In the 
last lecture, we were looking at a second
kind of a thought experiment where we were
taking a laser light source and then putting
in a shutter to control the light, the appearance
of the light after the shutter and we detected
it using a detector.
If you take a look at the schematic of the
last lecture and then continue that thought
experiment, all in the idea of trying to localize
the photon in time okay.
Now if we look at the schematic here, what
we see is that we 
have a laser light source, sending in the
light and then we have a shutter okay.
The shutter exists in 2 states, on state and
then off state, and whenever the shutter is
off, you have the light coming into the detector,
when the shutter is on, there is no light
okay.
If I plot the intensity of the light as a
function of time while I am actually operating
the laser, you would expect the light intensity
to fluctuate somewhat similar to this.
Similar to the blue line, where somewhat what
we would expect is to get this blue lines
depending on the lasers turning off and on,
right.
So now what I was asking or what we were talking
about is that this corresponds to the shutter
turning on and off with a time period indicated
by the blue line.
However if I start making the shutter go off
and on much more frequently, right, what happens
to that?
Is there a limit where I start operating the
shutter so really really fast beyond which
this does not obey?
If it does not obey, what happens to this?
It turns out that yes there is a limit, that
is if barring mechanical inabilities notwithstanding,
suppose we have since this is thought experiment,
I can actually have a shutter which can operate
at any speed.
So if you take a shutter that can close on
and off really really really fast down to
few picoseconds and femtosecond what you will
see is that we are turning on the shutter
and going off down to let us say microseconds,
nanoseconds, and then to picoseconds.
What you will see is that the 
light will not appear after the shutter, so
even though you are turning off and on, the
shutter now will act like a effective light
block, it does not let the light pass through,
why is that and can we actually estimate at
what time, I mean what are the parameters
that determine this or that governs this?
It turns out the answer is yes and it boils
down to the same fundamental mechanical principles
that we talked about and the first one being,
I mean that the principle first one being
the position and the momentum.
We looked at the position on the momentum
and then said we can use that to determine
what is the localization accuracy for a given
photon is, in here we are going to use uncertainty
principle in a different form.
The form that we are going to use is connecting
the uncertainty in the measurement of energy
delta E with the uncertainty in measurement
of time, alright.
So these two have to be they are coupled and
the product of uncertainty will be greater
than our uncertainty principle, the constant
h by 4 p.
Now what does it mean with respect to the
experiment that we just said?
We know that the light photons when they are
actually coming out from the laser have a
definite energy given by each and every photon
that is coming out from.
The laser will have an energy given by E = h
nu, where nu is your frequency of light of
electromagnetic radiation or the light radiation
we talked about, okay, and it is convenient
to think in terms of lambda, so you can actually
recast that into hc by lambda where c is the
velocity of the light and lambda is the wavelength
of the light.
So now this whole expression you could think
in terms of the delta E per se, you can think
in terms of delta E is h delta nu.
As a result if you substitute back into this
original expression, what we will have is
h delta nu times delta t will be greater than
or equal to h by 4 pi.
Now you can see the uncertainty associated
with the time, the time localization, right,
is very tightly linked to the energy itself
and in turn is related to the color of the
light.
So this in turn can be simplified and then
we can write it as delta nu or actually delta
t here.
We are interested in asking how narrow in
time can we localize the photon?
Can we measure a given photon being there?
Right.
So delta t so that is what this means right.
Remember the graph that we saw or coming out
from the photo detector.
The intensity goes up like this and then comes
down.
So what you are actually doing is we are actually
localizing the photon, this is the intensity
here.
We are localizing the photon within this time
window.
So that would correspond to my delta t.
What this tells you is that this delta t has
a lower limit, I mean it cannot be any lesser
than, that is it has to be greater than or
equal to, at the most it can be equal to,
no h here because h gets canceled, so is equal
to 1 by 4 pi delta nu, okay.
If you want to localize it better and better,
then you have to give away the ability to
tell which color photon that you are actually
looking at okay.
The delta nu has to go up.
What we call this delta nu as the bandwidth
okay of the light source.
What it tells you is that, we took the light
to be the HeNe helium-neon, right.
So let us say if it is a red HeNe 632, that
is operating at 632 nanometer, we need to
ask how well do we know this 632 nanometer
and it turns out this can be pretty narrow,
right.
If you plot the again intensity versus wavelength
here, so wavelength corresponding to the,
my wavelength measured in nanometers, so you
will see it is pretty sharp and then the spread
that we measure would be, we know this pretty
accurately to about 1 in 1000 part of the
632 nanometer, let us say to accuracy of about
0.001 nanometer okay.
