I started mentioning what the adjoint of an
operator was. We still have a little bit of
mathematical preliminaries to do, but then
we need to also get started with the quantum
mechanics. So let me do this in adjust position
as I need little more information about operators
I will take a little break and talk about
this like lenar something like that, then
we come back to the main physical point.
So one of the things we do need however is
the idea of linear vector space which is infinite
dimensional and I mentioned two notable examples
of infinite dimensional vector spaces one
of which was l2 and the other was L2 in some
specified interval. So, we talked about the
space of square summable sequences and the
space of square integrable functions defined
for some real variable x running between a
and b. Now these two spaces have a special
role to play in quantum mechanics because
you will see as we go along that, the states
of a system would be described by vectors
in a linear vector space and the linear vector
space would generally belong to one of these
two spaces. So this is the physical reason
why we are actually interested in these two
spaces in particular.
Now when we talk about infinite dimensional
spaces there is also the question of convergence.
I did not pay much attention to it, so let
me spend a few minutes and talk about convergence
of states in vector spaces.
Notice that if I give you a sequence of complex
numbers Zn, Z1, Z2, Z3, etc formed according
to some rule. The natural question that arises
is does this sequence converge to a limit
or not? As n tends to infinity, does it converge
to a limit or not. If it does, then that would
be the limit point of the sequence and then
one could ask what kind of sequences would
converge and these are called Cauchy sequences,
and Cauchy sequence 
implies that the modulus of Zn - Zm tends
to zero as n, m tends to infinity. In other
words, if the sequence is such that the difference
between Zn and Zm modulus goes to zero as
both n and m tend to infinity, then I call
it a Cauchy sequence. Such sequences converge
to limits definite limit points. This immediately
implies that limit n tends to infinity, Zn
exists. It implies that this sequence has
a limit point which is some complex number.
This concept is carried over to state vectors.
The idea of Cauchy sequence is what we need.
Not all sequences are Cauchy sequences. You
can easily construct sequences for example,
if Zn is minus one to the power n, then of
course even ones are plus one, odds ones are
minus one and this sequence does not converge
to any limit at all.
On the other hand if this is true, then this
suffices to show that the limit of this sequence
exits. Now as you know, a set of points which
includes all its limit points is called a
closed set. So if you take the sequence Zn
and you include the limit as n tends to infinity
Zn you include that limit point then that
set is a closed set or a closed sequence.
Now taking this over to the idea of vectors
in a linear vector space, this sequence in
a linear vector space psin is a Cauchy sequence
if limit of the norm of psin minus psim tends
to zero as n, m tends to infinity. So this
vector difference, psi n vector - psi m vector
must have zero length in the limit as n, m
tend to infinity. That sequence is called
a Cauchy sequence. Now a linear vector space
has a large number of vectors and there is
no guarantee that every Cauchy sequence has
a limit point which exists in the same space.
If we are guaranteed that Cauchy sequences
tend to limit points, this implies that the
limit as n tends to infinity psin is equal
to psi. This limit is the limiting vector.
So every Cauchy sequence converges to some
vector. But there is no guarantee that that
vector is in the linear space. If it is then
this space is called a complete linear vector
space.
If the limit vector of every Cauchy sequence
in a linear vector space V is also in V, then
V is a complete LVS. So we will see why we
need these ideas. For finite spaces this does
not matter at all. It will turn out that every
finite dimensional vector space is in fact
a complete vector space because you form all
sorts of linear combinations. You add them
and so on nothing will happen. It will never
go out of this space. On the other hand, in
the case of infinite dimensional spaces you
have to specifically make this requirement
that the limit point of every Cauchy sequence
remain in the linear vector space, because
we are going to serve operations with infinite
dimensional spaces. If this is so, then the
linear space is said to be a complete linear
vector space.
In quantum mechanics, we are going to use
complete linear vector spaces all the time.
In fact we also need the idea of an inner
product. We have already talked about the
scalar product. So a complete linear vector
space equipped with an inner product is the
space that we use in quantum mechanics and
it is called a Hilbert space. It will turn
out that these are the natural spaces in which
the state vectors of systems exist. The reason
you need these mathematical niceties, none
of which I am proving, is to make sure that
operations with infinite sequences and infinite
dimensional spaces do not lead to errors.
