We take off now from last semester where we
left off with quantum physics this semester.
Before we begin the course proper, I would
like to get a little background from you on
what you already know, how we should run this
course, what the modalities are and so on.
As far as the evaluation is concerned, it's
the usual two quizzes and a final exam. Philosophical
and interpretational aspects of quantum mechanics
are still open in the sense that, all questions
have not been answered. In spite of over a
hundred years of effort, there are unresolved
questions, there are deep mysteries, and there
are central mysteries and issues. Some of
them may turn out in the future to be non-questions
in the sense that, you yourself may turn out
to be asking the wrong question and so on.
But at the operational level, it's actually
easier than classical physics. The mathematics
is simpler than classical mechanics, for instance,
and it works over a huge wide range of phenomena.
As far as we know, there are no violations
of quantum mechanics. What we don't know
is its interpretation to complete satisfaction.
We don't know if it's a complete theory,
we don't know if it will answer all questions
and so on and so forth. But certainly as a
calculational device, it's extremely successful
and as a way of understanding natural phenomena,
it seems to be the most basic of all subjects,
in the sense that, everything at the ultimate
level turns out to obey quantum mechanics.
So with that kind of a very sketchy preamble,
let me go on to state a few properties in
quantum mechanics. Depending on what the response
is, whether it's familiar or not and so
on, I start as writing down the formulism.
Because otherwise I could start by simply
saying the postulates of quantum mechanics
are as follows and as we go along, when things
become inconvenient, I add one more postulate
and then we go on and so forth and then you
just say this is a set and that's it, these
are the rules and so on. I shouldn't want
to do that Since it is a conceptually very
rich subject, I would like to find out what
it is that you know, what you don't know
and so on before we decide how to approach
this subject.
Now the first point about quantum mechanics
is that, all of you have heard that quantum
mechanics applies in the microscopic domain.
It applies to electrons, atoms and things
of atomic sizes. This is only partly true.
It applies to everything. It applies to human
beings, galaxies, the universe presumably,
electrons, photons and so on. It is just that
the manifestations of quantum mechanics become
very dramatic when you look at extremely small
objects. They become even more dramatic when
you look at extremely small objects moving
very fast relativistically. So, that's the
reason why one feels that quantum mechanics
is applicable only in the microscopic domain.
Not true. It's applied everywhere. It has
some manifestations which are very dramatic
even in real life, even in the macroscopic
domain. For instance, my favorite example
is the fact that solids are rigid objects.
The fact that I don't go through the floor
here, the fact that matter doesn't penetrate
itself is due to quantum mechanics. It's
due to the Pauli Exclusion Principle which
follows from quantum mechanics. So there are
these very deep consequences. The phenomena
you are used to in daily life like magnetism
or electrical conduction or the propagation
of sound in solids are all quantum phenomena.
Not just super fluids, super conductors and
so on which are given as usual examples of
microscopic quantum mechanics but even ordinary
phenomena like electrical conduction, the
conduction of sound in solids, magnetism,
the existence of permanent magnets, the existence
of diamagnets, paramagnets etc depend on quantum
mechanical principles widely. They are not
explainable classically.
You know that in the early days of quantum
mechanics, there were several ways in which
people found that classical physics failed.
One of them was the black body radiation.
So the very fact that collection of photons
behaves quantum mechanically is an obvious
instance where you have quantum mechanics
for large systems. When you go to the atomic
domain, one of the consequences of quantum
mechanics become very apparent and it's
called the uncertainty principle.
Now I should like to find out from you what
you know about the uncertainty principle.
What's the uncertainty principle? Let me
take one statement at a time. The position
and momentum cannot be simultaneously and
accurately measured. So we will make this
precise now. Position and momentum of what?
Any particle One particle, two particles,
a system? Well, let's assume that systems
are made up of particles. We have individual
particles and let's look at the given particle,
an individual particle. Now let's take the
statement word by word.
Simultaneously means at the same mathematical
instant of time and by accurately, to what
precision? To arbitrary precision, to infinite
precision in principle. Cannot be measured.
Now what does that mean? That means a failure
of experiment? Is it just that we don't
have a good enough piece of equipment? It's
a property. It's fundamental. It's more
than just a statement about measurement. It's
nothing to do with measuring apparatus although
that's the way it's normally looked at.
The position and momentum cannot even be precisely
defined for such objects at a given instant
of time.
