So, welcome to this course on Advanced Quantum
Mechanics with Applications. In this course
ah we will learn ah of course, the basic quantum
mechanics and ah the various applications
that have been hm developing since the last
decade or so. hm And, various ah research
topics that are associated with it. So, the
applications part comes there, but of course,
we have to also ah know the basic quantum
mechanics which I am sure that all of you
have studied in your undergraduate or graduate
curriculum. So, this is a development on that
I will of course, start with something ah
simple and basic.
Ah. So, let us look at the historical development
ah of quantum mechanics ah. So, it ah gives
you a detailed description of the behavior
of matter ah and light which means photons
and all the phenomena that occur at the sub-atomic
level ah. So, that is inside the nucleus or
inside an atom ah the whatever phenomena goes
on hm they are governed by ah the quantum
mechanics and the laws of quantum mechanics
.
So, Newton initially thought that light ah
consists of particles because it reflects
from a mirror and this is this give rise to
the corpuscular theory of light. However,
ah towards the beginning of the twentieth
century, there are lots of experiments done
ah which have manifested the wave character
of light; which means that they show interference
refraction hm and ah reflection etcetera which
ah otherwise ah especially the interference
as a diffraction there otherwise absent in
the case of particles.
So, this gave rise to this famous wave particle
duality. So, the conclusion is that light
in some experiments ah show particle character
ah such as photoelectric effect and Compton
effect and so on. And in others they show
wave character such as interference and diffraction.
This gave rise to the wave particle duality
of matter ah and it is not only the ah priority
of ah photons, other particles sub-atomic
particles such as electrons and protons etcetera
they also show wave particle duality ah similar
duality as seen for light.
So, ah there is a large amount of information
that has started coming in for the atomic
systems and ah they have ah raised lot of
curiosity. And of course, have given rise
to lot of confusion because of contradicting
ah you know inference is coming out from various
experiments. And, hm well a lot of people
at that time they talking about the early
ah 1920s and the mid-1920s between 20 and
1920 and 1930 where a lot of people have contributed
to the birth and development of quantum mechanics.
A notable among them are are Schroedinger,
Heisenberg and Born and they have enriched
the field so much so, that ah there are you
know consistent description of ah matter ah
at very small lime scales that have emerged.
So, the basic ah pillars or the basic ideas
that they are based upon mainly two of them
one is called the probabilistic interpretation.
So, ah this has to be this is distinct from
the ah the interpretation or the deterministic
interpretation that we are familiar in classical
physics. ah As a pose to that ah in quantum
mechanics there is a probabilistic ah interpretation
that needs to be ah incorporated which gives
rise to a lot of satisfactory answers ah to
a lot of experiments that have been done and
phenomena that people have observed.
And the second one is the uncertainty principle
which is related to the measure[ment] measurement
of a particular measurement and it ah ah sort
of ah saved, these two put together have saved
a number of experiments that are done on subatomic
particles. So, let us take this ah ah brief
tour to how things have evolved and that will
make the study of quantum mechanics more interesting
and of course, we will go to the applications
part later on.
So, let us see some cartoons ah on a lighter
note. So, that we understand things ah ah
in ah graphics using graphics.
And there are some of them ah which are so,
I have drawn two pillars ah the pillars as
I said are one is the probabilistic interpretation,
the other is the uncertainty principle of
a quantum mechanics which is developed by
Heisenberg ah we will go through that.
And, these ah say we go through one after
another, the probabilistic interpretation
was enriched by a simple experiment which
is called as a double split experiment. And
it is a Young ah it is called as a Young's
double split experiment being performed in
1802 which is more than 200 years ago . So,
ah there are light waves these vertical lines
that you see are really the ah the light waves
that are coming, they are approaching these
blue ah barriers like these ones, these blue
ah barriers. These barriers have two slits
and let us call this as a slit 1 ah here and
this as a slit 2 here.