Now all finite dimensional linear vector spaces
are complete and once you define an inner
product on them, they actually are Hilbert
spaces. It is only the nontrivial cases that
are the infinite dimensional spaces.
Now even given a Hilbert space, we should
now ask what about basis vectors in such a
space. As you have seen, a basis set in a
linear vector space is a set of linearly independent
vectors which span the linear vector space
completely. Once you are given a basis you
can make it orthonormal and so on. Now given
an arbitrary infinite dimensional space, there
is no guarantee that you can find the basis
set which is actually denumerable, namely
you can count vectors in direction one, direction
two, direction three, all the way to infinity.
If you can then it is a countable or denumerable
basis. So one more, little piece of technicality
is needed.
A Hilbert space 
with a 
denumerable basis, let us call it phin, n
= 1, 2 till infinity is called a separable.
We are going to work specifically with separable
Hilbert spaces. There are technical cases
were we do not have separable spaces and this
is outside of course.
Now once you are given the fact that you are
going to look at a separable Hilbert space
of this kind, it means that any vector psi
in the space can be written down uniquely
in an orthonormal basis. If the basis is orthonormal
it can be written down in the form n equal
to one to infinity, cn phin. So the cns are
unique quantities. Given psi, you can find
cn uniquely and of course the inversion formula
cn is just phin psi because this is an orthonormal
basis. I will assume that whatever basis you
give me, it is denumerable and I will make
it orthonormal by the Gram Schmidt procedure.
Then this is the expansion and this is the
inversion formula. So the set of number cn
suffices actually to specify this vector uniquely
and moreover, the fact that we want these
vectors to have finite norm which means psi,
psi is nothing but n equal to one to infinity
mod cn squared. This quantity here is the
norm of the vector.
So by definition this is a square of a norm
and it means this is finite. What do you call
sequences in which this quantity is finite?
They are square summable sequences l2. So
you see how square summable sequences are
going to play a very fundamental role in everything.
We start by saying our system is going to
be described by a state vector in some separable
Hilbert space equipped with some orthonomal
basis. Since I am going to look at these vectors
which are going to have a physical meaning
and we are going to insist that they have
finite norm, this quantity should be less
than infinity, so our knowledge of the state
vector can actually be summarized by our knowledge
of these coefficients here in this basis and
these coefficients would belong to l2. So
that is how l2 is going to play a very basic
role in everything. Of course we also know
how to go from one basis to another. Basis
sets are not unique. We saw that the formula
for translating from one to another is very
straight forward. All you have to do is to
use the completeness of these basis vectors
and insert the identity operation at all times.
So in other words phi n phi m is equal to
delta nm and summation over n phi n phi n
is equal to identity operator. This is orthonormality
and this is completeness and to go from one
basis to another etc, it becomes very trivial
as long as you go on inserting the identity
operator in between.
Now coming to operators; Operators act on
vectors in a linear space and linear operators
are those which act in a linear manner. In
other words A is a linear operator. If A acting
on psi plus chi is equal to A acting on psi
plus A acting on chi and A acting on c times
psi where c is some scalar, which multiplies
this vector of a linear vector space, this
is the same as c A psi. Moreover (A + B) acting
on psi is the same as A on psi + B on psi,
if A and B linear operators and A acting on
B psi is the same as AB acting on psi by definition.
The product of two operators is successive
but not commutative.
Now let us put in a little bit of physics
into this and I am going to do this by defining
a whole set of postulates. These postulates
have come based on our experience with quantum
mechanics, ultimately based on experiments.
But at the moment we can write down a fairly
axiomatic formulation of quantum mechanics.
Then of course the test is finally to ask
if this way of calculation are compatible
with experiment or not and to our best of
our knowledge, over the last hundred years
quantum mechanics has stood the test of time.
There are still many unresolved questions
in it. There are very delicate questions of
interpretation which are not been taken care
of fully. But as a calculation tool, it is
in fact quite robust and has been tested over
and over again. As far as we know there are
no violations at all.