The property of momentum and the property
of position cannot exist, in the sense of
being measurable with infinite precision at
a given instant of time. So it is not a failure
of experiment. It's not a failure or a lack
of resolution or technology. It's an intrinsic
property of nature for these objects. Now
what sort position and what sort of momentum?
Is it linear momentum? Angular momentum? Position
I understand. Canonically conjugate. We already
looked classically at canonically conjugate
variables. It's the same thing. Canonically
conjugate coordinate and momentum are pairs.
They would obey an uncertainty principle.
What about the x component of the position
of a particle and the y component of the momentum?
Do they satisfy an uncertainty principle?
They don't since they are not canonically
conjugate to each other. x with px, y with
py and so on are canonically conjugate variables.
What about 
non-cartesian coordinates?
What about, for example, the radial distance
in spherical polar coordinates in some coordinate
system from the origin of a particle and the
radial component of the momentum along the
radius vector? These two are canonically conjugate
to each other and it's these two which in
quantum mechanics would satisfy an uncertainty
principle. If they are not canonical conjugate,
this is not true. So already you begin to
see that the framework of classical Hamiltonian
mechanics is being translated into quantum
mechanics.
So Hamiltonian's play a very special role
and the whole assumption is that you already
have a Hamiltonian for a system and you have
the Hamiltonian framework of classical mechanics
and then you translate to quantum mechanics.
The question of how do you do quantum mechanics
for systems which are not Hamiltonian in the
classical limit is a much deeper and much
harder question which would not be addressed
in this course. It's not at all clear how
one would do this. In particular, this immediately
means that if you have dissipation in a system,
classically you know that it cannot be an
autonomous Hamiltonian system. Then you run
into trouble and the corresponding quantum
mechanics is not clear at all. It doesn't
mean friction doesn't exist. It's just
that the extension of the usual formalism
of quantum mechanics to these dissipative
cases is not straight forward and the problem
of dissipation in quantum mechanics is still
an open problem. The question of what happens
in open systems which exchange energy and
momentum and so on with the surroundings is
still an open problem. In that sense, the
quantum mechanics we are going to do is really
very elementary for systems which are essentially
going to be at absolute zero of temperature.
So we won't bring in temperature and systems
which don't have any dissipation at all,
till fairly late in the course. So it's
for those kinds of coordinates in momentum
that you have the uncertainty principle. Now
what does the uncertainty principle actually
say?
Let's look at x and px for a particle. This
is a canonically conjugate pair. Classically
I know that this is equal to 1. In the sense
of Poisson brackets, this is 1 if I look at
a single particle and I say x is a generalized
coordinate and px is a corresponding conjugate
momentum, the Poisson bracket of x with px
is 1. Now there is a similar relation written
down quantum mechanically. What does the uncertainty
principle actually say?
It says that delta x times delta px is greater
than or equal to h cross over 2. This is certainly
true. h cross is what I use for h over 2 pi
. h is the Planck's constant. This is an
experimental fact. This has been around now
since 1925 or so and has not been contested.
I should like to first find out from you what
you mean by delta x.
Now what do you mean by uncertainty? Is it
the least count of some ruler or measuring
device? It cannot be the least count. Well,
in principle if I have a sufficiently good
microscope or an accurate length measuring
device with an arbitrarily good least count,
I can measure position to arbitrary accuracy.
So what do you mean by delta x?
I make many measurements and take the standard
deviation of them but I can still improve
the standard deviation by making many more
measurements. Thus we come to the fact that
it's not definable with arbitrary precision.
A particle of this kind we are talking about,
cannot have a sharp value of its position
and its conjugate momentum, be it any state
of the particle. So it has nothing to do with
the precision of the ruler or your experimental
measuring device. This is well beyond all
that. The quantity delta x is a standard deviation.
So it says that you take your measuring device,
no matter how accurate it is and continue
making these measurements. You would still
find a standard deviation no matter how many
measurements you make and that standard deviation
is delta x.
Now obviously you need a calculational device
for delta x. There is a rule which helps you
to calculate this delta x and similarly for
delta px. The moment I say standard deviation,
I have to define a probability distribution
so as to find the standard deviation with
respect to that probability distribution.
The whole point of quantum mechanics is that
it should tell you what those probability
distributions are. The equation which a quantity
related to the probability distribution satisfies
is called the Schrödinger equation.