So, there are these two slits and there are
ah these secondary these act as secondary
sources and when light passes through them
ah and they are ah allowed to fall on a screen
ah that is kept here. So, this is the screen
and when light falls on this ah light ah undergoes
a phase ah difference ah the from the coming
out from the two slits. And this, phase difference
gives rise to a interference pattern which
means that there are bright bands and dark
bands depending on the on the condition that
these ah way this path differences or the
phase differences that they satisfy and we
get nice bands that are appearing on the screen
as it is shown below.
And these are ah these are there in ah every
undergraduate lab ah double street experiment
ah which shows a interference patterns due
to these two slits. One thing that should
be remembered is that the slit width should
be comparable to the wave length of light,
which means that if wave length of light is
of the order of ah 4000 or 5000 or 6000 Angstrom
ah this should be of the order of a millimeter
or so ah. So, then these or even less than
a millimeter ah then these are ah the the
well-defined interference pattern ah could
be seen.
And of course, the interference ah the ah
the distance between the dark and the ah bright
ah bands ah that we see here depends upon
the wave length of light and the slit width
and also the difference between the barrier
and the screen that you see here which I write
as D. Now, these are matter of study in optics
and in ah particularly in the interference
chapter, we will not elaborate on that.
We have brought this for a particular reason
of demonstrating how quantum mechanics can
be ah derived or rather ah how ah the probabilistic
interpretation can be ah inferred from this
experiment. So, we will skip all the details
that how what are the conditions for path
difference or phase difference to have a constructive
or destructive interference and so on. And,
go on to ah the quantum mechanical aspects.
Now, light is of course, ah consist of particles
called a 'Photons'. So, ah since we are talking
about particles let us talk about bullets.
You see this character which is firing guns
through two slits, just like what we have
seen in the last slide and these bullet marks
are allowed to fall on a wall which is kept
at a distance, again just like the photons
were allowed to fall on a screen. Now, the
bullets are allowed to impinge upon a wall.
Now, do you think there will be interference
pattern? Certainly not ok.
So, this if these character keeps firing bullets
at the ah wall through the slits there will
be bullet marks that will form ah on on the
along the two ah slits ah that are there and
on on the wall. ah And there will be no interference
pattern, there will be no bright and dull
bands that are going to form. So, the question
is that if photons are particles the bullets
are particles why does not ah why do not bullets
ah give rise to interference pattern whereas,
the ah the photons do.
Ah Could it happen because, the the photons
ah come in larger number because you are firing
a light beam and the light beam has got a
very large number of photons. So, is it that
because they are coming in large number of
photons. So, let us reduce the number. It
is a thought experiment, we are not doing
that experiment. Nobody is firing photons
very slowly they are still at the velocity
of light, but it is just a thought experiments
you say that I am going to only release a
few photons at a time. And they would ah go
through the 2 slits, as shown here and fall
on a screen and would ultimately give rise
to an interference pattern as it is seen here.
So, if you allow for a long time then the
pattern will build up and this is important
thing the pattern builds up means that there
is interference pattern. So, even if you reduce
the number of photons by a large ah fraction
the phenomena, the basic phenomena that happened
earlier ah do not stop ok . So, ah if they
are coming in small numbers the question is
how does the interference pattern take place
or rather it it happens. So, how do they interfere
that is the question.
So, one solution is of course, that the photons
interact with themselves. ah It is not very
clear from the statement that we want to say,
but they inter what all we mean to say is
that they interfere with themselves. Or may
be ah they have a wave character associated
with each photon which we know that certainly
that is a possibility. And so, they can interact
the coming back to the first point they can
interact within the wavelength of the wave.
So, they can actually interfere mutually interfere
within the wave length of a wave and that
could give rise to the pattern.
Now, let us reduce the stream even farther
and send one photon at a time ok. Interference
it seem that the interference still takes
place, interference you one can still see
an interference pattern if you wait for long
enough, but there are two slits that are essential.
So, the question is that that how does a single
photon anticipate the existence of two slits.