So whenever I make a postulate let me put
a little dot here to show this is one of the
axioms. I start with a very general thing
and then of course we will make it very much
more specific. This
is
not the most general way of describing systems
in quantum mechanics, but for the moment we
start with this.
Every dynamical system is described by a state
vector, say psi for the notation that is an
element 
of a Hilbert space. Of course different systems
would have different Hilbert spaces. Now what
do I mean by a system? It could be one particle,
ten particles, collection of particles, a
human being, all the people in this room,
this earth, this galaxy etc. all these are
examples of physical systems. And the statement
is that every one of these systems is described
by its own state vector.
And the presumption is that all the information
you can get about the system is buried in
the state vector. And the subsequent rules
of quantum mechanics tell you how to extract
physical information given the state vector.
Since it is a dynamical system we are talking
about, this state vector psi is a function
of time. Right now we are talking about non-relativistic
quantum mechanics. So I am assuming what you
would assume in Newtonian mechanics namely,
there is some frame of reference, there is
an observer in the frame of reference etc.
And time is a variable in which dynamics occurs
and the state vector is a function of time.
And the next thing of course would be to say
it changes with time. As the state evolves,
it changes with time and the rule we need
is to say how it changes with time. The analog
of what you would say in Lagrangian or Hamilton
mechanics where you would give rules or equations
saying how the q's and p's of a system change
with time. This is what we will do next. But
we need a few more preliminaries. Every system
is described by a sate vector. Then the question
is what physical observables are in the system.
What are the physical properties of the system?
For this purpose it is convenient to think
of this system as a particle. That is the
way you do mechanics too by starting with
one particle and then you assign coordinates
and momenta to it or coordinates and velocities.
In a same way I think of a particle and it
is in interaction with the rest of the universe
and I assign a state vector to it. Then it
has physical attributes. This particle has
a momentum, velocity, position, angular momentum,
energy and so on. In quantum mechanics, physical
observables
also called measurables are associated with
operators. So for every physical observable,
there is an operator for the position, the
velocity, and the momentum and so on.
These operators need not necessarily have
to be independent because if you look at the
kinetic energy which is half mv squared for
a particle, once you give me the velocity
operator, I know the kinetic energy operator.
But with every observable I associate an operator.
Then the question is what values these operators
can have.
The natural way to do this is to talk about
the Eigen values of this operator. so given
a linear operator A 
in a linear vector space acting on a state
vector psi and this is equal to lambda times
psi for some special value of lambda and a
special value of psi, then lambda is an Eigen
value
of A with Eigen vector psi. So this is going
to be our basic equation.
Of course a given operator in an abstract
sense we have lots of Eigen values think of
a matrix think of a finite dimensional vector
space, then psi could be written as a column
vector in n rows and one column, then every
operator A can be written as n by n matrix.
And when it acts on this column vector, it
produces another column vector in general.
But there are special column vectors such
that when this matrix acts on those column
vectors, it reproduces the same vector multiplied
at best by a scalar. And those scalars are
called the Eigen values and these are the
eigenvectors, exactly as you have in matrices.
Now of course for a given Eigen value lambda,
you may have more than one eigenvector then
you would say this Eigen value is a repeated
Eigen value. If you have two or more linearly
independent Eigen vectors corresponding to
a given Eigen value, then the Eigen value
is a repeated Eigen value or a degenerate
Eigen value. It's also possible that the number
of Eigen values is infinite if the vector
space is infinite dimensional.
If it is an n by n matrix then you can at
best have n Eigen values which could be n
distinct complex numbers or maybe degenerate
partially but in an infinite dimensional space,
you may have an infinite number of Eigen values.
So this is an Eigen value equation and the
postulate we need is to make sure that the
Eigen values which in some sense will be the
only measurable values of an operator corresponding
to a physical observable should be real. If
the observable you are talking about is a
real physical observable then the Eigen values
should be real. And there is a theorem which
guarantees the kind of operators which have
real Eigen values and those operators are
self-adjoint operators. Square matrices have
real Eigen values. Let us talk about n by
n matrices. So an arbitrary n by n matrix
in general has n complex Eigen values. If
the entries are real and the matrix is symmetric,
you are guaranteed that the matrices have
real Eigen values. But if the entries are
complex, then symmetry is not enough. If a
matrix is equal to its Hermitian conjugate
then, the Eigen values are real.
