My name is Ramadevi.
I am a professor at department of physics
IIT, Bombay.
So I have been teaching this quantum mechanics
course.
The tools of quantum mechanics are essential
to perceive the microscopic world of atoms
and nucleus.
The main theme of this quantum mechanics course
is to give exposure to direct bra-ket notations.
Main emphasis will be on operator formalism
and their applications to harmonic oscillator,
hydrogen atom spectrum.
Every 4 lectures will be followed by a tutorial
session by my colleague, Dr. Jai More who
will discuss in detail some problems.
This will benefit audience to appreciate the
applications of the theory, lectures which
we will be uploading.
We will upload PDF and PPT files of video
of every lecture prior, so that these files
can be opened for clarity.
The references are the first 2 books.
The third can be kept as kind of a reference
book.
It is really a pad book.
So I would consider it to be like a reference
book.
It has everything.
But I will prefer the first book to be followed
by you which is Griffiths and the other books,
Sakurai then slowly get on to Sakurai also.
Shankar, you can use it as kind of a reference.
Okay, so this is the plan.
I want to take you slowly from what you have
learnt in your first year, into the second
year, first course, quantum mechanics.
So that will be review of quantum ideas.
Then linear vector spaces, operators, state
vector formalism of Harmonic oscillators.
I am going to concentrate on hydrogen atom,
angular momentum and then the concept of spin,
addition of angular momentum.
How does one add to angular momentums?
Then these things will force us to introduce
a coefficient which is called as a Clebesh-Gordan
coefficients.
So we will spend some time in looking at these
aspects at length, okay.
So this is the main things which will be done.
There will be some miscellaneous, okay.
So to start the introduction, why do you want
to do quantum.
This you have already seen in your first year
course that there are limitations of classical
ideas, right.
So what is the first one?
You have seen microscopic world.
Many of the experiments, you cannot explain
using your familiar Newton's law of, or any
classical law.
So the first thing which you would have seen
in the Black-Body spectrum which is the Planck's
law which helped you to fix exactly all the
points in the Black-Body spectrum.
Is that right?
Everybody is with me?
And then photoelectric effect cannot be explained
just using classical laws.
The more intense the beam, more photoelectric
current will flow is not correct, right.
You all know.
So other thing is the scattering of, which
is called as Compton scattering of photons
with electrons, which also you have seen,
how the wavelength changes.
And these things can be explained only using
quantum ideas, not classical ideas.
So one of the core important thing which went
into all the 3 aspects of quantum aspects
of photon, what is that?
Is the universal Plank constant?
You did not put a constant which was different
for Black-Body spectrum, did not put a different
constant for photoelectric effect for a different
constant for Compton scattering.
Is that right?
You all knew it is the same h, it gave you
an interesting, that is why it is universal.
The same h which you started with the Planck
constant which was introduced by Planck to
say that the energy of photons are quantized
in quantas of integral multiples of h nu.
That h is the same h which happens even in
photoelectric effect or Compton scattering.
It may be mechanically using it but you know
it is the universal nature, feature which
shows up in that Planck's constant which is
called as a Planck's constant h1, okay.
You do not use different values, use the same
value.
So needed introduction of a fundamental constant
leading to Quantum regime.
So h tending to 0 is a limit where you start
getting to the classical, okay.
So h not equal to 0 is the one which we call
it is the Quantum regime, okay.
So this is also familiar to you, Planck's
equation, E=h nu.
And you can use Einstein's relativistic energy
for massless particle.
What is that?
E=pc and you can try and write the momentum
of a photon to be h/lambda.
Everybody is familiar with this, right.
So for massless photon, E=pc which is h nu.
Is it visible?
And from here, you can write p=h nu/c=h/lambda,
right.
So this is what the first equation E=pc is
for the massless Einstein's energy momentum,
relativistic equation and Planck's hypothesis
is E=h nu from which you can show that the
momentum is h/lambda, okay.
So that is what I have written here.
So what was the proposal of de-Broglie?
He said why should it be that momentum, E=h/lambda,
it should be applicable for other particles
as well.
That was the daring proposal.
He said p=h/lambda should be true even for
other particles, like electron, protons and
so on.
But E-pc is not correct.
If you can write momentum to be h/lambda,
is the de-Broglie hypothesis.
Which means that there is a wavelength associated
with other particles, treat them as waves,
okay.
This p=h/lambda is the crucial point which
tells you that you can look at any object
both as particle, look at particle, you would
talk about momentum.