This quantity which you call the wave function,
which we will introduce shortly, will help
you calculate the probability distribution
and after you find the distribution, you compute
the average. Formally, standard deviation
is defined as x square minus x average square
to the power half. x square minus x average
square to the power half is the variance and
the square root of the variance is a standard
deviation. The only difference is that when
you normally do classical experiments or take
any statistical quantity and measure it repeatedly,
we obtain a certain histogram.
In quantum mechanics, the angular brackets
refer to averages with respect to a quantum
mechanical probability distribution. That
distribution will be specified through the
Schrödinger equation. Once you do that, the
uncertainty principle is certainly true. In
practice, the uncertainty principle for a
physical particle will find you an answer
which is very much bigger than h cross over
two. The reason is, over and above the quantum
mechanical uncertainty, which is intrinsic
in nature, there would be other uncertainties
such as, the fact that the system is in contact
with surroundings, thermal agitations, noise,
vibrations, other perturbations from outside,
a passing comet that will change your apparatus
a bit, etc would add up on top of quantum
mechanical uncertainty. But in its pristine
form, the inequality in the uncertainty principle
is the quantum mechanical product of quantum
uncertainties, calculated in a very precise
way. Notice it's an inequality, which means
that it is conceivable that you can arrange
matter such that it's an equality. That's
the least value it could take. It could never
become less than a certain non-zero positive
quantity and of course you know that this
is extremely small, in the units we are normally
used to. That's the reason you don't see
this very often unless you go to microscopic
objects because h cross over 2 here, is of
order 10 to the -- 34 in the standard international
units.
Since microscopic objects like a piece of
chalk or a person or a rock, the energy involved
is of the order of joules or tens of joules,
and that's ten to the power minus thirty
four joules, you don't see quantum uncertainties
unless the objects themselves are extremely
small, moving extremely fast.
Now this is just one form of this principle
and it's an inequality whereas quantum mechanics
is a very fairly precise subject in the sense
that it will give you equalities always. So
this inequality is a very weak form of quantum
mechanics. It definitely cannot be violated
but in any given situation, you should be
able to actually calculate what's delta
x, delta px and find out what the right hand
side is. So you don't need the inequalities.
Just that no matter what you do, this cannot
be violated.
Incidentally is it possible to have delta
x arbitrarily small from this inequality?
Yes, but the price you pay for it would be
that delta px would become proportionately
large. we are going derive this we are going
to derive this inequality from the Schrödinger
equation from the postulates of quantum mechanics
one of which would be the Schrödinger equation
we would derive this your this inequality
here um it's a fairly straight forward thing
Incidentally this is not the most general
form of the inequality. You should immediately
ask what if I have two arbitrary canonically
conjugate dynamical variables. Is there a
Poisson bracket? Is there an uncertainty principle
between them? The answer is yes. You could
in fact ask if I have two physical quantities,
doesn't have to be position and momentum,
two physical quantities associated with any
object, can I write the product of uncertainties.
The answer is yes you could. It's going
to be generalized.
What I meant by we can derive is that, no
matter what experiment you do, the inequality
is never violated. So in that sense, it's
an experimental fact. However, we know why
that is so because from the postulates of
quantum mechanics, this follows as a consequence
and is verified. So it essentially says that
we verified quantum mechanics or the predictions
of quantum mechanics. You can never verify
a theory. You can only verify its predictions.
Incidentally, this is not the most general
way of looking at it. It will turn out 
that Poisson brackets for physical quantities
or canonical variables are replaced in quantum
mechanics by operators. We will discuss this
in great detail .Once you have operators on
the right hand side; you need to know what
do you mean by expectation values or average
values of these operators. We will go through
that in detail.
It will turn out that if you had any physical
quantity A, and any other physical quantity
B, whether A's and B's are functions of
the canonical variables, the q's and p's
which we take over from classical mechanics,
then, quantum mechanics will tell you that
this is greater than or equal to on the right
hand side, a certain quantity which is associated
with A and B. It will turn out to be this
is greater than equal to one half the modulus
of the expectation value of the Poisson bracket
of quantum bracket, that is, Poisson bracket
of A with B, which turns out to be the commutator
and I am going to define what it means. So
that's the general uncertainty principle
and this is something we are going to show
explicitly. When you apply it all the way
back to a Cartesian component of position
and it's canonically conjugate momentum,
this right hand side will reduce to h cross
over two.