Now of course for matrices it's quite clear.
If you give me a matrix I can inspect and
find out if it is Hermitian or not. If it
is Hermitian, the Eigen values are guaranteed
to be real if not that's not true. But for
arbitrary operators, maybe differential or
integral operators and other complicated operators,
you need a generalization of the idea for
Hermitian conjugate and I am going to call
them self-adjoint.
If you notice on an L2 space, I defined the
operator d over dx, the derivative operator.
And then I said lets look at those vectors
which remain in the space when (d / dx) acts
on it. It then turned out that the adjoint
of (d / dx) was minus (d / dx). So that operator
is not self adjoint and there is no guarantee
the Eigen values are real. Now that is immediately
obvious because if I put e to the power ix
and I do (d / dx), I get i e to the power
ix. So there is an Eigen value which is i.
On the other hand you are absolutely guaranteed
that if the operator is self adjoint then
the Eigen values are real. So, physical observables
are associated with self adjoint operators.
A little later when we look at problems involving
one dimensional motion and so on then I will
point out carefully the difference between
a Hermitian operator and self adjoint operator.
Actually there is a certain difference and
it is not necessary that a Hermitian operator
be a self adjoint operator although in elementary
treatment on quantum mechanics, these two
words are used interchangeably because in
the case of matrices, they mean exactly the
same thing but we got to be a little more
careful.
At the moment let me be rigorous and correct
and say that the operators have to be self
adjoint operators. This would imply all Eigen
values are real. Also the number of Eigen
values of an operator maybe infinite. It's
also possible that the Eigen values are not
even countable. For instance if I look at
the position of a particle moving on a continuous
line - infinity to infinity, this position
can take continuous infinite number of values
.Therefore we must allow for the fact that
the set of Eigen values of an operator may
actually be a continuous set. The Eigen values
are not even countable. That is a very important
possibility and we need to take care of it.
So right away it says that matrices may not
be enough to do this whole thing because there
would be more general operators which would
have continuous sets of Eigen values. On the
other hand I look at the matrix I know its
Eigen values are all countable. Even if it
is infinite in number, it is still countable.
So we must allow for this possibility. At
moment I simply say all the Eigen values real.
The physical quantities are associated with
self adjoint operators. These operators will
act on the state vector and possibly produce
other state vectors and so on.
Then the result of the measurement of any
physical observable 
is always an eigenvalue of the corresponding
operator. So you make a measurement on a system
described by some state vector the measurement
has to be designed by you cleverly to measure
some physical quantity and the result that
you record will be a real number and it is
guaranteed to be one of the Eigen values of
the operator associated with this observable.
What is not clear and what is not definite
about quantum mechanics is that you cannot
say which eigenvalue is going to emerge. This
is where quantum uncertainty comes in. it
will be one of the eigenvalues but we cannot
say which one before the measurement. On the
other hand you can repeatedly make lots of
measurements, take the average over all the
eigenvalues and you would call that the mean
value of this observable.
So you can do this by preparing identical
imaginary copies of a system, making a measurement
on copy one, copy two, copy three, etc all
the way, take arithmetic average of the results.
That would be a distribution or an ensemble
and the result would be the mean value or
the expectation value of this operator. What
quantum mechanics does is to give you a rule
for calculating the expectation value.
So let me repeat this again because it is
a very basic point. It is imagined that every
system is described by a state vector. I would
like to take this system and measure, let
us say its angular momentum. In classical
physics, once I identify the generalized coordinates,
generalized momentum and so on I have a formula
for the angular momentum in terms of the q's
and p's and in principle there is nothing
to stop me, given initial conditions, finding
what these q's and p's are, at a later instant
of time. I can calculate its angular momentum.
You can then measure it and test it against
the experiment. If the system is a quantum
system, then if it is in some state, we do
not know what state exactly it is in. I measure
by an apparatus designed to measure angular
momentum. The answer I get is going to depend
on the state in which the system is but the
answer I get even for a given state of the
system in general cannot be predicted before
the measurement. Even with the knowledge of
that state, you cannot predict the answer.
