I will be giving a set of lectures, pertaining
to an introductory course in quantum mechanics.
So this is going to be quantum mechanics.
Many of you would be familiar with concepts
from classical physics, presumably you know
what is meant by independent degrees of freedom,
this is all from classical physics; generalized
coordinates of course, generalized momenta,
phase space and so on. Some of you might even
be aware of the richness of the structure
of phase space. Now, when it comes to quantum
mechanics these concepts could undergo some
alterations. The alterations were not obvious
under history of quantum mechanics is as revealing
as it was dramatic. The reason is that classical
mechanics could really not give any pointers
to understanding the quantum world; it made
mockery of intuition of human beings and it
became a challenge to the best and brightest
of minds to find out how exactly the quantum
world works.
The features relating to the quantum world
were really obvious only through the atomic
and sub microscopic phenomena that is not
to say that quantum mechanics does not hold
for macroscopic objects. In fact, several
macroscopic phenomena, I can think of ferromagnetism
right away can be explained only on the basis
of quantum physics. Having said that we normally
can understand most of things that happen
around us in daily life, most of the macroscopic
world just based on our physical intuition,
our prejudices and our common sense which
has really been in us due to our experiences
in daily life. So as long as, we are dealing
with objects of length scales, mass scales
and time scales that we are used to in daily
life. Most of the time we can get by with
our physical intuition and with classical
laws which seem to confirm predictions based
on our intuition.
Unfortunately, and interestingly we cannot
do that with quantum laws. So, it turns out
that in the history of quantum mechanics;
it is in the context of atomic phenomena and
sub microscopic particles that these laws
really came out into the open. Einstein is
supposed to have remarked that our physical
intuition is a collection of prejudices that
we have developed till age eighteen, and prejudices
and intuition do not really explain things
in the quantum world. Many concepts in quantum
physics are very dramatic, it is very difficult
for the human mind to comprehend it atleast,
it was at that stage in history and I am talking
about the end of the 19th century where most
classical laws were pretty well understood.
The first thing, that was very different when
it comes to quantum physics is the fact that;
an object can be thought of as a wave and
as a particle. This is normally called the
wave particle duality in most books and the
series of experiment showed that matter behaves
like waves and also like particles. I am normally
fond of two of this class of experiments;
Young’s double slit experiment which demonstrates
the wave nature of objects; and Einstein’s
discovery of the photo electric effect, and
this demonstrates the particle nature of objects.
So, I will just give a rather a brief description
of the Young’s double slit experiment first
and we will take it up from there.
Let us first look at it from a classical point
of view. Start with the source of bullets.
The bullets are shot out in various directions
and they hit a target; the target is a screen.
Now, as the bullet strike the target you could
perhaps use detectors and count the number
of bullets that have hit the target. Clearly,
there will be an integer number of bullets.
You could modify this experiment by putting
a screen out here with two slits. And now
see what is the pattern that emerges on the
target screen by pattern I mean that if you
plot the distance along this versus the number
of particles that strike the screen, you would
like to see how exactly this plot looks.
So from the source, there are bullets that
go and you would therefore, expect the plot
of distance versus number of bullets to have
a peak out there and similarly, out here because
the bullets that come here. The pattern would
be that of two peaks one there and one here
and nothing else in between. Now, this is
what you would expect from classical physics.
Let us replace this source of bullets with
an electron gun so that we do a quantum experiment,
now we start off with a source of electrons.
Once more, we have a screen on which these
electrons fall and if we have detectors and
counts the number of electrons that reach
the screen, we would count an integer number
of electrons. Once more, we can put a screen
out here with 2 slits through which the electrons
have to pass in order to hit the target. You
would normally expect once more a pattern
like that, if you plot a distance versus number
of electrons that strike if the electrons
behave like bullets or like particles.
On the other hand, something very dramatic
happens on this screen instead of a pattern
like this 
what one sees is a alternate bright and dark
fringes. This is the kind of property that
you would expect, if the electrons behave
like waves and if they were waves that emerged
out of these slits they could interfere and
depending on whether there is constructive
interference or destructive interference you
would expect an interference pattern like
this. This is intensity versus distance from
the screen: a more intense area than a less
intense area than a more intense area and
so on. This is very striking, because this
region for instance is a region where no electrons
can reach; if the electrons are thought of
as particles.
