we have learnt so far about the correspondence
principle that says that for n tending to
infinity that is the large quantum number
limit 
the quantum results go to classical results
in this lecture we are going to see how we
can use this principle to derive something
called the selection rules and also the a
coefficient for the harmonic oscillator recall
i took three examples in the previous lectures
and i just consider two of these the harmonic
oscillator and the particle in a box and in
the harmonic oscillator case i said that the
the displacement can be written as summation
c tau e raise to i tau omega t where omega
is the basic frequency which is nu times two
pi which was equal to two pi v over two l
on the other hand in the case of harmonic
oscillator x t has only one component and
i can write this as the amplitude e raise
to i omega t in fact i want to do better than
that this is usually written as a cosine of
omega t which is amplitude divided by two
e raise to i omega t plus e raise to minus
i omega t
thus you see that ah the particle moving in
a box has the fundamental frequency v over
two l and all its multiples and therefore
when it performs motion it will radiate all
kinds of frequencies the fundamental and it
is harmonics tau times that much where as
a particle performing motion in a harmonic
oscillator would give out only one frequency
let us see how we can use that so classically
just learnt that particle in a box will give
all frequencies by all frequency we do not
mean that it will give out frequencies which
are continuously varying but if it makes a
transition if the particle makes a transition
from nth level to say e n minus tau level
then the frequencies emitted would be tau
times omega n right where omega n is the angular
frequency related to motion in the nth level
so classically this frequency omega n or the
angular frequency connected with the motion
in the nth level will be the frequency of
radiation and what we saw last time is that
if i write its displacement verses time it
is like this it is not a pure harmonic so
this is what tells you that x t actually has
higher harmonic also which i wrote earlier
c c tau e raise to i tau omega lets call it
omega n t so it will give you frequency of
radiation omega n and since the motion also
contains other frequencies all the other frequencies
will also come out let us see what does it
teach us about quantum transitions so let
us now write quantum mechanically 
radiation does not come out because a particle
is oscillating with frequency nu rather it
comes out by jumps so radiation comes out
by jumps radiation comes out by jumps between
levels where as classically it is because
of the acceleration now if a radiation is
emitted due to jump from nth to n minus tau
th level the frequency nu n n minus tau is
given as en minus e n minus tau divided by
h this is given by finite difference right
where as we recall the previous lecture in
classically it is given by a differentiation
with respect to j
now as n tends to infinity nu n t n minus
tau go to tau nu n where nu n is the classical
frequency so you see that classically since
all these frequencies are observed right so
let us write this classically since all the
frequencies tau nu n are observed right that
means quantum mechanically 
a particle can jump to any n minus tau
let me write this again so this is nth level
n minus tau level and here is this jump coming
in and some radiation comes out so radiation
frequency n n minus tau is en minus en minus
tau over h and you see that classically all
tau nu n are observed because the particle
makes a motion periodic motion of basic frequency
nu n but then also the all this harmonics
exist so this means there is no restriction
on where that is n minus tau does the particle
jump to
if let us see a count example if we say that
n can go to n minus one only and this is just
a counter example if we say this then as n
goes to infinity only the fundamental 
frequency will be observed that means it is
just not n minus one it can go to any frequency
it likes is now the question arises question
is there a n critical beyond which we can
say that classical mechanics is good and below
which quantum mechanics is good
in other words is there a boundary between
classical and quantum mechanics the answer
is no actually the true theory is quantum
mechanics it is only as we saw in the previous
ah lecture only when we can replace delta
e by h by the derivative d e by d j times
n tau that classical theory holds good and
that means basically that n has to be large
enough and tau has to be much much smaller
than n that is the only condition but what
this principle is telling you is that you
should actually be approaching classical level
when you can replace that the difference by
derivative effectively and for that the condition
is n is very large and tau is much much smaller
than n
now lets take the other example the other
example is this harmonic oscillator 
and classically its motion is given by x t
equals a by two e raise to i omega t plus
e raise to minus i omega t that is it has
only one frequency 
let us look at it quantum mechanically there
are these levels h nu two h nu three h nu
four h nu and so on and the radiation is given
out when there is a jump 
now you see how we see by comparing with the
classical results there is a selection rule
suppose n to n minus tau transition takes
place then the corresponding frequency right
the corresponding frequency 
in classical limit 
will be tau times omega
however classically we see only omega this
implies tau at max can be one and this gives
you a restriction that in a harmonic oscillator
the only jumps that takes place are where
n goes the particle makes the jump n goes
to n minus one or it absorbs n plus one so
this gives what is called a selection rule
and that says selection rule says that the
jump is made or the transition takes place
for which delta n magnitude is one otherwise
for all of the transition there will be no
radiation
so there is no other transition that gives
out radiation and why is that let me repeat
because if other transitions also took place
in large n limit we will see other frequencies
also and that doesnt happens so to be consistent
with correspondence principle there is only
one transition allowed in quantum level
so a question that has arisen just now is
question if you need single frequency for
laser should you use 
harmonic oscillator confinement actually the
two questions are not related because in a
laser you choose the frequency that causes
the transition to be precisely that frequency
so you can choose any system and the stimulated
emission will take place only for that particular
frequency with which you are doing this stimulated
emission so this question ah doesnt really
arise