In any case, it's clear that you have the
modulus of some number or quantity here or
some complex number in general, and you have
the modulus of it, the modulus cannot be a
negative number. It could be zero. Now you
could ask from where Planck's constant comes
in. It will generally appear here in the calculation
of this object. Occasionally there are cases
where it would become zero. For example, if
you had x with py, this is 0.
In quantum mechanics, the corresponding term
would mean something like delta x delta py
is greater than 0 or equal to 0. Do we have
enough technology such that delta x and delta
px 
is exactly equal to h cross over 2? That's
a very good question. Do we have sufficiently
precise experiments where you could actually
take a given particle, take its x component
of the position and the x component of it's
momentum, measure delta x delta p, put it
in some state and measure this and then discover
that delta x delta p is actually equal to
h cross over two? The answer is yes. That's
called the minimum uncertainty state because
that's the least value the uncertainty could
have. In fact, we can go better. Not in terms
of x and p, but in terms of quantities which
satisfy relation similar to that.
If I schematically plot delta x here with
delta p here on this side, these are standard
deviations so they are not negative numbers.
They are both non-negative numbers. If you
plot this, the minimum uncertainty occurs
when delta x delta p is equal to h cross over
two and what sort of curve is that? It's
a rectangular hyperbola because you have x
y is equal to constant. That's a rectangular
hyperbola.
So, in fact, you have a situation like this.
On this curve, you have minimum uncertainty
in suitable units. If x is length, then p
is MLT inverse. So, they don't have the
same physical dimensions. If you reduce them
to both sufficiently dimensionless variables
by dividing through a momentum scale and dividing
by a length scale in the problem, then, it
will turn out that the symmetric point here,
where delta x is equal to delta p, in some
sense, is the best you can do.
The optimal thing you can do is to reduce
the error in x to the least possible, the
uncertainty in p to the least possible and
that's the least we can do. Such states
are realizable and we will talk about it in
detail. In fact now, technology is such that,
not in the context of positions of particles,
but in the context of quantum optics states
of the radiation field, where these two quantities
are a pair of variables which are analogous
to x and p; which are the counter parts, you
can actually move along this curve here. So
you could get to a stage when you are here,
which implies that delta p is below the uncertainty
limit but delta x is greater. These are possible
and they are called squeezed states. So it
means that you squeezed the uncertainty in
one of them. You are practically going to
zero but the price you pay is that the uncertainty
in the other one increases enormously. Theoretically,
you could even go down to zero in this case
but the other variable would go to infinity.
Most of the time, the states of the system
that you deal with are sitting out here, where
delta x delta p is greater than h cross over
two. This is where the product 
is exactly equal to this but you could go
to this region out to there. In general, that's
what happens unless you arrange matters extremely
carefully.
So, the uncertainty principle in quantum mechanics
has nothing to do with the precision we have
available. Even with the best of precision,
the uncertainty principle still operates and
it says that the product of uncertainties
in any variable and its conjugate variable
has a least possible value. But if the pair
of variables is canonically conjugate, classically,
then this Poisson bracket is not zero. It's
equal to 1 and correspondingly, it turns out
that you have a relation of this kind out
here.
Similarly, delta r delta pr is greater than
or equal to h cross over two and so on. The
crucial thing is that you cannot measure one
at one time, another at another time and then
say that I have accuracy. It must be simultaneous,
at the same instant of time. That's not
possible and it has nothing to do with equipment.
It is simply there as fact of nature. The
problem is you have to put on your quantum
thinking cap and think classically. The moment
you have a relation like this, it implies
that you cannot specify x and px to arbitrary
accuracy. This means that for a particle which
obeys quantum mechanics in the phase plane,
if this is x and px, I cannot say that the
position of that state of the particle is
there.
I cannot put a point down which flies in the
phase of all that we did in the previous course
where I talked about phase trajectories. The
phase trajectory is a phase space point moving
along and I said the point specifies the state
of the system and all the coordinates and
the all moment are specified. But you cannot
do that now because the moment I put a point
like this it implies that I know both x and
px to arbitrary precision. So states of a
system cannot be any longer represented by
points in phase space. We have to replace
it with something else which obeys the uncertainty
principle. Therefore, there is an intrinsic
fuzziness. You cannot do better than something
like this.