On the other hand, the fact that there is
an overlap between waves that interfere has
created this pattern the Young’s double
slit experiment therefore, demonstrates very
clearly the wave nature of the electron. Consider
on the other hand, the photo electric effect.
Now, in the photo electric effect light is
shown on matter perhaps a metal 
and electrons are emitted. Now, if the light
were thought of as electromagnetic waves and
electrons come out of this metal plate you
would normally expect the energy of the electron
that comes out to be related to the intensity
of the incident light, In other words, as
the intensity of the incident light changes,
you would expect the energy of the electron
to change, For light of very low intensity;
you would expect low energy electrons, for
light of high intensity; you would expect
high energy electrons, but that is not what
happens. And what happens, can be explained
only on the basis of the Corpuscular theory
in other words, you treat light as a collection
of particles namely; photons, each one with
energy given by h nu where nu is the frequency
of the light and h is Planck’s constant,
has the same dimensions as action rule second,
those could be the units.
So, it turns out that photons interact with
the electrons in the metal plate and transfer
energy to them and pull them out, So that,
the energy of the electron does not really
change with the intensity of the incident
light. The more intense the incident light;
the more the number of photons, the less intense
the incident light; the less the number of
photons and therefore, the energy of the emitted
electrons do not depend upon the intensity
of the light. On the other hand, they would
depend upon the frequency of the light the
higher the frequency the more the energy.
Now, these two statements have been experimentally
verified; clearly demonstrating that particles
and waves are both attributes of the same
object, the electron, it is not nearly the
electron if you look at photons, protons,
neutrons or any form of matter you can associate
waves with all forms of matter .And therefore,
even for large systems there are matter waves
associated with them it is simply that as
I said before you understand many things about
macroscopic systems based on classical laws..
Classical mechanics is really an approximation
it is not the exact theory and finally, when
everything was in place it was clear that
when the Planck’s constant, in the limit
planck’s constant going to zero and if the
limits are taken properly, you can retrieve
the classical laws from the quantum laws.
So, given these 2 we understand that in the
quantum world there is this wave particle
nature, attributed to all objects and we need
a theory to explain this we have to learn
to accept waves and particles being properties
of the same entity and not waves or particles
which is the kind of lesson that the classical
world has taught us. To explain this, was
difficult.
Einstein and Planck, and many other physicists
were involved in this explanation and something
called the old quantum theory was proposed
primarily by Bohr Jordan. However, it was
also realized soon enough that the old quantum
theory was rather adhoc in nature and cannot
be used as a strong edifice or a good mathematical
structure for explaining, what happens in
the quantum world, a new theory had to be
brought in. Think of it this way, physical
intuition is not helpful and in a manner of
speaking classical physics is merely shrugged
her shoulders. There is a certain helplessness,
there are no pointers as to what should be
the central objects around which the theory
should be built, what should be the central
objects which should be used in explaining
things like the wave particle nature of matter,
and then physicists brilliant minds working
in different parts largely in Europe Copenhagen
Berlin; got into aggressive debates about
what should be the right theory of the quantum
world.
This was a period in the early twentieth century,
which was very dramatic educator and revealing
because by the 1920s it culminated, in what
we now know as quantum mechanics. So, I will
not make much ado about the old quantum theory
was not very satisfactory and what we now
work with is quantum mechanics as was given
to us largely by 2 master minds; Heisenberg
and Schrodinger.
.
The primary architects of quantum mechanics
as we see in the text books today, there are
alternative formalisms of quantum mechanics
but during the course of my lectures I will
be only talking about Heisenberg’s; matrix
mechanics 
and Schrodinger’s; wave mechanics both being
equivalent formalisms of quantum mechanics.
Having got no pointers from classical physics,
as to what should be the central issues central
objects around which the quantum theory should
be built. Heisenberg fought that observables,
that are objects that you measure in experiments
should be the focus of attention and the theory
should be built around these objects.
.
In classical mechanics, we are familiar with
objects that can be experimentally measured
like position, momentum, energy, and so on.
Now, these objects are the observables in
an experiment. In principle, a clever experiment
on any physical system should be able to tell
us what is a value of observables and in quantum
mechanics Heisenberg said; that observables
are represented by matrices.
Matrix mechanics, itself came about because
of the inspiration that Heisenberg drew from
light atom interactions 
because there are atomic transitions that
result due to interaction of the atom with
light and parameters pertaining to these atomic
transitions, which can be measured. It was
realized by Heisenberg and others that, these
observables are best represented through matrices
because the Eigen values of the matrices,
are the outcomes of experiments.
