So, in this lecture, I shall actually start
with this question. The question is interesting,
because I am supposed to be talking to you
about the quantum chemistry; and the question
is, do we understand quantum mechanics?
But before I tell you why I ask such a question,
let us look at this figure. And if you look
at this figure, there obviously the… If
you look at the marks left by this skier;
this is something that is impossible. If you
look at the figure, this is definitely impossible.
And now with this, let us look at the original
two-slit experiment that I started with. So,
in the two-slit experiment, if you remember;
if you keep only one slit open, you will get…
If you keep only this slit open, you will
get this pattern. While if you keep both the
slits open, the pattern that you get is shown
here. See you think point here. At that point,
if you kept only one slit open; if you kept
only one silt open, at that point, you will
have electrons arriving. But, if you keep
both the slits open, none of the electrons
will arrive at that location. Therefore, the
conclusion is that, any electron passing through
the system knows that, I have kept both the
slits open. That is why none of the electrons
arrive at that location. So, any electron
passing through the system is able to feel
both the slits.
Now, in our classical if I had a particle,
you see what is going to happen is that, the
particle will pass through either one slit
or the other slit. Now, electron does have
characteristics of particles. But, in spite
of that, this electron is able to know that,
there are both the slits open that, we do
have an explanation for that; the explanation
is that, there is a wave and what is happening
is that, the wave passes through both the
slits and interferes destructively at that
particular point. That is why the electron
does not go.
But this is something that is very difficult
for us to understand, for us human beings,
because in our experience, there is nothing
that is able to do such a thing. And therefore,
how can the electron behave like that? That
is something that is difficult for us to understand.
And therefore, if you are asked, do you really
understand it? And the answer is that, you
see with our classical thinking, the only
answer that we can have is, we do not really
understand it. And in fact, R P Feynman, one
of the great scientist of last century; he
used to go around saying that, nobody understands
quantum mechanics.
So, the way you want – you or anybody has
to imagine of the two slit experiment is that,
the wave associated with the electron actually
goes through both the slits. So, it is as
if some part… I mean it is not really correct
to say a part goes through here and another
part goes through there, but it is something,
is going through both the slits even if you
have only one electron. And therefore, in
quantum mechanics, this is actually the way
things are happening even though such a thing
will not happen in our real world; it is as
if something has go on through one other slits
and other, something associated with the electron
has gone through both the slits. That is what
is happening.
So, then you have this very interesting question,
how does an electron propagate? See if I was…
If I have a chalk piece; imagine this chalk
piece; if I drop it, I can actually see the
path that it follows. So, I can use what is
referred as Newtonian mechanics to describe
its motion; I can actually see the path that
the electron will… not the electron, but
the chalk piece will follow. So, similarly,
if suppose I wanted to describe the motion
of this electron, how will I do it is the
question that I want to answer. So, in order
to do that, what I am going to do is… The
answer is already given; it says through all
possible paths… Let me explain this in a
slightly better fashion.
Imagine that instead of having a system of
just one wall with two slits. Imagine that,
I have three walls. So, these are three walls
as we can see: 1, 2, 3; each one of them has
two slits. And here is the electron; source
of electrons is the electron produced there
and it will pass through these system of slits.
And imagine I want to think of the possibility
or the probability that the electron will
arrive at that location. How will I calculate
this probability? In the case of the two-slit
experiment, what I said was, to calculate
the probability, I first have to calculate
the wave function. To calculate the wave function,
how will I do it? There are two possible ways
in which the electron could go maybe through
the first slit or through the second slit.
Each possibility makes a contribution to the
wave function. So, you have to first evaluate
that contribution and then add the two contributions
together to get the total wave function.
And then after that, you have to take the
magnitude of that wave function and square
it; you will get the intensity or the probability.
So, if you extrapolated that at this situation,
what will happen is that, you see the wave
associated with the electron could have perhaps
followed this green path; that is, definitely
one possible path. This red path is another
one; this blue path is yet another one. And
if you think about it, you would realize that,
there are 2 to the power of 3 – 8 different
possible paths that the wave associated with
electron could have taken to arrive at that
location.
So, the contribution on the first one; imagine
I have some wave; calculating it; I will calculate
and call it psi 1. Then I will have a second
path; I have not run all the paths; only three
of them I have drawn. But, the paths are there.