Now please note this is in the units of wavelength
lambda and in order to estimate how small
time localization can be, we need to express
that in terms of delta nu, the frequency,
so that is when we have written our expression
in terms of frequency, we can also write it
in terms of delta lambda, so let us do that
first.
So we have delta nu is basically you can write
it as C by lambda 2 difference lambda 1.
So what we generally tend to do is that we
take the central lambda 
and c and delta lambda, where delta lambda
is lambda 2 minus lambda 1.
So if we express the central lambda here would
be in our example it would be equivalent to
632 nanometer for red helium neon laser in
our example okay and the delta lambda is about
0.001 nanometer okay.
If you plug these things in and you can go
ahead and calculate, and it will turn out
to be of the order of about a gigahertz or
so, okay.
So this in turn we can actually plug in and
then you can estimate the delta t and if you
calculate that, that delta t will come down
to about 1, again we are talking in terms
of the order of magnitude here, this many
seconds, so about nanoseconds or so.
So you can operate the shutter such that the
shutter is on only for about a nanosecond
or so until that point, you go to microsecond,
you go to hundreds of nanoseconds, until that
point you would not feel any effect of this,
but once you start operating it down to about
a nanosecond are even lesser right, like a
picosecond or something.
Assuming that you can actually operate a shutter
in that high speed, you would start to see
no light coming out of the shutter if we were
to use a helium-neon laser that is having
a bandwidth of about 0.001 nanometer, that
is 1 in 1000.
So if you know it very very few, I mean the
wavelength, the color is extremely pure, then
your ability to form a short pulse is limited.
I mean we do not see it in a normal life because
nanoseconds and picosecond, we do not go down
to that in a regular basis.
To just give you a example camera flash that
you operate, that is one of the fastest light
pulse that you see in a day to day life.
The fastest one of late you can if you get
buy it in a market is about 1 in 10,000 of
a second okay.
A regular flash without any extra visions
or such it is either you get it at 1 by 60
or 1 by 90 of a second.
So compared to that, 10 raise to 9, 10 raise
to minus 9 seconds is tremendously low and
that is the reason why you do not see it and
secondly in a camera flash what you are actually
operating on is a white light.
It has a lot of color, so it is a pretty broad
spectrum and hence the bandwidth.
We will use this very principle to actually
when we actually start probing the system
at very fine time scale are using high intensities
because this is one way of actually generating
high intensities and yet keeping the overall
photon load to be small is be able to localize
the photons in time or can think of that as
a time focus.
There is a big bust of photons, but only for
a very short extremely brief period of time
and nothing at all for a long period of time,
right.
That is one trick that we can use to minimize
the photon load unto the sample which often
is a concern, but at the same time, we can
probe that with a very high intensity because
momentary instantaneous intensities in these
situations can be extremely high okay.
So if we have to generate a laser light of
that kind , then we need to have this in mind
and how do we generate them using the lasers
and all that we will be seeing in detail in
course, but my point here is to say these
principles, the quantum mechanics principles
are not one of esoteric existence.
It is not like you want to study it you know
using a pen and paper and it is really not
of much use, it is not of that kind but it
is actually very much applicable in the course
that we are actually doing and we do use it
and in fact quite often you will hear things
like okay what is the bandwidth of the laser
and what is your spot size of the focus beam
that you have focused in the microscope and
things like that.
For this, the governing principles start from
quantum mechanics, that is my point and it
is important to understand it not to the greatest
detail but sufficient enough for what we need
to do it in the course.
So this is all boiling down to one of the
ideas that Dirac had put forward which is
the absoluteness to the size which led to
2 principles, I mean which led to the uncertainty
principle, I kind of led you there and using
that I have looked at 2 examples and then
shared that how they are very much related.
Now the second principle that Dirac has formulated
is about superposition of states.
I am talking about this second for quite a
good reason because we will be dealing, I
mean we will be talking about this using an
example and then write after that immediately
we will use that in our formulation of obtaining
what an expression to describe what happens
when light interacts with the matter okay.
So what is this superposition principle?
So, let us look at that.
So what Dirac said is that in order to describe
matter, we would have to develop a new formalism
and in that formalism what he used is that
there can be different states the matter or
then object or or anything that you want to
describe can exist.
These states are represented by a vector,
let us call that, I mean he used a notation
called bracket notation, this essentially
he took a bracket like that and he split in
half and then said basically these form a
vector space, ket vector space and the bra
vector space.