So, suppose one way measuring energy in an
experiment: the value of the energy which
one would get on measurement would be one
of the set of Eigen values of the matrix that
represents energy. Energy is re-observable,
the state of the system itself would be the
Eigen vector corresponding to that Eigen value.
Of course, there are cases where for a given
Eigen value there is more than one Eigen vector
but such degeneracies will be dealt with by
me later in subsequent lectures. And therefore,
you have Eigen value equations there is an
observable.
I will put a hat on that to show that it is
an observable or an operator, it acts on an
Eigen vector. For the moment I, call that
psi it is a column vector because this is
a matrix that gives me a number may be a number
a; which is the Eigen value or a set of Eigen
values. This is a typical Eigen value equation,
where a matrix acts on a column vector or
Eigen vector and you have a number with the
same Eigen vector on that side. If, I diagonalize
the matrix the diagonal elements would be
the Eigen values and the outcome of the experimental
measurement would be one of this set of Eigen
values.
There is a crucial difference between classical
mechanics and quantum mechanics. It is a very
interesting difference and is being studied
even now in great detail in measurement theory.
In a classical system, if I make a measurement
of the energy and I get a certain value I
can insist, that before I made the measurement
the energy of the system was the same ,the
same value that I got on measurement .Whereas,
in quantum systems I cannot do that, the measurement
outcome nearly tells me that on measurement,
this is the energy of the system and this
is the state of the system. I cannot extrapolate
and say that before measurement, the energy
of the system was the same nor can I insist
that the state of the system after measurement,
is the same as the state of the system there
was before the measurement.
Now, this is another crucial difference between
classical and quantum physics. So while, Heisenberg
developed matrix mechanics by which observables
were given importance as central objects,
and outcomes of measurements are the Eigen
values of these observables. There was an
alternative formalism given by Schrodinger.
Wave mechanics: Let us go back to the Young’s
double slit experiment. In the Young’s double
slit experiment there was a non zero probability
of the electron being in the region between
the slits on the screen recall 
the experiment. Since, we had fringes here
also clearly, the electrons were there the
waves were interfering even in this region.
So, there is a non zero probability of finding
the electron here, here, here, here and of
course, all over there. This makes the whole
subject probabilistic in nature, you can only
talk of the probability of the electron being
there or the electron being here or the electron
being there or the electron being there, you
cannot say with certainty, that the electron
was there or was here and not here, because
if it were not here, one would not see fringes
out here.
It is this wave like nature of the electron
and any other physical system which perhaps
inspired Schrodinger, to develop wave mechanics,
because Schrodinger starts with the state
of the system, being a wave function. And
in fact, we will see in subsequent lectures
that psi could in general be complex we could,
work with psi as a function of space time
and even if we were working in a situation
where we were not worried about the time evolution
of the system, means the probability amplitude
of the system being; in the region x two x
plus d x, y two y plus d y, and z two z plus
d z, psi star psi is therefore, the probability
density and the total probability of seeing
the system anywhere in space will be one.
So, observables are represented by matrices
and the Eigen values of observables are the
outcomes of experiments but when you do a
measurement the answer must be a real number
and therefore, observables are represented
by Hermitian matrices because Eigen values
of Hermitian matrices are real.
It is possible, to make measurements of two
or more observables. Each observable, represented
by a Hermitian matrix and as I said earlier
the system collapses to an Eigen state, the
Eigen state is going to be a common Eigen
state of all these observables. It was soon
realized that matrices are really representations
of operators, in an appropriate linear vector
space. The state of the system, the quantum
state lives in this linear vector space.
Now, this needs some explanation: the linear
vector space should not be confused with the
usual physical space in which we live where
we have the usual Cartesian coordinates: x
y and z and time t .I will give an example,
to illustrate this point. Consider a free
particle, moving along the: x axis. So in
principle, the particle can be anywhere it
continuously moves from minus infinity to
infinity. It is pretty clear therefore, that
if I measure the position of the particle,
the observable is positioned; there will be
an infinity of Eigen values available in this
problem, which means that the Hermitian matrix
representing the position observable, should
be an infinite dimensional matrix and therefore,
the linear vector space in consideration is
an infinite dimensional linear vector space.