So, I will have to evaluate the second one
– contribution on the second; and then up
to psi 8, I will have to calculate and add
them together to get the total. And once I
get the total, I will take its magnitude and
square it; I will get the probability or intensity
if you like. I will speak a probability if
I have only a single electron. But, if I have
a beam of electrons, then I will speak of
intensity. So, that is what happens in this
case.
But, suppose I say, now, I am going to think
of even more complex experimental setups;
suppose I had an arrangement like this, again
these are all imaginary things. Nobody who
is sensible will ever do these things. But,
what has happened here is I have produced
the electron at this location and I am going
to calculate the wave function for it at that
location. And I think probably 12 walls in
between; and each one of them has 4 slits.
So, if we have 4 slits on each wall and 12
walls, then what will happen? How many paths?
The answer is 4 to the power of 12 parts are
there. So, you have to think of all these
4 to the power of 12 paths. Somehow calculate
the contribution from each one of them. After
having calculated that, you will have to add
all the contributions together; you will get
the total wave function and take this square
of wave function. That will give you the probability.
This is the prescription. Now, suppose you
make it even more complex. One second; let
me just go back. I think I did make a slight
mistake; I suspect here is a system with 12
walls and 4 slits. The one that we are looking
at earlier had more slits. Actually, this
has how many? I think 8 slits and 12 walls.
So, what we can do is, imagine you increase
the number of walls as well as the number
of slits. So, then what is going to happen
is, you have the source here; you have the
screen there; you have made many many many
walls. These are all imaginary walls in between.
And on each one of them, what you will do
is, you will go on making slits. So, for example,
these are the slits on the first one, second
one – second, third and so on. So, then
what will happen, you see you will have to
calculate contributions from paths like this.
This is one typical path eventually arriving
at this location. So, I can imagine the limit,
where I would have infinite number of walls;
of course, I mean that is purely imaginary,
not physically possible to have it.
Imagine I have infinite walls. And in each
one of them, I am going to make more and more
slits, so that the number of slits are progressed
infinity. Then how many paths would I have
to think of? Infinity to the power of infinity
– a huge number; all these paths are going
to contribute to the wave function. So, you
may have a path, which does this; something
like that. Even that will make a contribution
to the wave function. Of course, if you think
about it physically, you know that the contribution
from such a path may be small. But, in principle,
it will also make a contribution.
So, I will do the ultimate; I will imagine
that, I have infinite walls; in each one of
them, I have made so many slits – large
number – infinite number of slits, so that
the walls themselves disappear. So, I say
the walls are there; the slits are also there
only in my mind. Correct? So, I have made
so many slits that, the walls have disappeared.
But, of course, the paths that I am thinking
of – they have a real physical existence
and they are all there. So, in the ultimate
limit, where you have so many slits that,
each wall has disappeared. What are you actually
talking about? You are talking about the propagation
of the electron in free space. And how does
an electron in free space propagate? Here
is the answer. If you have…
If you start with an electron at this location;
now, at this instant, I have an electron at
this instant here. And imagine I want to calculate
the probability that the electron will be
there; maybe 10 nanoseconds later suppose.
What should I do? I have to think of all possible
paths that the electron could have taken.
They will all start at this location now and
the end to there after 10 nanoseconds. So,
you will have to think all of these paths
– this one, many many many paths. All these
paths are possible. You have to think of anyone,
all of them; each one will make a contribution
to the wave function and you have to evaluate
the contribution of any particular path or
all possible paths and then perform a sum
over all these paths. And that will give you
the wave function. So, this is the meaning
when I said an electron propagates through
all possible paths, because all paths make
a contribution to the wave function. This
is to be compared with Newtonian mechanics.
See I told you, if have this particle here
and if I drop it, I know exactly it follows
only one path. But, that is not the way things
are in quantum mechanics. If I have an electron
here now and if you ask me where will it be
after 10 nanoseconds; then I cannot actually
tell you where it will be. All that I can
do is to give you the wave function. What
is the use of the wave function? The wave
function will tell me the probability density
that, it may be located at this point after
10 nanoseconds. That probability I can calculate.
How will I calculate? I may think of a path,
which starts here and goes there; another
one, which again starts here now and goes
there after 10 nanoseconds. I have to think
of all the possible paths that the electron
could have taken; evaluate the contribution
from each; sum all these contributions. That
will give me the total wave function. Take
its magnitude and square it; that will give
you probability density that it may be located
at this point. That is the way it is.
So, these are shown in the slides actually.
The point, where you produce the electron;
the point, where… This is where you produce
the electron or you start with the electron.
