Now let's look at the situation when we
have a source of electrons or photons. So,
let's say it's a source of electrons. It's
an electron gun and again we have, you
know we have the same sort of set up with
the detector in the back here which we can
think of as, you know the screen in the
back is fluorescent so that whenever an
electron hits you get, you get little
burst of light. And now, we can ask what's
the intensity of the electrons arriving
at, at point x? How much light do we see?
How many do we detect? And the, the thing
that happens here is that, well, one thing
that you noticed is as you turn down the
intensity of the source of electrons, you
stop noticing that the electrons start
arriving at point x at discreet points in
time. So, you see a flash and then nothing
for a while and then another flash,
nothing for a while and so on. And as you turn
down the intensity, the flashes don't get
any, any less intense. What changes is the
frequency with what you see with the
flashes. So, again what we have to say is
that, you know that would seem to tell us
that these electrons are really particles.
You know what they are, they are particles
with, you know these charge particles and,
and you know as you turn down the
electron, intensity of the electron
source. The fewer and fewer electrons
going out per unit time and as they go out
and they are not through you know they're
deflected through the edges of these
slits. They randomly show up at some
point x. And the, the probability that you
see in electron here depends upon, you
know, you turned down the intensity of the
source you see them less and less
frequently but they arrive as discreet
objects as discreet lumps. And so, you can
talk about ah, the probability of
detecting the, the electron at point x,
you know, you can call it as intensity
i(x). And so, now, we would, you know,
given that, given that they are like
discreet particles or bullets, what kind of
behavior would be expect. And so, again,
if he, if he open only slit one,
the intensity as a function of x looks
like this. If you open only slit two, the
intensity of the function of x. The
probability that we see, see in electron
looks like this. And, since, since
electrons are behaving like particles like
bullets, you would imagine that if both
slits are open, you should see this curve
which is i12 which is sum of i1 and i2.
But in fact, what you end up seeing is
this interference pattern like we did in
the case of [inaudible]. I12 is not equal
to i1. + i2. And that's the strange thing
about quantum mechanics. So how could it
be if electrons are traveling, if they
are, if they are like particles, if they
are discrete objects, discrete you know,
indestructible objects, how could it be
that when both slits are open, you do not
get to see the sum of these two curves as
the probability of, of the electron ending
up at x. You could say, well, let's reason
about this a little more carefully and
say, well, clearly the electron was fired
through the source. It went through the
source, it called deflected. And then it
either went through slit one or through
slit two. And if it went through slit one,
it ended up at x with probability equal to
i1(x). If it went through slit two it
ended up at x with probability i2(x).
Surely, if both slits were open. It should
end up with x with probability i1 + i2(x).
Because after all if it went through slit
one, why should it matter to it whether
slit two was open or not. And now, the
answer is we don't know but, but when you
do the experiment, you get to see the
interference battle. Now, in quantum
mechanics, we have a way of explaining
this. What we can do is we can say, well,
actually, there's an amplitude which,
which the electrons gets up at one and
ends up at x. And that amplitude is a1(x).
And actually the, the, the probability
that we detect the proton at point x. I(x)
is actually the square of a1(x). And
similarly, there's an amplitude with which
it goes through slit two and ends up at x.
And if only slit two is open, i2(x) is
just the square of this amplitude. And
similarly, if both slits are open, then
a1,2 is a1. + a2. So that the amplitude
with which the photon ends up at x is just
a1(x) + a2(x). And of course the
probability that you've detect before turn
down is a12(x) squared. This is just like
the [inaudible] case where we have the
height of the water wave and the intensity
is the square of the height of the water
wave except that there's no height here.
So what is this amplitude? Well, we don't
know but this is how nature behaves. The
electrons behave as to there was some
amplitude with which it ends up at x and
this amplitude can be positive or negative
leading to this kind of interference
battle. That's the funny thing about
quantum mechanics. That's how electrons
and protons behave. So, let's summarize
what we've learned. So, we did this double
slit experiment three times in three
different settings. First, we considered
it. It was a source of particles or
bullets which we think of as bullets. Then
we repeated this experiment with waves,
with water waves. And finally, we repeated
it with quantum objects like with
elementary particles like photons,
electrons. So, of course, in the case of
bullets, we have discreet objects that
come, you know, bullets come as discreet
chunks ah, as UNS. In the case of waves,
the energy arrives not as discrete objects
but it's, it's continuous. And as we saw
in the case of protons and electrons, they
behave discreetly. They arrive in discreet
chunks which we think off as electrons or
photons which are particles of light so
discreet. In the case of bullets, we
talked about the probability of arrival
at x. In the case of waves, we measured
the intensity or energy. In the case of
electrons or photons, well again we
measure the probability of arrival. Which
we said is proportional to the intensity.
In the case of bullets, when we have both
slits open, we saw no interference. In
this case of waves, we saw interference.
In the case of protons and electrons, we
again, have interference and this is the
funny thing. So, even though photons and
electrons arrive as discreet entities and
we think they should have gone through
either slit one or slit two, we do get the
interference pattern and this is part of
the mystery. This is where we have this
strange behavior quantum mechanically. In
the case of bullets, when both slits are
open, n12 is n1 + n2. In the case of waves
i12 was not equal to i1 + i2 but what we had was
that each one to the height of the wave
did add. And the intensity was the square
of the height. In the case of photons or
electrons, again, we had, i12 is not equal
to i1 + i2. The probabilities did not add
but then we, we came up with this notion
of an amplitude which is just some
inverted notion and said, a1,2 is equal to
a1 + a2. And that the intensity or
probability is just the square of a, and
actually put the square inside absolute
values because in fact, the amplitude can
also be a complex number not just positive
and negative.