So, here we have a problem of a particle moving
in one dimensions but the linear vector space
in question is an infinite dimensional linear
vector space. There is another wrinkle, to
this problem the Eigen values themselves form
a continuous infinity it is not as if the
Eigen values are discrete, like this. It is
a continuous infinity of Eigen values because
the particle is gliding along the x axis and
therefore, we have to modify our opinions
of matrices and extend it and we have to allow
for objects, which have a continuous infinity
of rows and a continuous infinity of columns.
In other words, we have to accept the concept
of operators in linear vector spaces.
The state of the system in this problem is
of course,, an infinite column vector. In
a physical situation, we would like to track
the state of the system as it moves in space
time and therefore, there is a prescription
this prescription tells us, how to go from
this abstract Eigen vector in some linear
vector space to a function psi of space time
coordinates.
Now, given this prescription it is possible
to talk about the dynamics of the system as
time evolves and the dynamics is normally,
given in terms of differential equations.
Functional spaces, have in a very natural
fashion got into the picture because this
is a function of x y z and t. Thus, linear
vector spaces, function spaces, operators,
matrices differential equations, all these
have become a very natural ingredient of the
understanding of quantum mechanics. So, we
go into the next question how do we make measurements?
How do we find out what is the value of the
observable in question? Normally, when we
do an experiment we conduct several trials
on a system, repeated measurements, many trials
and then take the arithmetic mean of all the
values that we have got as outcomes of the
measurement, and that is going to be the value
of the observable. Now, in quantum mechanics
as I have already stated if you make one measurement
on a system, the state is no longer the original
state which was there pre measurement. Post
measurement the state collapses, to one of
the possible Eigen states of the system.
Immediately, after measurement the system
collapses to an Eigen state and therefore,
there is no point in repeating the measurement
on this Eigen state or the new state of the
system because this was not the system initially.
The only way to handle this problem, is to
make an ensemble: identical copies of the
system, on which we conduct the measurement
and then make one measurement on each copy.
Take the arithmetic mean and that is going
to be the value, the average value of the
observable in question. The average is represented
like this where x is the observable, this
is merely a notation and the average value
is always with reference to a particular state
of the system. So, this is the way you represent
quantum averages.
Now, we have an interesting question. Suppose,
we make the measurement and suppose we make
simultaneous measurement of 2 observables
or more observables. Is it possible in a simultaneous
measurement of these observables, the word
simultaneous is very important. Is it possible,
to get accurate values for both these observables.
The answer in quantum mechanics is generally
no, it is not possible! And you can trace
it back to the fact that the matrices corresponding
to the 2 observables need not commute with
each other. In general, they do not and when
2 matrices do not commute with each other,
we can show that it is not possible to get
accurate values for both the observables in
question.
Of course, we are allowing for and if we make
simultaneous measurements, that is a very
important thing. You allow for human errors,
you allow for instrumentation errors and after
that also it is not possible in a simultaneous
measurement of 2 observables in general, to
get accurate values for both of them. This
is an inherent feature of quantum mechanics,
this is quite unlike classical physics. It
is an inherent feature of quantum mechanics
and it has lead to a very many interesting
relations one of them being the: Heisenberg
uncertainty relation.
So, normally referred to as the uncertainty
principle, so what is the Heisenberg uncertainty
relation?
Suppose, we consider 2 observables: x and
y and delta x: is the variance in x I make
measurements in a certain state of the system.
So delta x, is simply equal to: x minus expectation
value of x the whole squared average. This
as most of you know is, expectation x squared
minus expectation x the whole squared that
is the variance in x. Similarly, I define
delta y in the particular case of a particle
moving along the x axis let x denote the position
of the particle 
and p sub x the momentum linear momentum,
the Heisenberg uncertainty principle in this
context says, that delta x, delta p sub x
should always be greater than or equal to
h cross by two, where h is the Planck’s
constant and h cross is h by two pi. In a
minimum uncertainty state, the equality is
satisfied.
In general, it is greater than this bound
there is no way by reducing instrumentation
errors improving human effort and bringing
this down to zero whereas, in classical physics
it is possible that the measurement gives
you delta x delta p x equal to zero such a
thing is not possible in quantum physics.
It is not just true for position and momentum
but for any 2 observables that do not commute
there are corresponding uncertainty relations.