What you can do however is to be sure of however
is it, when you make this measurement the
result that comes out will be one of the angular
momentum operator. Which Eigen value it would
be, you cannot say before the measurement.
So one imagines making an infinite number
of copies of the system, all in the same state
and then perform measurements or different
people perform measurements on different copies
of the system and then you record all the
answers.
The first measurement maybe Eigen value lambda
one emerges, in the second measurement Eigen
values lambda six emerged and so on. You add
up all these Eigen values and take the arithmetic
average in the limit in which the number of
measurements is infinite. That would be the
average value of this physical observable
and it is called the expectation value. And
quantum mechanics is going to give you a rule
for calculating the expectation value. This
is what it will do and that is going to be
our target. Of course it will also give you
a rule for calculating the scatter about the
expectation values. It is no good just giving
you one number. It is like giving an average.
It will give you the mean, the mean square,
the mean cube and so on. It gives you rules
for calculating everyone of the moments.
Now it is conceivable of course that if I
measure make a sufficient number of measurements
on my physical system and compute a sufficient
number of moments of each of the these observables.
Then in principle, I know everything I need
to know about the system and all information
I want about the system is acquirable. So
our target would be to find out what kind
of measurement should be made on a system
in order to find out all possible physical
information. Now this is the probabilistic
content of quantum mechanics namely it gives
you a rule for calculating average values.
So in that sense, it is a probabilistic theory.
Does it mean that quantum mechanics does not
provide with time evolution changes?
we are going to come to that
The rule that I am going to talk about is
going to depend on the state vector. It says
in the state, this is the expectation value.
Now of course if the state changes from one
instant of time to another, you need to know
what the rule is by which the state vector
changes. And that rule will be one of the
postulates and that rule is precisely the
Schrodinger equation.
Schrodinger equation is just the differential
equation that tells you how the state vector
of a system evolves with time. Therefore you
can calculate the state vector at any instant
of time given the initial state vector. But
even a knowledge of the state vector is not
enough to give you a hundred percent reliable
prediction on what the result of a measurement
is going to be. That's the way quantum physics
is. The point is it possible that the state
of the system is such that I can gain complete
information about the system?
Go back to classical mechanics. Look at a
single particle moving in 1 dimension. It's
described by generalized coordinate q and
a momentum p. therefore in phase space in
the q-p plane; I specify a point that tells
me everything I need to know above the system.
Those are the only independent attributes
of the system and I know it completely. In
quantum mechanics, I am saying that the system
is described by a state. Now suppose you take
the same system, same particle which is quantum
mechanical and lets say that the independent
dynamical variables are its position and its
momentum in 1 dimension.
Then the question is, is there a state of
the system such that the position has a sharp
value. So I give you the state and I immediately
know the position. In other words, the question
is, is it a position Eigen state. If it is
then of course I apply that position operator
on the state vector I get a number. There
is no distribution. The answer is yes you
have position Eigen states. You also have
momentum Eigen states but no state can be
simultaneously a position Eigen state and
a momentum Eigen state. Such a state doesn't
exist. So that is where the uncertainty comes
in. So it will turn out that for those variables
which are canonically conjugate in the classical
sense, namely the Poisson brackets are equal
to one for those variables in quantum mechanics,
it is impossible to find simultaneous Eigen
states.
There can be no state of a system of the particle
in which both and x and both x and p can be
simultaneously specified to infinite precision.
The question is, is it possible that I may
miss out on some of the Eigen values when
I make measurements. The point is, given the
states of the system; the idea is that the
states will cover all possible values of all
possible observables. Only then would it be
a complete Hilbert space.
So everything is included. All possible eigenvalues
may be highly improbable. You may need special
efforts to produce a state of the system corresponding
to a very large Eigen value of some observable.
But the idea is they are also in the same
space and nothing is left out. It is not a
question of without doing the experiment.
The point is that you want to describe the
results. The whole idea of physics is that
you want to describe the results of experiment.
So you need a formulism which will tell you
how to calculate and then you compare with
experiment.