A photon coming through this ah slit that
is ah left to mine is the photon that is passing
through that is slit, it when it is passing
through this it of course, does not know the
existence of the other slit. So, how does
the interference still take place if you reduce
the stream to such an extent.
And the solution is that give up the notion
of talking about location of the particle
until it hits the screen. So, this picture
is ah this middle ah picture where we actually
see that its passing through one of the slits
is misleading. Do not talk about measuring
the position of particle or the location of
the particle, until it hits the screen. If
it hits the screen it will leave a mark and
then you know that it is ah it is it has passed
through either of the slits ok.
So, talk about the probabilities of the photons
to pass through one slit or the other. So,
we do not know we we simply say that the photon
must have passed through either the left slit
or the right slit or call it slit 1 and slit
2, but we really do not know. And, if you
allow that to happen or allow that at or rather
you assume that then the interference pattern
can be understood. . So, that tells you that
is the properties of quantum objects are linked
to the measurement.
So, when you do the measurement, when you
talk about the measurement that is important.
So, do not talk about the measurement of the
particle when it is ah ah passing through
one of the slits; you really do not know which
slit it is passing through. ah Similarly,
the role of the observer is also crucial.
So, if there is an observer to carefully monitor
so, suppose there is a robot sitting ah at
one of the hm ah slits or at both the slits.
And, these robot actually knows exactly where
it passes through which slit it passes through
ah or say that this is one robot which counts
the number of photons passing through that
slit.
So, then immediately you are making a deterministic
and if you do that then the interference pattern
will go away. So, if the ah observer carefully
monitors which slit does the photon go through
will wipe up the interference pattern. So,
there is an entangle relationship this is
what the we talk about entanglement in ah
daily life that everything is you know knotted.
And so, there is a ah nice ah entangled relationship
between the observer and subatomic objects.
This is what one of the basic principles of
quantum mechanics.
Ah Let us now, assess the observer's role.
So, this is called as Schrodinger's cat ah.
So, there is a box in which this cat is enclosed
and ah there is a hammer which is ah hanging
or rather which is tied to a a device or a
pulley or some kind of a sort of lever. And,
this is right over a green colored ah liquid
in a bottle or in a vessel and this is assumable
the poison. So, if you leave the cat inside
the room and lock from outside and you do
not know what is happening to the cat.
Either the cat could get very inquisitive
or very curious about what is this green colored
liquid and it would sort of ah disturb this
whole apparatus and this ah hammer could drop
down and break the poison. And if it breaks
the poison then the cat dies and if the cat
dies then of course, ah ah we would open the
door and find the cat to be dead. And suppose,
it could also happen that the cat is in such
ah strange hm ambience for the first time
and it does not want to do any mischief with
this green colored liquid and it stays away
from it nothing happens, open the door and
you find the cat to be alive.
So, let us hope the cat to be alive hm ah,
but there is all all always the possibility
that cat could be dead because of the reason
that we just say. So, once you are out of
the room and not watching it then in your
mind there is a 50-50 possibility of the cat
to be dead or the cat to be alive. And, ah
as soon as you open the door your ah the whole
ah state of the room collapses into one of
the ah possible states said to the say the
cat is alive, which is an optimistic scenario
. So so, it is the observer's role is important
observer's role only comes when you make an
observation or you make a measurement. And,
the system actually collapses into one of
its available states these are called eigenstates
of the system.
So, the eigenstates it is the cat is either
its dead or alive. So, the ah if you want
to write it mathematically then we could write
it as it is a 1 by root 2 is a normalization
factor and for the psi of the cat to be alive
ah and its psi dead. And the moment you made
the measurement ah it it collapses into one
of the states. So, the observer's role is
very important here.
Ah Let us now go to the other pillar that
we have talked about of quantum mechanics
that is Heisenberg's Uncertainty Principle.