That is where you are detecting the electron.
So, in the element of infinite walls and infinite
slits, you are actually speaking of free propagation;
and then you find that, all the paths contribute.
But, then of course, you will object to this;
you see you… because I have not actually
given you a prescription for evaluating the
contribution from a particular path. See I
just said evaluate the contribution from any
given path. How will I evaluate the contribution
from a particular path, is the next question.
And for that, there is a prescription.
You see if you have this particular path or
any paths, imagine electron is here; I am
thinking of calculating the probability density
at this point. You think of a path, which
may be a straight line. That makes a contribution
to the wave function. Or, maybe a path, which
is say a parabola connecting the two. Even
that makes a contribution. Therefore, you
think of any particular path; I want to know
how much is the contribution of that path.
And the answer is that, if you give me this
path; that means I know the position of the
electron at any instant if it followed that
path. The moment I say I have a path; that
means I know the position of the electron
at each instant of time along that path.
So, if you say… It can be a simple one dimensional
though I have been talking of the things in
3 dimensions; but let me say, I am thinking
of a simple 1 dimensional situation, where
the position of the particle is denoted by
x. And the moment I say I know the path, it
simply means that, I know x as a function
of time. So, if I knew the position as a function
of time, then what I can do, I can calculate
the velocity of the particle, because velocity
is obtained by differentiating this with respect
to time. And if I knew the velocity, I can
calculate the kinetic energy. Therefore, if
I knew the position, I can actually calculate
the kinetic energy of the particle. And not
only that, you see maybe the particle is moving
subjected to some potential. Therefore, I
potentially say, is a function of position.
So, it is possible for me to also calculate
the potential energy of the particle if it
followed that path. If it is following a definite
path, I can always calculate its kinetic energy
at each instant of time. Also, it is possible
to calculate the potential energy of the particle
at each instant of time.
And, if I can do that, I can take the difference
between the two, not the sum, because some
of you might have expected the sum to be there.
But, that is not how it is. If you take the
difference between the two and integrate it
from the initial time to the final time; the
limits are from initial time to the final
time; this is something that is very familiar
to physicists; it is referred to as action
and it is denoted by the symbol S. See if
you give me any path, it is possible for me
to calculate the action associated with that
path. And interestingly, what happens is that,
the contribution of any path to the wave function
in quantum mechanics is determined by this
object, which is referred to as the action.
If you give me any path, I will be able to
give you a number, which is the action for
that path. And once I get that number, what
I will do is, I will take e to the power of
i into S divided by h cross. What is i? i
is square root of minus 1. I mean you may
say this is very arbitrary. In fact, it is
the arbitrary. In some sense, it may continue
and then it will become clearer. So, given
any path, what you do, you take e to the power
of i S by h cross; h cross is the Planck’s
constant divided by 2 pi. It is going to occur
in all over discussions.
And, this will determine the contribution
of a particular path. And then if you have
all the possible paths, what you will have
to do is you will have to sum over all the
paths – all the possible paths. But these
paths have the condition that, they will all
be starting at this location at the initial
time maybe now. And then there are the final
time maybe after 10 nanoseconds. That is the
condition. So, you have to sum over all the
possible paths. And this object is going to
determine 
the wave function. Now, wave functions – if
you have said it a little bit in quantum mechanics,
then you know that, they have to be normalized.
So, there will be a multiplicative factor
there, which ensures that, the results are
physically acceptable; acceptable in the sense
that… See because wave function is related
to probability, psi square will give you something,
which may be referred to as probability density.
And if you integrate it over the entire space,
answer has to be 1. This is anticipating some
parts of my lecture later. But, this is a
normalization fact; that is, all that you
need to know now; and you have to sum all
the other possible paths.
So, that is the prescription. Now, as I said,
you may ask me, where did I get this prescription
from? You see this may be taken as a postulate.
There is no other way in which this can be
justified. This is… You can say this is
how the nature is. This can make people unhappy,
because you see I suppose to be doing science;
and suddenly out of the hat, I pull out something
like this, which cannot be derived. Unfortunately,
this formula cannot be derived from anything
which is more fundamental.
So, if I like, I can take this as a postulate
of quantum mechanics and develop the subject
of quantum mechanics there from. Now, as I
said, maybe this will make you unhappy, but
then I would like to remind you that, in Newtonian
mechanics, when you start, you have studied
force is equal to mass into acceleration.