That's the best you can do because you cannot
say there is a point in phase space. The moment
you cannot say there is a point in phase space,
as time evolves, you cannot talk about a phase
trajectory either because the earlier classical
picture was that, this point moved around
as a function of time and gave you a trajectory.
At every instant of time, you identified a
phase space point with the state of the system
or the particle. You cannot do that now. You
have a little fuzzy ball and this fuzzy ball
is moving around, distorting its shape and
you don't know what it's going to do.
So this entire classical description of particles
with specific trajectories goes out of the
window. You simply cannot talk about it which
is why; when you study the hydrogen atom for
instance, you cannot talk about an orbit for
the electron because the moment you say something
is orbiting, you have immediately told me
its position and its momentum.
Because if you write down an orbit, here is
a center of attraction and here is an orbit,
then at this instant of time you said this
is the position with respect to the center
of attraction and the tangent here will give
you the direction of the momentum. You multiply
the velocity by the mass and you get the momentum.
This is not possible. So the idea of orbits
is gone completely. It may be a very convenient
picturisation even in quantum mechanics under
certain circumstances but the fact of the
matter is electrons don't have orbits.
How do we measure delta x and delta px simultaneously?
We will come to that. We will talk about how
we can possibly think of measuring it experimentally.
Does this property arise because particles
have dimensions? No, it's not because particles
have dimensions. That's a good question
he says you cannot specify a given point maybe
because it's got a finite extent no. Even
if it were a point particle, even if it were
a mathematical ideal point mass, this is true.
So it has nothing to do with finite extent
of the particle. It is an intrinsic property.
That's the way it is. So we can no longer
speak about orbits. In quantum chemistry,
one is used to talking about molecular orbital
theory. The s orbitals, p orbitals, pi orbitals
and so on. They are convenient picturizations.
That's not the way the electron actually
behaves in an atom. For instance, when you
do the Bohr model of the hydrogen atom, you
are told that in the ground state of the hydrogen
atom, the orbital distance is one times the
Bohr radius and it's moving in a circular
orbit. That cannot be true.
Because if that were true, you have this particle
here with the center of attraction O and something
moving at the Bohr radius a0, the orbital
angular momentum of this particle about the
center of attraction cannot be zero. It's
not going through the particle at all. So
you calculate r cross p, it's supposed to
be uniform circular motion and of course,
the orbital angular momentum is non-zero.
But at the same breath you are told that in
the ground state of the hydrogen atom, l = 0.You
have the hydrogen atom in which you described
the state of the electron by a principle quantum
number m, orbital angular momentum quantum
number l and the magnetic quantum number m,
then in the ground state i.e. the lowest energy
state, n = 1, l = 0 and m = 0.
And you are also told that if you look at
the component of the orbital angular momentum
perpendicular to the plane of the orbit, it's
l times h cross. But if l is 0, that is zero.
That cannot be true if it's orbiting at
a finite distance. So that alone should tell
you that this picture is wrong immediately.
This is certainly true but they cannot be
supported by a picture of this kind. On the
other hand, it is true that if I calculate
the average distance or the most probable
distance, is in fact a zero. But that's
a statistical and a quantum mechanical average
here.
So the real fact of the matter is that the
electron does not have a position and a conjugate
momentum simultaneously. It is not a point
particle in that classical sense of the word.
You have to get out of this classical way
of thinking. In some kind of crazy extended
object, you need something to describe it
by what's called as the state of the system,
the state vector which is a generalization
of the wave function. We will discuss it in
great detail and I will tell you how the state
vector is defined and how to calculate its
properties here. As far as I am concerned
that's what the electron is in reality.
I don't answer questions like is it a wave,
is it a particle, is it a particle on some
days and a particle on another days etc. These
are terms which are meaningless when you apply
it in the context of quantum mechanical particles.
The failure is not on the part of the quantum
mechanical particle. The failure is not on
the part of the experimentalist. The failure
is on the part of the English language. Words
like wave particle and so on have been coined
by us to paraphrase a set of properties. These
properties are the properties of objects with
which we have daily experienced. And then
when you say a particle in the mental picture,
you have a particle like a billiard ball,
a hard rigid object, very compact, sometimes
idealized even to a point, massive, something
which is localized, carries energy, momentum
and so on. On the other hand, the word wave
is mentally based on our experience with them
in nature which we see around us like water
waves or sound waves.
These are diffused objects that are not localized.