Each of these vectors would represent a state
of a system okay.
So in his description, this is what he is
going to say, we will give a name, so let
us say and this is some g and this is some
e of whatever it is.
So this for him a or g would represent state
of a system, okay.
What he realized is that the state of a system
have unique properties in that you would be
able to use general principles of linear algebra
to express various different operations or
various different measurements that you can
actually make out of the system.
For example if you want to know what is the
polarization of a photon that I am actually
looking at, okay.
So in this example, my measurement process
is measuring the polarization itself and then
the observable is one of the following which
is you would like to collapse them into, so
let us say there is a light source 
okay and this light source passes through
filter okay.
These represent, these are special kind of
filters, let us call them as polarization
filters, so I am going to change that into
a circular object.
The properties of this filter is such that
the photons that are coming out from here
once it passes through this filter will be
of having a particular polarization okay.
I mean that is defined with respect to the
axis of this crystal axis.
Optical crystal is nothing but any substance
that has some unique optical properties, in
this case it is actually exhibiting a specific
directional property wherein we see that the
light photons that are coming out after passing
through these filters have defined polarization
okay.
What the polarization itself is you can right
now think of that as a simple property, but
later on, we will see what that could be and
all that stuff, so but right now if you do
not know what the polarization is, just think
of that as an inherent property of a light
photon, okay.
So given that what we can actually see is
that so this polarization is defined with
respect to the crystal axis, once you define
it with respect to the crystal axis, then
what we can actually do is that we can probe
or we can ask with a different filter.
Basically rotate this filter 
to about 90 degree, then the property of the
polarization is such that when this direction
and this direction are mutually orthogonal,
meaning they are different by 90 degree you
get no photons at all.
However, if you have the same light source
and then we pass it through same filters,
now if you had to probe with filter whose
orientation is parallel to the original filter,
I would get the intensities here and here,
right, the I, so let us call this as I input
and let us call this as I1 and let us call
it as I0.
I0, I1, and I2.
You will see that the I1 is approximately
equal to I2, which is the number of photons
coming out from here, I mean there is hardly
any change in the photon intensity, albeit
some reflection losses and absorption losses,
but more or less, you will have the maximum
pretty good transmission efficiency of this
filters here, okay.
However, if I do this which has so same I1
and then I point my filter at an angle alright,
so then the I2 seem to have a definite relationship,
even the other one it will explain but I2
seem to have a definite relationship with
respect to the I1 which basically is proportional
to cos square alpha.
Meaning when it is completely aligned, you
have I1 almost equal to I2, when completely
orthogonal, nothing comes out, but somewhere
in between proportional to the angle, proportional
to the cos square of that angle, you have
this thing, right.
That is, this is a more general statement
of the special cases that I have told you,
but the cos square representation is more
general.
So this is no surprising at all, it is pretty
simple and one can explain this as just a
property of the filter and doing that.
However, what becomes perplexing is that let
us take a situation where you have the light
that went through this polarizer and I have
another filter, right, which is like this,
and according to our description you would
see that there is no light at all coming out,
right, I2 would be pretty close to 0, right.
Now if I were to add another filter here okay,
that is at an angle, you suddenly start to
see light coming in from there.
Now in this new scenario, I2 is not equal
to 0, and in fact, I2 will be much much larger
than this blue I2 that I have drawn here.
Now how do you explain this?
So, there is nothing fancy here, you can actually
think of this as a series of measurements,
so clearly while it can look paradoxical when
you say that by adding an extra filter you
are suddenly seemed to be having increased
the intensity throughput.
But what actually you are doing is that you
would be able to say that by putting in this
filter, you have generated some amount of
intensity here, alright.
Now that photons have a peculiar behavior,
peculiar character which let them pass through
this extra filter.
Now how do you get that peculiar character?
Classical, I mean there is simple ways of
explaining it through wave theory of light,
but you will see that the wave theory fails
when you have to explain the fundamental nature
of the light itself, either be photoelectric
effect exhibited by PMTs day today in our
life will be using that to detect light very
regularly later on in the course.
So to be able to have a description that is
consistent throughout where we do not have
to switch at whims and wills of what the experiment
it is.
So it turns out that Dirac had a very relaxed
formalism seem to provide a pretty nice continuum.
Wit, that I hope that we will see you in the
next lecture and we will talk more on the
Dirac’s bra and ket and their properties
and how we can use to explain the polarization
dependence.
Thank you.