So these are two different things. The system
is a part of nature. It is acting in some
fashion, quantum mechanics is a formulism
which helps you to describe the system in
precise terms. So I would always like to have
rules for computation for physical quantities
which I can compare against the results of
measurements. So we won't confuse the formulism
with the actual experimental observation or
the measurement. These are two different things
all together. The statement I made was, given
an arbitrary state of a system, if I look
at some particular physical observable corresponding
to the system such as the position of a particle,
if I measure the position of the particle
the answer I get is guaranteed to be one of
the eigenvalues of the position operator.
But, I cannot tell you a priori which eigenvalue
will emerge on the basis of one measurement
even if I know the state of a system. What
I can do is to assign probabilities with which
different eigenvalues would emerge given that
state of the system.
So suppose the position you take some physical
observable corresponding to a system and lets
say this observable can take the values one,
two and three. Those are the only Eigen values
allowed no matter what state you are in. Now
I put the system in an arbitrary state and
I make a measurement. I may get the value
two. I make another measurement on an identically
prepared copy and may get the value three
and so on.
Given the state of the system I cannot tell
you whether I am going to get one, two or
three in a given measurement but even before
you make the measurement, I can tell you the
probability with which you are going to get
eigenvalues one as p1 and similarly for p2
and p3 based on a knowledge of the state of
the system. Therefore it follows that one
time p1 plus two times p two plus three times
p three is after all the average value of
this variable. If I took an infinite number
of copies of the system, identically prepared
and made measurements, I would get one, two,
three etc a large number of times.
I draw histograms to denote how many times
I get one, two, and three and so on. Then
the relative fraction of times I get p1 is
of course p1 and similarly p2 and p3. Now
what I can tell you from knowledge of the
state of the system is I can give the numbers
p1, p2, p 3, etc. And of course once I give
you p1, p2, p3, you will say one time p1 plus
two times p2 plus three times p3 is in fact
the average value and that is precisely what
you will get by making this infinite number
of measurements and taking the arithmetic
mean. So that's a sense in which quantum mechanics
is probabilistic.
Will the average value be one of the Eigen
values? It could be but the average value
of the height of the students in this room
need not be one of the heights of the students.
The average value of a set of integers need
not be an integer. It depends on the distribution.
The question is, is this a short coming of
quantum mechanics or is nature inherently
random. Our belief is that nature is this
way. Firstly, it is not a failure of experiment.
So the uncertainty in quantum mechanics is
not due to the fact that you do not have sufficiently
precise measuring instruments. Even if you
had infinitely precise measuring instruments,
this would still be the case.
Second, it is not an artifact of our formulism
of quantum mechanics. We believe it is because
there exists a fundamental constant called
Planck's constant which is non-zero. So the
fact its numerical value is irrelevant that
is 10 to the power -- 34, its numerical value
is linked with the fact that objects as big
as us do not have many quantum corrections.
I mean we know we have essentially classical
objects like electrons and the quantum corrections
would be very significant. That is what the
numerical value of Planck's constant is it
determines.
Had Planck's constant been thirty four orders
of magnitude greater, the quantum uncertainties
would be so wild that we would not have the
classical world we have but the fact that
its non-zero is what's leading to quantum
mechanics and we think thats what it is. I
do not know if we can ever know the why of
things at an ultimate level always because
this ignorance will simply be pushed back
one step further to whatever. if in future
some other theories subsumes quantum mechanics
and it is superior theory which actually includes
quantum mechanics as a special case, even
then the question would persist why that.
What do you actually mean by copies of a system?
Well, suppose I asked you to measure the resistance
of one meter of copper wire. There are two
ways in which you do this. You take a long
spool of copper wire, a kilometer long, cut
it into a thousand meter lengths and have
thousand different people make measurements
one each. That would be an ensemble average.
The assumption is that the copper wire is
homogeneous and therefore whatever is one
meter here is the same as one meter there
and properties have not changed. Then that
gives you the arithmetic average. So that
is the sense in which I shall make copies
of the system and in principle you need an
infinite number before you get the true arithmetic
average.
Of course in dynamics, remember that we talked
about ergodic systems. There we said that
the time average is equal to the ensemble
average. So the idea was you do not have a
thousand pieces of copper wire. You have just
one piece and you repeatedly go on making
measurements on this one piece. And the presumption
is that all the values or realizations that
these thousand pieces of wire would have given
of this random process would already be realized
as time goes on in the single piece of wire
and that is what we called ergodicity where
I said the ensemble average is equal to the
time average here. So that's the specific
assumption in dynamical systems. We are not
making any such assumption here at the momentum.