Have you studied a derivation of that? See
what is happening is that, you have observed
the field; and then from that observation,
you would say this is how it is. Or, in classical
electrodynamics, there are four equations;
which again has abstracted out of experimental
observations. And the good thing is once you
assume these four equations, then you can
make lots of predictions; and all of them
in agreement with experiments. So, similarly,
if you make this assumption, this you will
have to think of it as an assumption. Once
you make this assumption, then you can develop
the whole subject of quantum mechanics from
here.
There is a nice analogy though this does not
concern the subject of quantum mechanics itself;
whereas, there will be lot of similarities
between the way an electron behaves, quantum
mechanical particle behaves and that of a
drunken walker. Essentially, what I am saying
is that, there is lot of similarities between
the way an electron moves or lot of connections
actually, not… I wish they should be completely
similar. The mathematics used to describe
the motion of an electron, is very similar
to the mathematics that is used to describe
the walking of a drunken person. That is the
precise way to put it. You may wonder why;
let me make clear. I suppose you are all familiar
with Tintin comics; even if you are not familiar,
it does not matter. This is a character in
Tintin comics; his name is captain Haddock
and he likes to drink.
So, if suppose this person – captain Haddock
– in the evening, he always gets drunk.
So, imagine he goes to the pub maybe around
9 O’clock in the night; stays there till
10 O’clock in the night every day. And then
10 O’clock, he will come out of the pub
and he is completely drunk; he wants to of
course, go home, but he has absolutely no
sense of direction. So, what he does is he
will execute what is referred to as a random
walk. Now, a mathematician will describe the
random walk in discussion. He will say that,
each step that the person takes each in a
direction independent of the direction of
the previous step. See if he takes a step
like this, then after having taken the step,
he has absolutely no memory in which direction
he has to go. So, the next step will be independent
of the previous step. So, that is the characteristics
of a random walk.
Suppose I am going to study his walking. So,
this is the bar and he comes out at 10 O’clock
in the night. And on the first day, suppose
I observe him; then maybe he would have taken
such a path, which I suppose is not very likely,
because he will be executing all kinds of
walks. But, maybe this is one of the possible
paths. And he will be at this location maybe
perhaps after 1 hour. The next day if I observe
him; and he will not definitely follow that
path, but another one; and the third day,
maybe another one; fourth day, another one
and so on. Suppose I am persistent I have
studied him, I have looked at his behavior
for 1 year; then on the 366-th day, I am going
to observe him let us say. And suppose one
of you ask me, where will he be after 1 hour;
will you able to answer the question? Definitely
not; you can say, maybe there is a large probability
that, he will be at this area. The probability
that he will be 10 kilometers away; that is
0. That kind of statements you can make. Therefore,
it is like the electron. See if I had an electron
here now and if you ask me, where will be
after 10 nanoseconds; I can only give you
the probability. Similarly, in this case also,
I can only give you the probability.
Now, suppose I want to calculate the probability;
how will I do is; suppose I want to calculate
the probability. In fact, I want to calculate
the probability that the electron… Not the
electron, but captain Haddock is at this point;
I want to calculate the probability density
that the random walker is at that location
after 1 hour. How will I do that? The answer
is here in this picture. See you will say
that, it is not very likely that, he would
have followed perhaps a path like this; but
maybe a path like this is more probable than
that. Therefore, you see what should I do
is, I should think of all the possible paths
that he could have taken starting at the initial
location and ending there at the final location.
You have to think of all the possible paths.
He could have taken in any one of them. What
happens is that, some of them would be more
probable than others. Therefore, you have
to assign probabilities to different paths.
And then for each path, you have to do a calculation
of the probability and then sum over all the
possible paths you are going to get the probability
– total probability.
See this is very much analogous to the case
of the electron. But, in the case of electron,
there is a large difference – huge difference;
here you are calculate… for each path, you
are assigning a probability; whereas, in the
case of the electron, for each path you are
assigning not a probability, but a contribution
to the wave function; psi 1, psi 2, psi 3,
etcetera remember; they were wave functions.
Therefore, for example, in here you have psi
1 plus psi 2 plus psi 3 etcetera until psi
8 perhaps. So, these things – wave functions
– they are in general, complex; they may
be positive or negative. Therefore, if you
added complex numbers, you see the final answer
can be 0.
Whereas, in the case of the random walker,
what happens is that, you are adding not wave
functions, but probabilities. Probabilities
are constrained in that; they have to be real
first of all; and the second thing is that,
they are all positive, they are never negative.