They don't have rigid boundaries. They are
not hard. They are smooth, gently undulating
everywhere, delocalized in space and time
etc. they carry wavelength, frequency and
properties of that kind. We put all classical
objects into these two bins. One of them would
consist of particles which are compact, localized,
hard objects, carrying properties like energy,
momentum etc and the other is waves which
carry properties like wavelength, frequency
etc. they are delocalized extended objects.
And we know that in daily life, these are
mutually exclusive. The wave is not a particle;
the particle is not a wave. We can see and
measure properties and decide whether it is
falls in this basket or the other basket.
What we shouldn't do is to extrapolate this
categorization of objects into waves and particles
to the microscopic domain because in that
domain it is conceivable that there exist
objects which have some of these properties
and some of those properties. It's also
conceivable that we have objects which have
these properties or those properties depending
on how you probe them or how you measure these
properties. That's what happens for electrons
or microscopic particles. So the failure is
not on the part of the electron or on the
part of our ability to probe nature. It's
a semantic failure. We shouldn't use terms
which are mutually exclusive sets of properties
in the macroscopic world to describe the microscopic
world.
If you understand that, there is no wave particle
duality mystery. It is just that this description
fails completely and you need a better description.
It so happens, not unfortunately but fortunately
that the description is not in terms of ordinary
language. It is intrinsically mathematical
and that's the language you need to describe
these objects in. It's completely unambiguous.
It's just that you cannot put it back in
the normal words to 100% because these words
have been coined by us to understand the world
of macroscopic phenomenon. This is the reason
why we need mathematics. Therefore, the question
of something being physical and something
being unphysical or purely mathematical, if
I say it in equations shouldn't be there.
It is just that the words are insufficient.
Hence we took a longtime to recognize this
but once you recognize it, it's like learning
another language and that's the appropriate
language for it. It's not that we haven't
coined enough words for this. It's not that
either because you will just be coining arbitrary
words to translate mathematical terms into
this. it's not that at all it's not that
the language doesn't have enough words is
nothing much is gained if I say something
which is neither a wave nor a particle is
a wavicle [Noise] that's being used in the
in the literature people have said this nothing
is gained by it
I would rather say that is it's a state
vector. In fact it's much more precise that
way. Its not just at the level of semantics
alone. It goes much deeper than that. The
fundamental thing about quantum mechanics
will come out in the fact that, even ordinary
words like "and" and "or" have different
meanings in classical and quantum physics,
and you have to get used to it. If you think
a little bit about it, the word "and"
in ordinary language is always with respect
to multiplication of probabilities. You want
this to happen and that to happen, then the
probability is multiply. The word "or"
is always with respect to addition of probabilities.
You have two mutually exclusive possibilities
and you want this or that to happen, the total
probability is addition. So I would like to
say that "or" means addition and "and"
means multiply. That's the classical way
of looking at it. In quantum mechanics the
same thing is true except it's not true
for probabilities but for objects called probability
amplitudes. So it's even a change of classical
probability. Quantum mechanics could be looked
at as a change in the rules of classical probability
completely.
Now, of course the immediate hard question
that arises is when does something stop being
quantum mechanical and start being classical?
I know that a single electron is obviously
a quantum mechanical object. Ten electrons
are still quantum mechanical. A molecule is
quantum mechanical. On the other hand, a piece
of chalk looks classical which can be describes
by classical physics. So when does it stop?
Is there some sharp boundary? The answer is
no. there is no sharp boundary. This is not
a question of whether there is a precise boundary
between classical and quantum physics. It's
just that, as you go towards the quantum domain
from the classical domain, the quantum corrections
to classical physics become more and more
significant. Then it's a question of what
accuracy you want. There will be a stage when
the classical description breaks down completely.
There is no smooth boundary at all. It's
a much diffused boundary. This is like some
of these paintings you see. You start with
red on the right hand side and blue on the
left hand side and as you go, there is no
sharp boundary. It's just that things become
bluer as you move to the left and right. Eventually
there is no red. Everything is blue. It's
in the same sense the classical quantum boundary
is diffused. It's not a sharp boundary.
So all the quantum phenomena when started
with something fully quantum mechanical will
start becoming more and more classical as
you move towards the classical region. But
I must say here that classical mechanics is
an approximation to quantum mechanics not
the other way around. The more fundamental
theory is quantum mechanics and classical
mechanics is a limiting case, an approximation
to it. Just as relativistic physics is believed
by us to be the right formalism and non-relativistic
physics are an approximation when v over c
becomes negligible and so on.