Now all we have said is physical observables
which are real observables should really have
real values when you make measurements and
therefore we looked for all those operators
which have real eigenvalues and they are self
adjoint operators. Now, the spectrum of an
operator may be continuous or discrete. It
could be integer valued, non-integer valued,
continuous, partially discrete, and partially
continuous and so on. Let me give you an example.
The energy of an electron in a hydrogen atom
in quantum mechanics as you know is minus
one over n squared in Rydberg units for the
bound states. But then those are negative
energy states and they are all discrete minus
one over n squared where n runs one, two,
three, and four. As n becomes larger, these
states come closer to zero and then all possible
positive values are also allowed.
So that spectrum consists of the set of minus
one, minus one fourth, minus one over nine,
minus one over sixteen till zero and then
all positive numbers. That is the total spectrum
of an electron interacting with the proton
by Coulomb interaction in quantum mechanics.
So the negative numbers are the bound states
of the hydrogen atom and the positive numbers
correspond to a free electron scattering of
a proton. They are still part of the spectrum.
What will become significant is that the state
of the electron in this part of a spectrum
is very different from the state of system
there. And that is what will distinguish the
scattering states from the bound states.
Is it possible to make a measurement without
changing the state of the system?
This is again a very deep question does it
change the state of a system when you make
a measurement. This is very deep question
and this is where the problem of measurement
in quantum mechanics arises. The answer of
course is, in general, when you make a measurement
and produce an eigen value as the answer,
it implies that the measurement has somehow
converted your state into an eigen state.
If I make a measurement on this and I get
lambda one as the eigen value and psi one
as the eigen vector corresponding to the eigen
value lambda one, it implies that this operation
of measurements has somehow affected the system
and instantaneously changed its state from
whatever it was to the eigen vector psi one.
So definitely it is made a change. On the
other hand, if the system was already in the
eigen state psi one, I make a measurement
it remains where it is.
So measurement in general changes the state
of the system quite drastically and it is
called the collapse of the wave function in
quantum mechanics. And one interpretation
of quantum mechanics assumes that, by some
mysterious as yet unknown process, any measurement
of physical observable on a system collapses
the state of a system instantaneously to one
of the eigen states after which the Schrodinger
equation will take over and system will evolve
once again. So the idea is that if you make
an instantaneous measurement immediately after
you made a measurement and got the eigen value
lambda 1, you will still see lambda 1. How
long it will take to decay here and become
something else is dependent of the Hamiltonian
of the system.
On the other hand, there are interpretations
of quantum mechanics which do not talk about
instantaneous collapse at all but use totally
different premises to explain the measurement
process. So the details of the measurement
process itself are something we will push
under the carpet. We will not talk about it
here because that is the part of quantum mechanics
which is tricky and is opened to interpretation.
Do all states of the system have to be eigen
states?
Absolutely not!
Let me give you a specific example of what
will happen and this is in fact going to tell
us how to calculate expectation values. It
will also lead us to the probabilistic interpretation
of quantum mechanics. We could have started
with the way it is done in the final lectures
and then work back to expectation values.
But I would like to start from here and go
back because I am going to assume that it's
easier to understand classical probability
theory and then I point out where quantum
probability is. It differs from classical
probability. The whole thing can be looked
at in yet another way which is a small change
in the rules of probability theory. Now that
takes you from classical to quantum mechanics.
Suppose you have a physical observable, associated
with which is the operator A and the state
of your system. For the moment let me assume
everything is being done at one particular
instant of time. So I would not put in the
time variable. So let us suppose this is the
operator and the state is psi. Then the question
is, what is the average value that I am talking
about which you get when you measure A repeatedly
over an ensemble or collection. I call that
the expectation value. I am going to denote
it by two angular brackets. And as I mentioned,
quantum mechanics gives you a formula for
this quantity.