How we can ah understand Heisenberg's uncertainty
principle simply, this is what a simple sine
wave that you see. So, it is a sin x versus
x ok so, this is your ah y axis in which we
plot sin x and this x axis you plot ah x.
So, it is a one sine wave, these are two sine
waves. So, the top panel shows two sine waves
with frequencies f 1 and f 2 with the green
and ah red line and below that is is a superposition
of these two sine waves ah names ah which
consists of f 1 and f 2.
And what happens we when we superpose many
sine waves? It you see by superposing two
sine waves the there is getting shrunk ah
ah on the ah you know, these period that is
the periodicity is getting ah shrunk. And
so, these two are at the same phase and and
of course, ah we are ah we are getting that
this ah ah that the ah the particle is getting
localized. ah ah And, if you superpose many
such sine waves it will be absolutely localized
and we will see what ah what is seen here
which is like a delta function.
So, if you take many sine waves sin ah k 1
x plus k 2 x plus sin k 3 x plus sin k 4 x
and so on and then try to plot them ah where,
k is equal to you know 2 pi over lambda and
lambda is related to the frequencies and so
on ah. So, these are will give rise to a ah
delta function type ah structure where, the
particle is a if the particle is associated
with this sin ah of this wave packet which
we call it as wave packet where we superpose
many sin waves, ah these are ah these will
look like this. So, what it means is that
ah this is the essence of uncertainty principle
that if a you try to ah localize the particle,
if you try to ah do ah precise measurement
of its position you have to give up upon the
momentum uncertainties; as here the momentum
uncertainty is really infinity.
Because, a very large number of sine waves
have been ah superposed and they are ah over
ah range over a spectrum. So, that the delta
p is very large at the expense of delta x
to be very small. So, the relationship is
that the delta p ah and delta x this should
be order of h cross or h we will see in a
in just a short while that what h means or
this is h cross is equal to h over 2 pi ah
where h is known as a Planck's constant.
So, this is the essence of the uncertainty
principle I will show ah an interesting ah
ah picture ah. So, ah this is written there
that superpose many sine waves and the wave
becomes more localized and this localization
leads to a very large uncertainty in the momentum.
So, particle is localized at the cost of huge
uncertainty and the value of the momentum
and this is what the statement of the uncertainty
principle is.
So, this is a famous joke ah which was so,
apparently these are two cars ah. So, there
is a ah police car which is chasing this black
car, the police car is in red and white. And,
apparently ah this is been driven by Heisenberg
and the police ah apprehension for speeding.
And ah when he ah is caught speeding he says
that the police asks him that do you know
how fast are you driving because, you have
broken all those speed limits.
Heisenberg says no, but I know exactly where
I am. So, this he says my delta x is equal
to 0. ah However, ah since I do not know how
fast I was driving so, my delta v or my delta
p is unknown to me or it is large. ah But,
I know exactly where I means my delta x is
very small between which means my position
uncertainty is very small. This is just a
a cartoon or ah just on a lighter note ah
what it is ah what is essence of uncertainty
principle. Now, let us ask a very pertinent
question. So, we are talking about quantum
mechanics right from the beginning it is applicable
to whom. And, the answer is that to everything
in the universe.
And ah so, for majority of the things that
we see around are really the limiting cases
of the quantum world that we live in ok. So,
we will see that that how ah so, what is the
meaning of then studying classical physics
at ah at all levels. I mean in the sense starting
from your school level and then your undergraduate
level and then your graduate level and so
on; the classical mechanics, classical physics,
classical electrodynamics that we talk about
then what is the existence of that or rather
what is the utility of that. And, that we
will show that these are the classical world
is really a limit of the quantum world .
ah So, a quantum mechanics as said earlier
ah ah number of times that its applicable
to subatomic particles. And, most familiar
subatomic particles are fermions and bosons
and we do not want to get into details about
the properties, but just for your information
they are ah the fermions. So, the fermions
of statistics ah they have anti symmetric
wave function, they have half integer spins.
