say even more-- whenever Einstein
used the word realism to say if I measure a
spin and I find it up, it's because before I
measured it, the spin was up. It's almost like learning
something about an object. If I look at this page
and I find the color red, it's because before I
looked at it, it was red. But then in quantum
mechanics, that doesn't seem to be the case. The state is this mix. And it was this mix
before you measured. And after you
measure, it's this. So there is no such
thing as you learn by doing one measurement what
the state of the particle was. So we will not resolve
Einstein's paradox completely here because we would
have to learn more about spins, which you will do soon enough. But here's the catch
that actually happens. If, instead of having an
ensemble of quantum states, you would have an ensemble of
those states that half of them are up and half
of them are down, you could now decide to measure
the spin of the particle along the x direction. You take these particles,
and you measure along x. And what you will calculate
with quantum mechanics later in this course-- if you measure along
x, in this state, you will find all of them to
be pointing along plus, up along x, all of them. While on this Einstein ensemble
of 50% up and 50% down, you would find 50% up along
x and 50% down along x. So there is an experiment
that can tell the difference, but you have to look
in another direction. And that experiment,
of course, can be done. And it's a calculation
that can be done, and you can decide whether
these quantum states exist. And they really seem to exist.
