so what we have done in the lecture so far
is that established through experiments and
that is very important to understand experiments
that number one black body radiation and that
means its spectral density can be explained
by taking the energy levels of an oscillator
quantized and what does that mean that means
if an oscillator has frequency nu it can take
energies zero h nu two h nu and so on n that
is 
n quanta of h nu and through this we obtained
a spectral density curve that fit at the experiments
perfectly and it also went to in the two limits
nu over t being very large and very small
it went to the respective wiens and rayleigh
jeans formula
number two not only this what we also saw
is that radiation itself can be thought of
as composed of lets call radiation molecules
and i am using that term the literately to
show you because it came from the considerations
of entropy change of radiation when it expanded
keeping the energy same and the volume increase
from v one to v two and it was the same as
an ideal gas so radiation molecule and then
called photons that have energy h nu for radiation
of frequency nu
so on one hand i have oscillators that radiate
that have energy h nu two h nu and so on on
the other radiation itself has energy h nu
now the question that arises and let me write
that question is quantization that means energy
coming in units of h nu two h nu and so on
limited to oscillators that radiate and radiation
explain now let me explain that oscillators
that radiate are charged particles and as
they accelerate and go back and forth they
start giving out radiation of the same frequency
and radiation of course as light or whatever
so is it limited to this or is it more general
or so lets that question or is quantization
a law of nature and that means does does it
apply to other systems like mechanical 
oscillators and we will see atoms and all
that
and it turns out that this is more general
and it becomes a law of nature so all this
built up slowly so now in this lecture what
we are going to discuss is is that the quantization
rule applies to mechanical oscillators also
one thing you must keep in mind through all
this is that test of any theory or hypothesis
is its confirmation by experiments thats what
science says whatever i can propose should
be very far fiable by experiments or must
explain an experiment and experiments based
on the experiments or experiments to do that
check that theory
so a background background i am going to give
you a specific heat of insulators and what
was known in this was something call that
dulong petit law that said that specific heat
of insulators is three r per mole and this
was based on equipartition theorem recall
what equipartition theorem is equipartition
theorem says that energy per degree of freedom
is k t by two for a molecule so per mole energy
is going to be n avogadro times k t by two
which is nothing but r t by two what dulong
and petit law observed is that for atoms in
a solid energy is one half m v square average
kinetic energy and one half k x square average
potential energy both are quadratic and both
average r t by two r t by two for each degree
of freedom
now this is in one dimensional case in three
d what we going to have so if this is three
d solid by three d i mean three dimensional
each atom has three degrees of freedom so
each atom has energies kinetic energy is three
by two m v square which average value by this
this bracket i mean average is three r t by
two per mole and potential energy is three
by two k x square average is going to be three
r t by two per mole so the total energy is
going to be three r t
and therefore specific heat that means specific
heat per mole is going to be three r t d by
d t equals three r which is roughly twenty
four joules per mole r is roughly a joules
now this is fine this is what was observed
that if you plot specific heat versus t it
was roughly twenty four joules however as
t went down as t was reduced what one observed
was that the specific heat went down and become
zero like this for t tending to zero and this
could not be explained this could not be explained
by classical equipartition theorem could quantum
theory come to rescue and thats precisely
what happens
now we have already seen earlier while discussing
the development of quantum theory that equipartition
was theorem did not hold in case of black
body radiation if it did rayleigh jeans formula
would have been alright so one should not
expect that it would hold in other systems
if this is the general theory so let us see
now how quantum mechanics ideas quantum mechanical
ideas at that time explained the specific
heat going to zero as t goes to zero now we
have already calculated that average 
energy per oscillator is equal to h nu equals
divided by e h nu by k t minus one this is
what we had calculated just to recap how did
we do that we said that an oscillator 
can have energies zero h nu two h nu and so
on and probability of having energy n h nu
is nothing but e raise to minus n h nu by
k t divided by summation over n equals zero
to infinity e raise to minus n h nu by k t
and we multiply it by the probability with
the energy and we got the average energy average
e average at temperature t was equal to n
h nu e raise to minus n h nu over k t summed
over n equal zero to infinity divided by summation
n equals zero to infinity e raise to minus
n h nu over k t and this gave me h nu over
e raise to h nu over k t minus one
therefore for n particles 
i am going to have three n that is three n
degrees of freedom and each degree of freedom
has energy e raise to h nu over e raise to
h nu over k t minus one this is going to be
the energy of the system e t so for a solid
that has n atoms each degree of freedom has
frequency nu the total energy e t is given
by three n h nu over e raise to h nu divided
by k t minus one so we found applying quantum
ideas energy for n particles is going to be
three n h nu over e raise to h nu over k t
minus one and therefore specific heat i am
not going to the of whether is constant volume
or constant pressure for solids it doesnt
really matter if their compressibility is
very small
so c p and c b dont really differ by much
is going to be d e by d t which is going to
be three n h nu d by d t of one over e raise
to h nu over k t minus one which is nothing
but three n h nu divided by e raise to h nu
over k t minus one square times e raise to
h nu over k t times h nu over k t square so
this is equal to three n h square nu square
over k t square e raise h nu over k t over
e raise to h nu over k t minus one whole square
and if you plot it as a function of temperature
as t goes down this fellow goes to zero exponentially
to see this let t go to zero then e raise
to h nu over k t goes to a very large number
and therefore i can write c as three h square
nu square over k t square e raise to minus
h nu over k t and that goes to zero
so that explained that specific heat should
go to zero as t goes to zero if you follow
quantum ideas and that again established the
trust in quantum theory let us see if it needs
to dulong petit law also so what we have calculated
is c equals three h square nu square over
k t square e raise to h nu over k t divided
by e raise to h nu over k t minus one whole
square now as t goes to a large value i can
write e raise to h nu over k t roughly equal
to one plus h nu over k t and therefore c
is three h square nu square over k t square
times one times one over one plus h nu over
k t minus one whole square and that becomes
three h square nu square over k t square times
one times k square t square over h square
nu square and if we cancel terms h square
nu square rows one of the ks goes with one
of the ks here and t square is also cancels
and you get the result which is three k times
n or three k per atom which is indeed three
r
so not only when i apply quantum ideas to
mechanical oscillators i explain that c goes
to zero as t goes to zero it also goes to
the correct limit in high temperature by the
way this model of of calculating c is known
as einstein model so i conclude this this
section by emphasizing that einstein applied
the ideas of an oscillator having quantized
values of energies to explain the vanishing
of specific heat of solids as temperature
goes to zero and that gave a little more trust
in quantum theory
in the coming ah two lectures this week i
am going to now ah discuss how quantum ideas
were also applied to other systems to explain
ah say hydrogen spectrum and the energy level
of an oscillator in general and this sort
of started building up quantum theory