If you look for instance at the ground state
of a quantized linear harmonic oscillator
that turns out to be a minimum uncertainty
state in the position and momentum variables
there are very interesting states in optics,
states of light which obey the minimum uncertainty
relation. You look at ideal, laser light,
it is a coherent state of light and there
in terms of another pair of variables which
I will call the quadrature variables, such
a relation is satisfied, such an equality
relation is satisfied and we should remember,
that there x does not mean position and certainly
the momentum is not involved but they are
2 pairs of a pair of observables for that
context. Then there is squeeze state of light:
squeeze state of light is one where either
delta x or delta p x is less than root of
h cross by two.
So squeeze light, squeezed state of light.
Once again, I should emphasize that x and
piece of x would not stand for position and
linear momentum but for 2 appropriate observables.
So, in quantum physics it is possible to have
different types of states satisfying uncertainty
relation. In a subsequent lecture, I will
talk about the Heisenberg uncertainty relation
and also about variations and extensions of
the uncertainty principle.
So already, we have seen some unfamiliar things
we have seen wave particle duality. Quantum
mechanics, is inherently probabilistic 
in contrast to classical mechanics which is,
inherently deterministic. And in this context,
I have to mention something else in classical
physics, suppose there were a particle which
is kept on this side of a barrier, an impenetrable
barrier the probability of seeing the particle
on that side of the barrier is, strictly zero
whereas, in quantum physics because of the
wave nature there is a non zero probability
of seeing the particle on this side of the
barrier as well if it is a quantum particle.
This phenomenon, that there is a non zero
probability of seeing the particle there here
there and there is due to something called
tunneling this again is a third very important
difference which manifests itself in the quantum
world which you do not see in the classical
world.
It is very important that, states of a system
can be tampered with particularly in the context
of quantum optics you will see very many interesting
states of light. In atom light interactions,
it is possible to tune the state of the atom,
it is possible to make atomic transitions,
various types of light sources could be used
for this purpose.
.
And even in active areas like quantum information
and computation, we work with quantum states,
specific quantum states, tinker with them,
use them for logic gate operations like you
would do logic gate operations with classical
systems with bits. You could use quantum states,
to store quantum information, use it for teleporting
information from one place to another and
these are areas of extreme research activity
even now. All this, leads to a certain picture
of the state of the system.
.
In classical physics, the state of the system
is completely specified. Classical mechanics,
the state is completely specified if, the
relevant generalized coordinates and momenta
are specified and if you have a phase space,
of these coordinates and momenta every point
is a state of the system. Time is a parameter,
and these coordinates and momenta; evolve
in time. These become the observables, in
quantum physics. Time is a parameter both
in classical and quantum physics.
Observables are represented by matrices and
the state of the system in quantum mechanics
is prescribed by the wave function, psi. It
is pretty clear, that this is dramatically
different from classical physics. When all
intuition fails, and you want to create a
theory you have to take recourse to something
very exact, something that will speak the
truth.
.
Help comes, only in the form of mathematics
and the various branches of mathematics involved
in the understanding of quantum mechanics
are linear vector spaces, differential equations,
matrix algebra, group theory. So, an understanding
of the quantum world really amounts to seeing
how exactly, mathematical structures are useful
in describing the physics in the quantum world
and in explaining quantum phenomena. And therefore,
in subsequent lectures I will be first visiting
some of these branches of mathematics.
Starting from, something we know and explaining
some of the essential features of linear vector
spaces in the next lecture and taking recourse
to all these wherever necessary in order to
explain the quantum phenomenon.
In this lecture, I have attempted to give
you a first glimpse of the quantum world.
I have tried to emphasize the contrast between
a world of classical laws and a world of quantum
laws, the surprises and the unexpected. I
proposed to do the following in subsequent
lectures: As I have already indicated, the
state of the system lives in an abstract linear
vector space and therefore, for something
like the first half of my series of lectures
I would be only looking, at the abstract state
of the system. In order to, illustrate various
examples I would work out exercises as we
go along, rather explicitly taking into account
the fact that perhaps you are heterogeneous
audience of listeners and therefore, I will
try to work out a lot of the mathematics explicitly.
Now, after about 20 lectures I will move on
to the concept of the function space and talk
about the wave function, psi: as a function
of space. We will study that for a while and
then proceed to do dynamics where psi is also
an explicit function of time. Now and then
I will intersperse my lectures with exercises,
in order to illustrate the salient features.
This is a good place to stop by way of introduction.
In the next lecture I would begin with finite
dimensional linear vector spaces calling upon
ideas that we already have from vectors in
2 dimensions, vectors in 3 dimensions and
so on and proceed from there.
