so far we have discussed the heisenberg and
schrodinger pictures of quantum mechanics
and what i have shown you is that both are
equivalent both are equivalent and in heisenbergs
picture 
things are defined through matrices and in
the schrodinger picture 
we have the wave function and when you want
to define a quantity xop it is done through
operators for each quantity
so we had for example x mn matrix element
of heisenberg picture as psi m star x operator
on n dx and when this operator x operator
acts on psi n this means it just multiples
by x psi m star x times psi n x dx on the
other hand we also defined p mn which was
equal to minus infinity to infinity psi m
star p operator acting on psi n dx and p operator
when it operates on psi is a differential
operator so it actually differentiates it
so you get psi m star h cross over i d psi
over dx dx and then we also define the expectation
values for these quantities so expectation
value for x and expectation value of p were
define as minus infinity to infinity psi x
square x dx and expectation value for p was
defined as the integration over psi star h
cross over i d psi over dx integrated over
and we also understood this as the average
values of these quantities not only that we
could also see that operators corresponding
to p square was nothing but p operator square
and this became minus h cross square d two
by dx square operator corresponding to x square
was nothing but x square multiplying the wave
function and similarly we can define higher
powers of these operators and also their own
multiplication
what we are going to learn in this lecture
through all this is a very important principle
of quantum mechanics known as the uncertainty
principle 
which states that quantity is in quantum mechanics
cannot be measured in general with very high
precision in fact two quantities which are
called conjugate to each other have a relationship
given by uncertainty relation if you measure
one quantity to very high accuracy the other
quantity becomes less accurate and that is
what we are going to see why it happens in
quantum mechanics it actually is a result
of the quantum condition and therefore this
is a principle which is very unique to quantum
mechanics you may have learned about uncertainty
principle in your twelve ththis is the statement
so let me just write it and then we will explore
it further you may have learned that for a
given particle delta x delta px is greater
than or equal to h cross by two
so what does it mean how do we define delta
x how do we define delta px we are going to
see it in a mathematical rigorous manner in
this lecture so let us see if i am making
a measurement of quantity x then i do get
an average value x and if i measure x square
i get an average value of that if i subtract
expectation value or average value of x from
this and take square root this is defined
as delta x similarly if i take p square expectation
value of that subtract the square of the expectation
value of p and take square root this is delta
p in general and let me write it on the right
hand side in general for any quantity right
lets call it o or operator o the uncertainty
in o will be equal to square root of o square
expectation value minus expectation value
of o square and that makes perfect sense let
me explain how so suppose i have a quantity
or say take some numbers lets say i measure
and i get one two two three one four and so
on lets restrict ourselves to one two three
four five six numbers and take the average
so lets say this is x the average will be
one plus two plus two p[lus three plus one
plus four divided by six and that comes out
to be eight plus five thirteen over six roughly
two
and therefore x square is roughly four let
us take x square average and that comes out
to be one plus four plus four plus nine plus
one plus sixteen over six which is nothing
but five plus four nine eighteen plus sixteen
thirty four plus thirty five over six which
is roughly six so you see average of x square
is greater than x square and therefore two
are not the same because there is a spread
in the values on the other hand if all xs
were the same then these two would have been
equal so what this delta x or delta p or delta
o in general gives you is the spread in any
quantity if i make several measurements and
find that they are different this delta x
defined in the manner given here gives me
how much is the spread and what uncertainty
principle tells you is that the spread multiplied
by the spread in the conjugate quantity is
greater than h cross by two and let us now
show that before that i just want to tell
you what happens 
for an eigen value in terms of its spread
so for example suppose i have an operator
o which operates on a wave function psi and
gives me the eigenvalue o n psi n then the
expectation value of o will be given by integration
psi n star o psi n dx which is nothing but
o n which comes out which is a number psi
n star psi n dx which is o n
on the other hand if i calculate o square
this will be integration psi n star o square
psi n dx which is nothing but o n comes out
you left with psi n star o operating again
on psi n dx and you get o n square integration
psi n star psi n dx which is o n square so
in this case what you see is that o square
is same as expectation value of o square and
therefore delta o is zero so if i take the
expectation value of an operator with respect
to the wave function that are eigen functions
and then calculate the spread in it or delta
x or delta o in it it comes out to be zero
therefore in stationary state we can write
write a wave in a stationary state delta e
will come out to be zero thats a fixed energy
state all right so with this background i
am now going to prove the uncertainty principle
for that i need something called the cauchy
schwartz schwartz will be swa inequality which
if i simply put in terms of vectors it says
nothing but a dot b magnitude is less than
or equal to magnitude of a and magnitude of
b product which is very easy to understand
because a dot b is nothing but magnitude of
a magnitude of b times cosine of theta and
this is always less than or equal to mod a
mod b product because cos theta mod is always
less than or equal to one
in terms of functions what it tells you is
if i have an operator o one and its product
with o two and take its expectation value
what that means is i am taking psi star x
o one operator o two operator psi x dx this
and take its modulus this would be less than
or equal to the individual expectation values
that is psi star o one psi d x times psi star
integration o two psi dx which is nothing
but this whole quantity is nothing but expectation