It's exactly in the same way because Planck's
constant is not zero but when the effects
that you are looking at or in some sense much
bigger than Planck's constant then you would
say it's a in the classical region. Can
you have quantum formalism without a classical
formalism at all? This is again a very deep
question and the answer is, it looks like
no. you cannot. And it's different from
relativity because it's conceivable that
you live in a universe where everything moves
at speeds comparable to the speed of light
in vacuum. You don't have a non-relativistic
region at all. It's conceivable and it will
be perfectly consistent. But the very interpretation
of quantum mechanics, by us at least, because
we have macroscopic objects, looks like you
need to have a classical limit. Otherwise
the interpretation becomes meaningless. But
I don't know the deep answer to the question
of is it consistent to have a universe in
which you have only quantum physics and no
classical region at all. But for us it's
a mood point. The reason is we are large objects
and we are in the classical region in many
respects. Therefore, for us to be able to
probe that universe of quantum physics and
understand it, it looks likes it's essential
for us to have a classical limit. The very
meaning of words like measurement, recording,
observation, etc is intrinsically classical
notions. So it looks like we need it. But
whether it's consistent to have a fully
quantum universe is not known.
Sir, today there are so many developments?
Why don't we have quantum gravity? Why is
this theory so distinct and complicated? We
don't know. There are deep reasons as to
why we have to go far down into length scales
and time scales and so on, before you see
the effects of quantum gravity. The reason
for it is simply that since we are spending
time talking about deep problems today, let's
continue with that. You see the fundamental
constants of nature that we know about, are
not constants in mass, length and time. That's
not the way the universe is built.
It turns out that what we have available to
us in the universe is a quantum of action,
h; a c and Newton's gravitational constant
G. These are the fundamental constants of
nature. These are the most basic constants
of nature. Actually, there are just two of
them. We could really say h cross and c. we
could set equal one and say they are fundamental.
These are really basic choice of units and
there is only G. but quite apart from that,
you have these three fundamental constants
of nature. You can construct from them, quantities
of dimensions; mass, length and time, by taking
suitable combinations of h, c and G. So I
request to do this. In combinations of h to
the power alpha, G to the power beta, c to
the power gamma, find alpha, beta and gamma
such that, you get a length or a mass or a
time and these are the Planck mass, Planck
length, Planck time, etc.
Now clearly, those would be the natural length
scales and time scales, when gravity as well
as quantum mechanics will play a role because
gravity is represented by this and quantum
mechanics by that. And it turns out that the
length scale that you talk about, l Planck,
lp is of the order of ten to the power minus
thirty five meters. That's the way these
numbers are in our universe. The Planck time
tp is of the order of 10 to the power - 42
seconds. This is the reason why we don't
know what's happening below that. Because
concepts like length, time and so on may not
even be definable continuously up to zero,
below these lengths scales and time scales.
So the problem of quantizing gravity, which
is what I presumed you are referring to, is
a difficult one. We don't know what goes
on there. part of the difficulty is that,
even concepts like length, mass and time,
length and time in particular or space time,
whether it's continuous all the way down
to zero or not, we don't know because the
laws of physics, as we know today, don't
presumably operate below this. So we don't
know what replaces it. This is the difficulty.
There are other technical difficulties in
quantizing gravity and so on. The fundamental
difficulty is simply that our notion of space-time
breaks down.
You said that the semantics is not enough.
Are we sure that the mathematics is enough?
well, to the extent that, if you make a set
of postulates based on some observations which
could not be explained by classical physics,
this is how quantum mechanics was initially
put in, and then you ask what are it's predictions,
what are the results of it's calculations
and then you observe once again, do those
experiments and test it out and so on, you
are testing this theory and this formalism
and the mathematics associated with it. It
looks like there are no violations, as far
as we know. But you can never prove a physical
theory to 100% accuracy. It doesn't exist.
There will be a domain where it might perhaps
breakdown at some stage. We don't know at
the moment. Now as to whether the mathematics
is sufficient, as far as we know, yes this
is sufficient but we don't know what happens
when you go to these regions.
When you go to quantum field theory, when
you include relativity, general relativity
plus quantum mechanics and then try to do
make calculations, even the formalism of quantum
field theory breaks down. It yields nonsense.