If you look at wave, you will talk about wavelength.
And this is that equation which relates both,
okay.
So this is the daring proposal of de-Broglie.
Equation of wavelength of matter waves must
obey the same equations of photon momentum,
okay.
So matter waves, once you start looking at
matter waves, you can look at them moving
on an orbit and look at the circular orbit
with constructive interference and the constructive
interference will give you a condition for
a wave as the circumference to be for the
integral multiple of lambda.
And once you substitute, the de-Broglie proposal
for lambda, you can also rewrite angular momentum
quantization.
This is your famous Bohr quantization which
you would have read as a proposal.
I am just trying to say that once you start
taking it as a wave and look at constructive
interference on a circular orbit, you automatically
get quantization of orbital angular momentum.
These are the famous experiments, Davisson-Germer
and these show that you can also look at electron
beam and dissections of the electron beam.
And in that sense, it confirms what de-Broglie
said that there is a wave nature for particles
like electron.
Usually, when we are given an electron or
when are given a photon, we see photon as
an electromagnetic light.
We see electron as a point particle.
But in this course, you should remember that
a particle and the wave are part of every
object and what we try to tap is what you
will get, okay.
So depends on the situation.
If you are doing photoelectric effect, you
will get only the particle nature of photon.
If you are doing interference experiments,
you will get wave nature of photon.
Similarly, electrons, if you doing collision
experiments in your classical physics, it
is like a particle nature.
But if you are doing dissection experiment,
it shows the wavelength, okay.
Depending on the experimental setup and its
energy, things are going to be different and
that is what is happening in the microscopic
world.
What we see in our day to day life, is the
microscopic effect.
But what is happening inside an atom or inside
a proton, are governed by quantum ideas, okay.
You cannot use your classical ideas and say
that what I see here should happen in the
microscopic world.
We do see violations, that is why we started
introducing the quantum ideas.
Like for example tunnelling, I cannot just
go over, across this wall outside but you
can still get, if it is a wave, a sound wave
could cross across, right.
That is once it is a wave, things can cross.
That information can be there in a region
where a particle cannot go out but a wave
can go out, okay.
This is your famous experiments which you
would have seen Young double slit experiment,
right.
And here you have 2 sources and if suppose
you shut one of the source, you will have
this blue line for light coming through S1,
similar intensity.
Similarly, if I shut S1, you will have a green
line due to the light coming through the source
S2, the slit S2.
But if both are open, okay, what is your expectation?
It has to be like the wave.
This is what you will expect, constructive
dyscalculia indulgence.
But if you try to see which slit through which
it went?
Then what happens?
The interference pattern disappears.
You will get, if you try to see how, which
way it goes?
Then this is the pattern.
So which way it goes if you are trying to
see, then you are looking for the particle
nature.
Trying to look for the particle nature, it
is just the sum of the intensities.
If you do not worry about which slit it went
through and just look at the screen, it will
have a superposition.
Not intensity adding.
It is the amplitude added and then what spread
of it and you will get an interference, okay.
So this is the experimental data, they have
tried to find out by closely putting a, kind
of, detector here to see whether the beam
is coming through this or coming through that
in that instant of time which they try to
do that, then they will get the screen B.
They do not try to do that by putting some
detectors here, disturbing the system.
They will get this screen C. So you can get
wave nature if you want to tap the particle
nature, the interference pattern disappears,
okay.
So it was done even for the electron beam,
not only for the photon electromagnetic field.
Of course, you could do this but this is done
for the electron beam and it was proven that
it has wave nature besides being a particle.
So attempting which slit the electron came
from gives plot B which is a particle info
and neither particle of wave nature can be
observed, is what I have tried to give you
from this plot.
Plot C gives the electron beam which is the
wave info, okay.
So I have slowly got you on to the wave particle
duality which is how many of you have read
Jekyll and Hyde.
So the character is just, so same person seems
to have 2 characters, right.
Both the Jekyll characters are very good character
and the same, similar thing but here, you
know, every object seems to have wave nature
and particle nature but if you try to probe
what you want to find, it shows that character,
okay.
So in some sense, all of us have both wave
character and particle character.
Why are our wave character not seen?
Do you know why?
Wavelength is very small which cannot be measured
in the present day.
So that is why you do not see the wave character.
But if you take in electron whose mass is
nearly small, then the wavelength is a measurable
regime.