value of o one expectation value of o two
this is known as the cauchy schwartz inequality
and we are going to use this now to prove
the uncertainty principle for that i choose
the quantity x minus mod x as one operator
so this is going to be o one and o two is
going to be this operator p minus expectation
value of p so i am going to have from cauchy
schwartz inequality so i am going to use the
cauchy schwartz inequality and take the so
what i am going to show is that take x minus
mod x and p minus not expectation value of
p expectation value and this should be less
than or equal to expectation value of x minus
x square 
square root times square root of p minus mod
p square
you may wonder why i am taking the square
root because you see expectation value of
x minus mod x is zero and expectation value
of p minus expectation value p is zero so
what i am really actually calculating is this
spread or delta x delta p on the right hand
side this is kind of rms deviation let us
work on the left hand side so expectation
value of x minus expectation value x p minus
this quantity is nothing but expectation value
of xp minus x p average minus x average p
plus x average p average and i am taking expectation
value of this which is nothing but expectation
value of xp minus xp minus xp plus xp and
i am going to cancel a few terms so this fellow
cancels with this and i get this equal to
expectation value of x p minus x p in all
this keep in mind that whenever i am writing
this angular bracket or the expectation value
that is a number right so this quantity is
a number and this quantity is a number if
you are uncomfortable in using x minus expectation
value of x and p minus expectation value of
p expectation value less than equal to square
root of x minus x expectation value square
expectation value times square root of p minus
expectation value of p square expectation
value then you can instead use for o one this
itself square and for o two the square of
p minus p expectation value square and write
the cauchy schwartz inequality as x minus
mod x square p minus mod p square expectation
value would be less than or equal to x minus
expectation value of x square expectation
value times the expectation value of p minus
p expectation value square expectation value
leading to the same result as derived earlier
so we have calculated x minus x p minus p
expectation value to be expectation value
of x times p minus x p expectation value product
i am going to use this and do and check i
will right this as x plus px expectation value
divided by two plus x p minus p x expectation
value divided by two minus x p so what have
i done in this what i have done is i have
written x p product as xp plus px divided
by two plus x p minus px divided by two even
i take the expectation value on the two sides
this is what i get and i will use this here
and here this then i know is nothing but expectation
value of xp plus px divided by two plus recall
from the quantum condition 
that xp minus px this is nothing but i h cross
so therefore this becomes i h cross by two
minus xp
now i leave this as an exercise for you to
show that expectation value xp plus px is
real all right this is very easy to show if
you write p in the operator form so this actually
become integration psi star x times h cross
over d psi over dx plus h cross over i d over
dx of x psi dx and you can do the manipulations
you can do integration by part and show that
this comes out to be real so what you have
is that xp expectation value is expectation
value of x p plus px divided by two plus i
h cross divided by two minus x p which is
real quantity a plus an imaginary quantity
h cross by two where a is defined to be xp
plus px divided by two minus x p so to collect
all this together what have we done we have
found that expectation value x minus mod x
p minus mod p modulus is less than or equal
to delta x delta p which were defined as let
me remind you delta x was defined as square
root of expectation value of x square minus
mod x square and this was define as square
root of x p square minus p square which is
nothing but let me also write this as x minus
expectation value x square expectation value
same thing for the other term this is nothing
but p minus expectation value of p square
expectation value they are all one and the
same thing
so on the right hand side i have delta x delta
p which is defined through all this operations
and on the left hand side now i have modulus
of a plus i h cross over two so what do we
have we now have a modulus of a plus i h cross
by two is less than or equal to delta x delta
p modulus of a plus i h cross by two is nothing
but a square plus h cross square by four square
root which is certainly greater than h cross
by two because a square is a positive quantity
and therefore from here i have my result that
h cross by two is less than or equal to delta
x delta p and this is the uncertainty principle
which says that if you measure for a wave
function x many many times and p many many
times there will be a spread but the spread
is going to be such that its product will
be greater than or equal to h cross by two
it is not a limitation of your measurement
but this is fundamentally inherent in quantum
mechanics i had alluded to it i hinted about
it when we constructed a wave packet for a
particles few lectures back and what i had
said there that if i have a wave packet which
has a spread of delta x the corresponding
mixture of plane waves that gives you this
has delta k spread in k space such that delta
x delta k is of the order of one you multiply
by h cross on both sides you get delta x delta
p of the order of h cross
so the moment you try to represent a particle
by a wave you get this sort of uncertainty
relationship because the moment you try to
show it has a wave it cannot be localized
to the accuracy which you wish to have there
is a limitation and then the momentum also
gets a spread correspondingly and which is
given by this uncertainty relationship you
can see if i make delta x smaller and smaller
delta p as becomes larger and larger if i
make delta x larger and larger delta p x becomes
smaller and smaller smaller so this is the
kind of relationship that these two spreads
have as i said earlier the wave functions
that are eigen functions do not have a spread
so lets ask a question 
are there wave functions that have either
delta x is equal to zero and these will be
the eigen functions of x operator or delta
p equal to zero and they will correspond to
eigen functions of p operator so you can physically
think that for eigen functions of x operator
x times its fx should give you a definite
x x zero this is the only way it can happen