Now how are we are going to calculate this
quantity. Buried in here in this formula is
in fact the probabilistic interpretation of
quantum mechanics. Let us assume that this
physical quantity, corresponding to the operator
A has a set of eigenvalues lambda 1, lambda
2, etc for the moment. Let me assume the eigenvalues
are countable. So let lambda 1, lambda 2,
be eigenvalues of A with eigenvectors phi
1, phi 2 and so on. For ease of notation,
let me assume these are all non-degenerate.
They are real numbers because A is a physical
observable, so the eigen operator is self
adjoint and let me go a step further and say
that these phi's can be made into an orthonormal
basis set. Now the immediate question is,
given a physical observable, can you always
make an orthonormal basis set out of its observables?
For ease of presentation, let me assume I
can for a moment. We will see what happens
when you cannot later on.
So this set here forms a basis set. Now what
does that imply? That immediately implies
that psi can be written as summation n equal
one to infinity, some coefficient say cn phi
n. So
I am going to assume that they form orthonormal
basis. Then I can write this psi in the form
cn phi n. psi is the state of the system.
Then this cn we know is equal to phi n psi.
Now key question is what the interpretation
of cn is. It is just a coefficient in the
expansion of a vector.
So it tells you how much of the vector psi
is pointing along the vector phi n. that's
all it is I mean if symbolically if this is
psi and this is for example phi 1, then this
quantity here is nothing but the projection
of the psi into this direction here. So it
is this vector if I multiply by phi 1 on the
right hand side that will in fact give me
the component of this vector along this. That
is mathematics it just tells you that you
projected this and only the coefficient tells
you how much operate this along that.
Now the interpretation in quantum mechanics
comes by saying this quantity is the probability
amplitude that when the system is in the state
psi, it is the state phi n. So it is the probability
amplitude and not probability. That's the
term that is introduced and it's a non-classical
term. It's a specifically quantum mechanical
term that is introduced to say that it is
the probability amplitude that the state is
actually phi n. 
The next postulate in quantum mechanic says
the mod squared of this quantity is the probability
and that is the postulate. But this itself
is a complex number but you are guaranteed
for any complex number the mod squared is
always a nonnegative real number and therefore
that is associated with the probability.
So this is one more way of introducing quantum
mechanics. You could start by saying that
these quantities and inner products are probability
amplitudes such that it is the probability
amplitude that this event is in fact and it
will work even with time. So if I have a state
at some instant of time and I have another
state at another instant of time, this inner
product the initial state here and the final
state here is the probability amplitude that
this guy has an overlap with that and that
will tell us compute these numbers. So let
me do this next time. We will start at this
point and I will show you how this definition
of this quantity, as a probability amplitude
immediately tells you what the probability
distribution itself is.
yeah
yes mod Cn squared
yes so there is a sum of nonnegative numbers
so each mod Cn squared must be less than 1
and therefore is a probability and that's
the whole point.
yeah so the whole idea of quantum mechanics
says the state of the system is some abstract
object and everything about the system is
buried in there.
ah the state of a system there is nothing
that's the way nature is
The state of the system is not an observable
by itself. It is something from which you
derive expectation values of observables but
it's not an observable itself. It's a false
analogy but it gives you a good picture to
say the state of the system plays the role
of something like the probability distribution
of a random variable. The probability distribution
is not an observable of a random variable.
It is not observable only the mean square
etc are observables but the distribution helps
you compute these quantities.
anything you can see anything you
can measure
yeah I will write this down clearly and then
you will see what Cn is
it's the probability amplitude that indeed
or better still I should call it an overlap
between the state psi and state phi n which
is what the geometrical interpretation is
but the postulate of quantum mechanic says
the mod Cn squared is the probability that
the system when it is in the state psi is
in the eigen state phi n. that's the exact
statement
yeah
that's the only quantity which has the desired
properties they are real number they are between
0 and 1 they normalized to unity.
why not mod Cn itself that's the way quantum
mechanic says
squared. That is the way it is. That is the
only consistent way we can write down quantum
mechanics. that's the way it is. its not mod
Cn its not mod Cn to the power one by three.
No its not mod Cn.
Yeah because of the inner product equation
but yeah surely but you know the point is
you could have started with the different
formulism all together and that's not the
way it operates this is the way it is.
Thank you!
