PROFESSOR: Superposition is very
unusual and very interesting. Now we've said
about superposition that in classical physics, when
we talk about superposition we have electric fields, and
you add the electric fields, and the total electric field
is the sum of electric fields, and it's an electric field. And there's nothing
strange about it. The nature of superposition
in quantum mechanics is very strange. So nature of superposition-- I will illustrate it in a
couple of different ways. One way is with a device that
we will get accustomed to. It it's called the
Mach-Zehnder interferometer, which is a device with
a beam splitter in here. You send in a beam of light-- input- beam splitter
and then the light-- indeed half of it gets
reflected, half of it gets transmitted. Then you put the mirror here-- mirror 1, you put
the mirror 2 here, and this gets recombined
into another beam splitter. And then if there would
be just a light going in, here there would be
two things going out. There's another one
coming from the bottom. There will be two. There will be interference. So you put a detector
D0 here and a detector E1 here to detect the light. So that's the sketch of the
Mach-Zehnder interferometer-- beam splitters and mirrors. Take a beam, spit the light, go
down, up, and then recombine it and go into detectors. This was invented
by these two people, independently, in the 1890s-- '91 to '92 apparently. And people did this with light-- beams of light before they
realized they're photons. And what happens with a beam
of light-- it's interesting-- comes a beam of light. The beam splitter sends half
of the light one way half of the light the other way. You already know with
quantum mechanics that's going to be probabilistic
some photons will go up maybe some photons will go down
or something more strange can happen. If you have a
superposition, some photons may go both up and down. So that's what can happen
in quantum mechanics. If you send the beam,
classical physics, it divides half and
half and then combines. And there's an
interference effect here. And we will design this
interferometer in such a way that sometimes we can produce
an interference that everything goes to D0 or
everything goes to D1 or we can produce suitable
interferences that we can get fractions of the
power going into D1 and D2-- D0 and D1. So we can do it
in different ways, but we should think of
this as a single photon. Single photos going
one at a time. You see, whatever light you
put in here, experimentally, the same frequency
goes out here. So what is interference? You might think, intuitively,
that interference is one photon interfering with
another one, but it can't be. If two photos would interfere
in a canceling, destructive interference, you will
have a bunch of energy. It goes into nothing. It's impossible. If they would interfere
constructively, you would add the
electric fields and the amplitude would be
four times as big because it's proportional to the square. But two photos are not
going to go to four photons. It cannot conserve energy. So first of all, when you
get light interference, each photon is
interfering with itself. It sounds crazy, but it's
the only possibility. They cannot interfere
with each other. You can send the
photons one at a time and, therefore, each
photon will have to be in both beams
at the same time. And then, each photon
as it goes along, there will be an
interference effect, and the photon may end
up here or end up there in a probabilistic way. So you have an example
of superposition. Superposition. A single photon
state a single photon is equal to superposition of
a photon in the upper beam and a photon in the lower beam. It's like two different states-- a little different
from here, you had photons in two
different polarizations states superposed. Here you have photons
in two different beams-- a single photon is in both
beams at the same time. And unless you have that, you
cannot get a superposition and an interference that is
consistent with experiment. So what does that mean
for superpositions? Well, it means something
that we can discuss, and I can say things
that, at this moment, may not make too
much sense, but it would be a good idea that you
think about them a little bit. We associated
states with vectors. States and vectors
are the same thing. And it so happens that
when you have vectors, you can write them as
the sum of other vectors. So the sum of these two
vectors may be this vector. But you can also write it as
the sum of these two vectors-- these two vectors
add to the state. And you can write any vector
as a sum of different vectors, and that's, actually,
quite relevant. You will be doing that
during the semester-- writing a state a superposition
of different things. And in that way
you will understand the physics of those states. So for example, we can
think of two states-- A and B. And you see, as I
said, states wave functions, vectors-- we're all calling
them the same thing. If you have a superposition
of the states A and B, what can happen? All right, we'll do
it the following way. Let's assume if you
measure some property on A, you always get value A. So you
measure something-- position, momentum, angular momentum,
spin, energy, something-- on A, it states
that you always get A. Suppose you measured the
same property on B. You always get B as the value. And now suppose you have a
quantum mechanical state, and the state is
alpha A plus beta B-- it's a superposition. This is your state. You superimpose A and B. And
now you measure that property. That same property you
could measure here, you measure it in your state. The question is,
what will you get? You've now superimpose
those states. On the first state, you always
get A; on the second state, you always get B. What do you
get on the superimposed states, where alpha and
beta are numbers-- complex numbers in general? Well the most, perhaps,
immediate guess is that you would get
something in-between maybe alpha A plus beta B
or an average or something. But no, that's not what
happens in quantum mechanics. In quantum mechanics, you always
get A or you always get B. So you can do the
experiment many times, and you will get A many times,
and you may get B many times. But you never get
something intermediate. So this is very different
than in classical physics. If a wave has some
amplitudes and you add another wave of
different amplitudes, you measure the energy you
get something intermediate. Here not! You make the superposition and
as you measure you will either get the little a
or the little b but with different probabilities. So roughly speaking, the
probability to get little a is proportional to the number in
front of here is alpha squared, and the probability
to measure little b is proportional to beta squared. So in a quantum superposition,
a single measurement doesn't yield an average result
or an intermediate result. It leads one or the other. And this should
connect with this. Think of the photon we
were talking about before. If you think of the photon
that was at an angle alpha in this way, you could say
that the polarizer is measuring the polarization of the object. And therefore, what
is the possible result it may measure the
polarizations say oh, if it's in the x direction
you get it right, and what is the
probability that you get it to be in the x direction is
proportional to cosine squared alpha-- the coefficient
here squared. So the probability that
you find the photon after measuring
in the x direction is closer in squared
alpha, and the probability that you'll find that here
is sine squared alpha. And after you measure,
you get this state which is to say the following thing. The probability to get the
value A is alpha squared, but if you get A,
the state becomes A because this whole state
of the system becomes that. Because successive measurements
will keep giving you the value A. If you get B,
the state becomes B. So this is what is called
the postulate of measurement and the nature of superposition. This is perhaps the
most sophisticated idea we've discussed today, in which
in a quantum superposition the results are
not intermediate. So when you want to figure
out what state you have, you have to prepare many
copies of your state in this quantum system and
do the experiment many times. Because sometimes you'll
get A, sometimes you'll get B. After you've
measured many times, you can assess the probabilities
and reconstruct the state.