The occupation is restricted ah and ah basic
they obey highly exclusion principle and then
they are indistinguishable and the examples
are electrons, proton, neutron, quark etcetera.
Whereas, bosons obey ah Bose Einstein statistics.
They have symmetric function, ah the integer
spins, ah there is no ah restriction on occupation
of bosons in a given energy state or a given
quantum level. They are of course, also indistinguishable
just like the fermions and the example of
photons, phonons, pi-mesons etcetera .
So, just another cartoon of showing that bosons
can actually be ah in any of the states, a
here it is shown as the ground state that
phenomena has a particular name which is called
as Bose Einstein condensation. Whereas, the
fermions ah two fermions cannot occupy the
same level. So, they are distributed over
all these ah excited state energy levels as
well or rather all the energy levels that
appear here .
So, how the quantum mechanics ah was thought
to be ah you know you invented or people have
started talking about it. How did that happen?
So, it happened with ah a simple experiment
which is called as a Blackbody radiation.
Now, again I will not go into what the what
is the definition of a blackbody but, a blackbody
is a body which absorbs all the radiation
that incidents on it. So, we are talking about
a the radiation from the sun and radiation
from other stellar bodies. So, suppose you
have a body which ah absorbs all the radiation
and emits nothing then that is called as a
perfect blackbody.
So, such blackbodies one was interested in
seeing that how the intensity of the ah the
hm spectra emitted ah. So, we we are talking
about also emission spectrum. So, it is it
emit some ah radiation ah so, does not absorb
everything. So, the emission spectra and as
a function of frequency and at a given temperature
ok. And, what was shown is that it shows the
nice; ah I will use another color to give
a guide to your eye. So, this is that experimentally
it is found that it has a non-monotonic dependency.
Whereas, the classical theory predicts that
it should go like this and then there was
another theory that was made by Rayleigh and
Jeans which says that it goes like this.
So, nobody actually predicted that it goes
up and then goes down and this was a problem
ah at hand with Planck. And Planck almost
ah like as he explained later that as a matter
of desperation, he ah had proposed something
which we will just see in a while. And so,
this is called as the ultraviolet catastrophe
that this classical kinetic theory hm re ah
predicts energy radiated to increase the square
of the frequency and so on. So, these are
all wrong and what you see this non-monotonic
thing or these ah thing here ah that you see
here are the once that are correct and experimentally
observable.
And this is shown as a function of frequency,
this is shown as a function of wavelength.
However, they all carry similar meaning or
rather ah it just says that at a given temperature,
if you ah look at the intensity of the ah
radiation ah a then ah either the function
of frequency or wavelength one sees a non-monatomic
nature which cannot be explained by classical
theory.
So, Planck solution was that that if said
that look ah he made a bold assumption that
light is emitted or absorbed in a some discrete
packets or quantum ah which is given by E
equal to h nu. When nu is the frequency of
the light and this h is called as the Planck's
constant. We will just ah detail these ah
little ah later and this h has a value which
is 6.62 7 into 10 to the power minus ah 27
erg-second or equivalently it is 6.6 into
10 to the power minus 34 joules second and
so on.
So, E equal to h nu. So, energy of this emitted
radiation is not a continuous stream, this
happens happens in packets and each packet
has an energy which is h nu; nu is the spectral
frequency and h is of course, here looks like
a constant of proportionality that is what
we will ah call, but this is known as Planck's
constant. Planck initially called this theory
"as an act of desperation". He says that "I
knew that the problem is is of fundamental
significance for physics; I knew the formula
that reproduces the energy distribution in
normal spectrum; a theoretical interpretation
had to be found, no matter how high".
So, it leads to the consequence that light
comes only in certain packets of quantum quanta
ah. So, ah this a complete breakdown on classical
physics of all the physical quantities ah
are known to be always continuous. ah This
ah the Indian scientist S. N. Bose had a ah
large contribution here, ah what Planck did
on an ad hoc basis. He actually wrote down
the correct distribution for the photons ah
which is known as the Bose Einstein distribution.