Therefore, if you add it to probabilities,
you see they cannot cancel each other. And
therefore, in the case of the random walker,
there is no interference. Whereas, in the
case of the electron, because you are adding
together different contributions to the wave
function, the wave function can cancel each
other. And therefore, you have interference
happening in the case of the electron. But,
interference will never happen in the case
of this random walker. And that is the major
difference. But, otherwise, mathematically
speaking, it is the same kind of object; and
in fact, instead of S, which is there in quantum
mechanics, the object that happens – they
do not worry about that equation. It is not
very understandable.
But, what happens is that, instead of this
action, what happens in the case of the random
walker is, you have this object, which actually
resembles the action. The contribution of
any given path is determined not by the action,
but something, which is similar to the action.
And what happens is that, you have to take
the exponent; you have to calculate the exponent
of that. And so what I want to say is that,
the mathematics. So, if a quantum mechanics
and the mathematics; describing the motion
of a random walker is very very similar. And
of course, we are very familiar with random
walks. Where do they occur in science? Answer
is that, if you had a colloidal particle,
which is immersed in a solution; and if you
observe it under the microscope, then you
will see that it executes random motion. It
executes random motion much similar to the
walking of a random walker or a drunken person.
So, this is what is shown in this slide. This
is the colloidal particle. It is big enough
for you see under the microscope let us say.
And then if you observe it, you will find
that, it seems to follow some trajectory,
which may be something like this and which
resembles the random walk of a drunken person.
And this is how a colloidal particle will
diffuse in solution, because if you put a
colloidal particle in solution… I wonder
whether you are familiar with diffusion equations
themselves. If you have a particle – colloidal
particle put in a solution, what is going
to happen is that, there are molecules of
the medium surrounding it. These molecules
will come and hit the ((Refer Time: 31:38))
and all. The colloidal particle, at any given
instant of time, you see there may be more
particles hitting from this side than from
the other side. And the momentum imparted
in this direction would be more than the momentum
imparted from the other direction; and then
the particle will go in this direction. And
when it has reached here, maybe the momentum
of the particles imparting from this side
is more than the momentum of particle imparting
from this side. So, then the particle will
go in that direction and then it will go like
that randomly in direction. And that is what
happens. This is essentially the thing that
is responsible for diffusion of a gas or diffusion
of a liquid or diffusion of colloidal particles
in a solution.
So, this we have already seen. And as I have
told you, in the case of the electron, things
are relatively bit weird, because you are
not adding the probabilities, but instead
you are adding the wave functions. And wave
functions are actually complex. And therefore,
they can cancel each other. The signs may
be positive or negative. Therefore, there
may be cancelations. Now, if he found likes,
it is possible to develop the whole subject
of quantum mechanics; you see starting from
this equation. In fact, as I told you earlier,
it is possible to take this as a postulate
and develop the whole subject of quantum mechanics
from this formula. That is possible, but mathematically,
it is a little bit difficult. It is little
bit involved.
If anyone is interested, they should have
a look at this book. See if you look into
this book, you will find that, the subject
of quantum mechanics is developed starting
from this particular formula. The whole thing,
the whole book is based upon this formula.
It starts from here and develops everything
or almost everything. Now, mathematically
speaking, this is the rather difficult. And
I do not know if any chemistry, quantum chemistry
books, where the approach is described. But,
it is a very beautiful and very nice approach.
Many of the physics book these days do make
use of this approach. If you want to get the
details of this approach, you will have to
read this book or some of the good quantum
mechanics books. For example, there is a book
by R. Shankar – very very nice book entitled
principles of quantum mechanics. It has a
chapter on this approach. But, because it
is mathematically involved, I shall not continue
to discuss this.
But, what I will do is, I will switch over
to the usual procedure of introducing quantum
mechanics, which is to introduce it as a set
of postulates. So, the way one would go about
is this. If you have wave phenomenon after
all, we are seeing that, there are waves.
So, if we have waves, the argument is that,
there should be a wave equation, because people
are very familiar with waves; and waves are
always described by wave equations. Therefore,
if we have wave phenomenon, there must be
a wave equation. And it so happens that, the
wave equation that is appropriate for the
description of motion of electrons or the
wave equation of quantum mechanics, is an
equation that was introduced by Schrodinger.
This is Schrodinger. And matter waves obey
his equation. So, we will see this equation
later.