So that is why you can have both wave character
and; for everyone, it is there.
But which is dominant in our case is only
the particle character.
But wave character needs a measurement of
wavelength which is within the experimental
regime.
And that happens only in a microscopic world,
okay.
So all the planets when you study, you will
not do quantum.
Why?
So point, it is really a massive object and
you will do Kepler laws and it is correct.
There is no need to worry about the wave character.
So that is what you should appreciate that.
Because I am teaching quantum regime, does
not mean that day to day life requires but
there are situations where you have to apply.
So wave particle duality.
So I am slowly taking you from classical mechanics
when you do Newton's law, you write all the
Newton's law where you do not worry about,
you can have simultaneous values for position
as well as momentum, right.
You do write the first equation or the second
equation, S=ut+1/2gt square when you write
or 1/2At square, you know both, the position
as well as the momentum, okay.
It is not that if you know the position, you
do not know the momentum.
You can measure the velocity or momentum.
So that is why that is a very important point
that x and p can be simultaneously found in
your classical laws.
Equation is Newton's law.
What is your aim in the Newton's law, you
determine x as a function of p once you have
x as a function of pd/dt, will give you momentum?
So situation completely changes once you go
into the quantum microscopic world and you
want quantum mechanics.
You cannot determine x as a one which gives
you the position.
P by de=Broglie hypothesis gives you the wave
nature, x gives you the particle nature.
As I have said, you cannot get both the particle
and wave natures simultaneously, right, x,
and I say the position of the object and looking
at it as a particle.
When I say momentum of the object by de-Broglie
hypothesis, I am seeing the wavelength which
means I am trying to look at the wave nature.
But I cannot simultaneously, we have seen
in the Young's double slit experiment, you
want the particle info, you try to probe by
putting a kind of a detector at the slits
to see which slit it went through.
But then you lose the interference pattern.
But if you do not see which one it went through,
you get the interference pattern.
You cannot go and which one it went through
and the interference pattern.
That is not possible.
Is that right?
So x and p cannot be simultaneously determined,
is one of the core important things which
I have tried motivating from the earlier slides.
And this is something which you all mechanical,
you learn in your courses, earlier courses
which is the famous Heisenberg's uncertainty
principle that position and momentum cannot
be precisely measured.
Is that right?
So here the equation is the Newton's law in
classical mechanics.
In quantum mechanics, the equation which you
have to solve is the Schrodinger equation
and the main theme of the Schrodinger equation
is to find a solution for a wave function
or it is also called probability amplitude.
In one dimension, it is written as psi as
a function of x.
It can also be dependent on time.
It is a complex; general it can be complex.
And whose modulus squared will give you the
probability.
It is something which you have learnt in your
first year.
So this is the distinction between classical
mechanics and quantum mechanics.
Classical mechanics course when you are doing,
you will be in this regime.
Now quantum mechanics course, you will be
in this setting.
So the familiar Schrodinger equation, the
left hand side I have written is the partial
derivative with respect to time and here,
the right hand side I have written the energy
operator or Hamiltonian we call which can
be rewritten as double derivative of the position.
I have written it in one dimension just for
simplicity.
V is the potential energy which the particle
is facing and we would like to solve this
equation to find psi of xt which is the wave
function, okay.
So that is the theme.
This is the equation and you know, it is the
proposal, this equation.
There are some indicative ways in which you
can derive it by looking at their electromagnetic
wave and then we propose for a nontrivial
potential energy and that is the Schrodinger
equation.
And many experimental curves, many experimental
data are all explained using this which means
this equation should be accurate.
It is correct, okay.
Just like Newton's law, there was no violation
when you try to verify many things.
Same way the Schrodinger equation is also
useful, okay.
So just to summarize, I will put it as a flow
chart.
So you have seen the dual nature of light.
Light is usually a wave but we can see the
particle nature from black body radiation,
photoelectric effect and Compton effect.
Usually, particles are particles.
You can look at wave nature of particles.
They are called matter waves or de-Broglie
waves.
To explain them, you can look at diffraction
experiment, uncertainty principle.
You can also look at the superposition of
various waves.
You look at the group velocity to see the
wave packet.
So these are ways of looking at the wave nature
of the particles.
And Schrodinger equation is the equation which
describes the matter wave and some of the
concepts which you will, which you would have
already studied as the probability finding
expectation values or they are also called
average values of position, momentum, energy
and so on, okay.
So this is quantum mechanics and we will see
this in detail.