So, they they more advances into the subject
of quantum theory is required to look at this
origin of this quantity called h. It is also
called as a quantum of actions popularly of
course, known as Planck's constant and it
is as I told that it can be thought of as
a proportionality constant of this equation
where, gamma is a spectral frequency. So,
let us see that when is this h important.
So, let us take an example . So, consider
a a macroscopic oscillator, a classical oscillator
that is I mean ah of mass 0.01 kg and the
maximum velocity of the oscillator is equal
to 0.1 meter per second and the amplitude.
And so, basically the amplitude of oscillation
is ah amplitude equal to 0.01 meter.
Now, of course we know that this does not
fall into the real mof quantum physics, but
we will show that explicitly. So, what is
the ah the solution of ah ah of a classical
oscillator? It is x x equal to A sin omega
t A sin or cosine does not matter, which gives
you a dx dt equal to a A omega cos omega t;
which tells you that it is equal to A omega
into 1 minus x by A whole square which is
equal to omega into A square minus x square.
So, that is the velocity of the oscillator,
the maximum velocity of the oscillator is
when x equal to 0 that is when the oscillator
is at the mean position.
And so, dx dt max ah which is equal to v max
ah happens when x equal to 0 which is the
same as the mean position and this is equal
to omega into ah root over. So, we just put
this equal to 0 and this becomes equal to
A omega. So, at x equal to 0 x equal to A
omega which is equal to a 2 pi A nu where,
nu is equal to v max divided by 2 pi into
amplitude. So, if you do this 0.1 2 pi 6.28
into 0.01 which is the amplitude this comes
out to be 1.6 hertz. So, that is the frequency
of oscillation of this thing and the time
that is associated with this frequency which
is equal to say T equal to 1 over nu, which
is ah 1 divided by 1.6 which is equal to approximately
0.63 second ok.
So, what is the energy associated with this
oscillator energy E we are talking about the
total energy ah which is of course, ah the
kinetic energy, but at the potential energy.
But, when it is at the mean position ah this
is what the total energy is; ah if you put
in all these values ah of m etcetera and v
max that we have written earlier it comes
out as 5 point ah 5 into 10 to the power minus
joules. So, this energy E into T which is
ah so, the product ah the product of ah the
E and T ah that is ah is 10 to the power ah
of the order of 10 to the power minus 5 ah
or 10 to the power minus ah whatever 4 or
something. So, this E T is certainly much
much greater than h.
So, these canonical variables such as energy
and time they are when they are multiplied
they give values which are much much greater
than h. Because, h is of the order of 10 to
the power minus 34 joule second ah and this
is only coming out to be 10 to the power minus
4 or 10 to the power minus 5. So, the conclusion
is that the quantum effects are negligible
. So, this quantum effects are negligible
and so, we are in the classical regime and
we say safely ah employ formula of classical
physics. So, that is the importance of h that
we want to bring out.
Now, Bohr realized that this wave particle
duality not a particles it is wave particle
duality is not the monopoly of photons. It
is applicable to massive particles as well,
what we mean by massive is that which has
a mass, it necessarily does not need to be
heavy. Such as electrons they show interference
diffraction and a variety of other phenomena
which are applicable to waves.
Thus, there is a general question: if the
particle has a wave description and wave has
a corpuscular description means particle description,
how are the properties interlinked? And de
Broglie solved this problem, he said that
look the momentum of the particle is related
to the wavelength of the wave. So, single
thing or a single particle ah or a single
object can have both the descriptions, both
ah particle and ah wave description. The wavelength
of the wave is related to the momentum of
the particle and this is by this relation
p equal to h over lambda, this called as de
Broglie relationship.
So if h is zero, no matter what the momentum
of the particle is, the wavelength associated
with wave should identically be equal to zero.
And, we are should be in the classical regime
as we have just seen in the example or the
illustration just in the last slide or the
earlier slide.