Now, I said that, in the case of a colloidal
particle, which undergoes diffusional motion,
there is mathematically speaking the motion
of the Brownian particle. And the motion of
the electron, are mathematically speaking
similar. And the motion of a Brownian particle
is actually described by what is referred
to as a diffusion equation. Again, I will
not go into the details of this equation.
But, in the case of a Brownian particle, you
are concerned with the probability of finding
the particle at a particular location x at
the time t. So, this is the kind of equation
that a diffusing particle will obey. And this
D there is known as the diffusion coefficient.
So, we know that, in the case of Brownian
motion, this kind of diffusion equation…
Do not worry about this equation, if you are
not familiar. But, there is an equation, which
describes a diffusional motion of a particle;
and it is that equation.
Now, I have argued that, mathematically, diffusion
and the motion of an electron in quantum mechanics
are similar. So, if diffusion is governed
by such an equation, then it is very natural
that, for the electron also, there should
be an equation, which resembles that. And
that equation is nothing but the Schrodinger
equation. It is this equation. You can see
the similarity if you look at it carefully.
This is for a free particle – particle moving
freely in space; you have d by dt of probability
here. You have the diffusion coefficient;
then you have del square. Del square involves
differentiation with respect to position.
And you can see that, in the case of electron,
you do not have the probability, but instead
you have the wave function. And the wave function
changes with time. How does it change? It
changes. According to this equation, instead
of the diffusion coefficient there, you have
this say, minus h cross square by 2m and then
of course del square; there you have… here
also you have del square. So, this is a structure…
Mathematically speaking, at least the structure
of it is very similar to the case of diffusional
motion.
So, the basic equation of quantum mechanics
is actually this equation, which is the Schrodinger
equation. In comparison with what I had written
earlier, we will look at this in more detail
later. In comparison with what I had written
earlier, there is an additional term. This
is the potential that the particle feels.
Earlier I was ((Refer Time: 39:23)) of a free
particle, which is not subjected to any potential.
But, if you have a potential, this is the
equation that it obeys. And not only that,
if you know the wave function at any initial
instant of time, it is possible to use this
equation. If you know the wave function psi
at an initial position x at the time 0, then
it is possible to use this equation and find
its value at the position x itself if you
want at a later time. Or, at any position,
we can calculate at a later time. That is
the idea. But, we will see more details of
this later.
My introductory lectures; I said first 3 or
4 lectures will be in introductory in nature.
This is the last one. So, it is natural that,
I should give you some references. The last
part of what I have told you; those are the
things that we will be discussing in more
detail later. And therefore, things were not
clear to now. It will become clear as we proceed.
But, let me give you some references to this
part. There are very very nice, very beautiful
books; you should have a look at them; they
are non-mathematical; I have selected only
non-mathematical books. The first book is
actually by G. Gamov. It is called Tompkins
in Paperback. This is a book that was written
in the… I do not remember the years, but
probably 1930s. Between 1930 and 1940, this
book was written.
Gamov was a Russian scientist, who migrated
to the west during the communist era. And
he wrote series of articles in one of the
British newspapers. And these articles were
collected together into the form of a book.
So, the book is very interesting if you have
not read; or, not seen it or read it. The
story is this actually. It is kind of friction.
Mr. Tompkins is a clerk in an office in London.
He has absolutely no interest in physics.
But, it so happens that, his girlfriend is
the daughter of the physics professor at the
university. So, the physics professor gave
some lectures – some popular lectures on
quantum mechanics, relativity and so on. And
just to please his girlfriend and hopefully
the future father-in-law, he goes and listens
to the talks of this professor. And as I said,
he has absolutely no interest in physics.
The professor speaks about quantum mechanics,
relativity theory and so on. Tompkins does
here a few things, but dozes off while he
is listening. So, he goes to sleep. And when
he is sleeping, he dreams. And the subject
matter of the book are his dreams.
So, in our universe, the value of h cross
is 
approximately 10 to the power minus 34 Joules
second; and this is extremely small – 10
to the power minus 34. Therefore, if you think
of a wave or if you think of a particle, the
wave length associated with that particle
will be lambda, which is equal to the h divided
by the momentum. So, because the value of
h is h cross and h is quite small, the wave
associated with any particle is very very
very small. And if you think of the uncertainty
principle – delta x into delta p; I told
you what it is; greater than or equal to h
divided by 4pi. This number is very very small.