is if fx is delta x minus x zero there is
also if you recall from previous lectures
is nothing but one over two pi integration
e raise to i k x minus x zero dk
since the amplitudes of all plane waves are
the same that means all momentum come with
the same probability so you can see that delta
p this is the representation gives you the
delta p is infinite where as delta x in this
case is zero so the product i know remains
finite how about eigen function 
of p operator for this i am going to have
h cross over i d psi over dx which is a p
operator operating on psi is going to be a
definite momentum p psi and this immediately
gives you that psi is e raise to i p x over
h cross so what i have shown you is that eigen
function of p is e raise to i p x over h cross
which i can write as e raise to i k x
this has value of momentum for a particle
in this status p fixed on the other hand this
is a plane wave so it goes all the way from
plus infinity to minus infinity so delta x
tends to infinity in this whereas since its
momentum is fixed delta p is zero these are
two ideal cases in one case the wave function
is a delta function in the other case wave
function is a plane wave spreading from minus
infinity in a for a particle these are two
ideal cases two extreme cases which are not
same what is seen is for example particle
in a box where wave function is between zero
and l or you also see a particle in a a simple
harmonic motion potential where if this is
the potential wave function is like e raise
to minus x square so it has some spread and
this is where delta x is not going to be zero
and delta p is not going to be zero but the
product will always be greater than or equal
to h cross by two similarly an particle in
a box delta x is not going to be zero delta
p is not going to be zero in fact i am going
to give you the problems in the assignment
where i would ask you to calculate delta x
and delta p for these and see what their product
is it should always come out to be greater
than h cross by two as we have seen rigorously
in our derivation
so what you have learnt so far is the uncertainty
principle and i have confined myself to one
dimension which says delta x delta p x is
greater than or equal to h cross by two and
it comes in very handy for quick back of the
analog calculations for example now i know
that delta x going to zero means delta p goes
to infinity and recall from the previous lecturer
for a bound state p expectation value itself
is zero so delta p going to infinity means
that p square expectation value is becoming
very large and this implies that p square
over two m is becoming very large so if you
confine a particle more and more its kinetic
energy tends to increase because of the uncertainty
principle if let it spread it tends to become
smaller and you seen this earlier again we
go back to particle in a box problem where
a particle was confined in a box of length
l and if you recall this energy en was proportional
to one over l square so as l becomes larger
the energy becomes smaller this is precisely
a demonstration of uncertainty principle
we can use it also to estimate energies of
different systems for example again i will
go to the example one for using 
principle for estimating ground state or lowest
energy of a system so first example i take
is this particle in a box of length l and
you can see in this case delta x is of the
order of l and delta p which is defined as
square root of p square minus expectation
value of p square is nothing but the square
root of p square because for bound state expectation
value of p is zero so delta p which is equal
to square root of p square expectation value
times l is going to be greater than or equal
to h cross by two and therefore p square itself
is going to be greater than or equal to h
cross square over four l square p square over
two m which is the energy is going to be greater
than or equal to h cross square over eight
m l square which you indeed find is the case
in the second example 
i will take the particle in a simple harmonic
potential and in this case if this is centered
at x equal to zero i know the expectation
value from symmetry of x is going to be zero
and let me assume that x square square root
is of the order of a so that we can write
the potential energy as one half k a square
which is one half m omega square a square
how about the kinetic energy now again p expectation
value zero therefore square root of p square
minus p expectation value square which is
same as square root of p square is going to
be of the order of h cross over two a which
is nothing but h cross over delta x and therefore
the kinetic energy 
is going to be p square expectation value
over two m which is h cross square over eight
m a square and the total energy e is therefore
going to be h cross square over eight m a
square plus one half m omega square a square
and to find the lowest energy i minimize this
energy with respect to the only parameter
that is available to me and that is a so i
do de over da is equal to zero which gives
me h cross square over four ma cube with the
minus sign in front plus m omega square a
is equal to zero or you solve this you get
a square is equal to h cross over two m omega
so what we have found my minimizing energy
with respect to a that a square is h cross
over two m omega and the total energy e was
given as h cross square over eight m a square
plus one half m omega square a square this
will come out to be h cross over a eight m
times two m omega over h cross cross square
plus one half m omega square times h cross
over two m omega which comes out to be h cross
omega by two precisely the same answer as
we got by solving schrodinger equation so
let me comment here since the minimum energy
is the same as by solving the schrodinger
equation the uncertainty delta x delta p for
the ground state of simple harmonic motion
should also be minimum or all other states
is going to be greater than h cross by two
so this is an application to get an order
of magnitude estimate of the energy using
the uncertainty principle i will also give
you some problems related to this in your
assignment
so with this let me conclude this lecture
by saying that delta x is the expectation
value of x x square expectation value when
you subtract delta x from this and take square
root this is how we define delta x similarly
if i take p square expectation value and subtract
the square of expectation value of p this
is how we define delta p and using the quantum
condition along with the cauchy schwartz inequality
we got delta x delta p to be greater than
or equal to h cross by two and thats the uncertainty
principle and then we applied it to estimate
energies of two systems