You have to replace it with some other formalism.
There are many candidates, one of which is
called string theory. Another one called loop
gravity, quantum gravity and so on. But these
are at a very deep conceptual level. Its very
likely that this problem 
that our present understanding of space-time
itself as well as our understanding of quantum
mechanics will undergo drastic revisions before
we solve this problem. The answer is sort
of not well established. The answer is that
it looks like we have not succeeded with existing
technology. Now it's purely mathematical.
The indications are the technical difficulties
that arise in this program and they suggest
that our very ideas of space- time, what we
mean by dimension, etc, as well as what we
mean by normal quantum mechanics I think,
will undergo drastic revision. So it's a
house under construction. It looks like the
existing frame work is not powerful enough
to do this.
You could ask now what about experiment. That
we lag far behind. The shortest distances
we can probe are of the order of ten to the
power minus seventeen or eighteen. This is
all we have been able to probe so far in the
highest energy accelerators. There are sixteen
orders of magnitude or seventeen orders of
magnitude between these two. We don't know
what hat can happen in there.
Similarly, the shortest length scales we have
been able to probe which are of the order
of 10 to the -- 23 seconds. There are twenty
orders of magnitude in between. We don't
know how things will change. So we keep an
open mind completely. That's the whole point
about physics. It's not mathematics. So
it is not a set of axioms from which you derive
results. The rules of the game we are prepared
to change and subsume in an even broader frame
work and then say what we did so far is a
special case, a limiting case, etc and try
to increase the range of applicability of
whatever theory you have.
So in that sense, there is no dogma completely
open to it. But people have been thinking
very hard for many years now. Physics is four
hundred years old in its modern form and in
the last hundred years, of course it really
accelerated. But one thing is for sure that,
just as Newtonian mechanics is now known to
be an appropriate approximation under certain
circumstances, so is it true that today's
quantum mechanics and our calculational tools
will be applicable in a certain region of
physical parameters. The understanding of
why this is so should get deeper, broader
and so on. The understanding of why this is
an approximation, why it works in this range
and not in some other range will become sharper.
But I think some fundamental issues themselves
will become clearer namely; what's the meaning
of space time, what determines the dimensions
of space-time, what's this business about
gravity, what are the fundamental forces of
nature and so on.
After all, it's a sobering thought that
we don't know 95% of the constituents of
our own universe. We can only explain 5% of
it 
in terms of matter. After so many millennia
of astronomy and the real explosion in the
last 50 years or so, the conclusion is that
we really don't know 95% of our own universe
and we don't know if it is one of an infinite
number of possible universes. So it's just
that like Wheeler said our knowledge is like
an island and the boundary of this island
is, of course the frontier of ignorance. So
the island is getting bigger all the time
but then its parameter is also increasing.
The level of ignorance is also increasing
at the same time. May be they will work but
we have to keep trying.
I should end by what Hilbert said in 1900
when he gave these famous twenty three unsolved
problems in mathematics in The International
Congress of Mathematicians. He listed what
was then, the major unsolved problem, of which
today, only the Riemann Hypothesis remains
unproved. Everything else has been taken care
of, one way or another. Some of them are not
problems. One of them was the axiomatization
of physics. Could all of physics we written
in terms of a set of axioms? The answer to
that is no. Physics is not axiomatizable.
But he made the statement and then it looked
like there was a lot of problem ahead and
so on, but he made a very optimistic statement.
So I think one should have this note of optimism.
It is essential for us to be curious. many
reasons why it's not part of the reason
is simply that all physical laws, as far as
we know, apply in certain regions of physical
parameters and when you go out of that, it's
a different set of rules.
Looks like it's very general but its interpretation
and the way it's set up is not satisfying.
So it already looks like there are big black
boxes in it which we have pushed under the
rug but they exist nevertheless. We have not
understood quantum non-locality. Except in
some mathematical form of a framework, we
prove our satisfaction. It really hasn't
come to grips with the meaning of measurement.
Now what happens to a system when you measure
it? So there are large gaps conceptual gaps
here. So it's clear it's an incomplete
theory in that sense at least that understanding
is not as deep as we would like it to be the
framework of quantum mechanics. You could
ask may be this is the best you can do how
do you know we don't know that too we don't
know but we have to keep pushing so today
has been a sort of general talk. We will start
of with some specific issues and so on next time.