So, the effect of the answer in the principle
simply cannot be seen in our world. In our
world meaning in our everyday life. In our
everyday life, we do not see any effect of
these things.
But, suppose you had a universe, where the
value of h cross is not 10 to the power of
minus 34; but suppose it is 1 Joules seconds,
then what will happen? You will be able to
see quantum phenomena with your eyes. And
he dreams of a world in which value of h cross
– this is of the order of 1 Joules second,
so that they can see wave phenomena happening
in his everyday life. So, one of the things
that Gamov describes is this. He has a car;
he puts the car into the car shed in the night;
closes the door; and then goes to sleep in
his room. Now, there is a typical quantum
phenomenon, which is referred to as tunneling.
So, the next day, he goes to sleep; the next
day, he comes out of his room and finds that,
the car has behaved like a wave. And when
it behaves like a wave, it actually can go
pass through even barriers.
So, it passes through the door, which is closed
of course, and comes to the outside leaving
the door intact. So, this is a typical quantum
phenomenon, which can be observed if the value
of h cross is not so small, but maybe 1 Joule
second. And of course, in chemistry, there
are many many effects of this kind of behavior,
which is referred to as tunneling. He also…
I mean if you look into the book by Gamov,
you will find that, he also describes relativity
theory. I will not go into description of
that; some part… little bit of chemistry
for example, combination of chlorine and sodium
and so on. So, please do have a look at this.
Then, there is another book, which is that
one; it is Robert Gilmore. He has written
a book, which is called Alice in Quantum Land.
You know this story, where Alice goes to the
wonderland. But, here instead of going to
the wonderland, she goes to the quantum land
and sees quantum phenomenon. There is another
one written by Gribbin; very nice book, very
strongly recommended. It is available to purchase
in most book house. In fact, in the airport,
I sold the book today; not today, but yesterday.
So, this is called Schrodinger’s Cat. There
is a very famous paradox, which is referred
to as a Schrodinger’s cat paradox; referred
to Schrodinger’s cat paradox. The book contains
a description of this paradox; but not only
the paradox, the whole of quantum mechanics
is described in a non-mathematical language.
So, very very nice book.
Then, this is… This book is by Cropper.
It is the biography of a people like Schrodinger,
Planck, Icenberg, who have been involved in
development of quantum mechanics. And not
only he gives you biography, but he also gives
you an introduction to the subject itself;
written very nicely. You see if you have cats,
then naturally you will have kittens. And
that is the subject of… I mean that is the
title of this book by Gribbin. See since quantum
mechanics was formulated, people have been
trying to disprove it or trying to find exceptions
to quantum mechanics. And so people have been
trying to all kinds of experiments; trying
to find conditions in which quantum mechanics
is not obeyed. And as of today, nobody has
been able to find any exceptions. So, this
gives you a history of all these experiments
that have been carried out till 1995.
This book by Gamov is very old; and therefore,
it was rewritten by somebody else called Stannard.
And it is called The New World of Tompkins.
So, it is necessary for you to read either
this or that. I spoke about Feynman. There
is a very nice and very beautiful book written
by Feynman. The spelling in the name is not
correct. The title is QED; QED stands for
quantum electro dynamics. But, you should
not be frightened by the name. This is a book,
which is very readable and very understandable.
I am sure you will enjoy reading this book.
This is a little bit more mathematical book.
Only if you are mathematically minded, this
book is advised; otherwise, it is not such
a great book.
You should also look at this website, which
I have referred to earlier. This is the famous
quotation that I have shown you earlier. He
said that, it is safe to say that, nobody
understands quantum mechanics. And you also
look at this quotation, which is from Wheeler.
Wheeler happened to be the PhD supervisor
of Feynman. And finally, I would also like
to add that, there is a book by… There are
3 books by R. Venkataraman available in India;
beautifully written books these actually.
There are 3 of them. I will give you the reference
to it maybe tomorrow perhaps. Very nice books,
not expensive; it is very readable.
So, here is the last slide. I have spoken
only for 3 hours and it is surely not enough
to convince you that, it is such a beautiful
subject. But, I hope by the end of the course,
which will be roughly 40 lectures, I hope
you will be convinced that, quantum mechanics
is actually a very beautiful subject. What
you should do is you should think of the physics;
you should try to understand the subject in
the physical fashion rather than approaching
it mathematically. Once you have understood
the physics, then understanding the mathematics
actually becomes easy. I think I should stop
now.
