so we have been going over the details of
quantum computing and the various aspects
of bit and we have also looked at some of
the relevant considerations which are required
for its implementation however one of the
basic issues which has come about is that
we have not really looked into some of the
foundations or the basic aspects of quantum
ah part of this problem so i thought in today's
lecture will back a little bit in ah looking
into the basics of quantum mechanics and take
it from their
so this particular ah lecture set would be
more on the basics of quantum mechanics and
looking at some of the aspects related to
the quantum phenomena so in terms of basics
of quantum mechanics we have already realized
that it is quite distinct and different from
the classical mechanics which we label as
the newtons mechanics or the newtonian mechanics
following the laws of newton and maxwells
equations which are for electromagnetic theory
so when we consider newtonian mechanics and
maxwells equations we can explain macroscopic
phenomena such as motion of billiard balls
or rockets
however when we are looking at microscopic
phenomena such as photon atom scattering and
flow of electrons in a semi conductor or the
kinds of discussions where been having until
recently which is miniaturization of computing
to try to do quantum computing and other things
quantum mechanics is the main idea so quantum
mechanics is a collection of postulates based
on a huge number of experimental observations
the difference between classical and quantum
mechanics can be understood by examining both
the classical point of view as well as the
quantum point of view
so let us see so the classical point of view
the newtonian one they ah isaac newtons one
is ah the one where the laws are written in
terms of particle trajectories a particle
is considered as an in divisible mass point
that has a variety of properties that can
be measured all of which we call observables
the observables specify the state of the particle
position and momentum and here please note
that we are mentioning both position and momentum
so this is what we call the classical particle
and often it is represented in terms of point
objects just for the just to make show things
simple point objects allow drawing of path
between this between difference points and
that's why its easier to mention in terms
of classical point objects and here both the
momentum as well as position so these like
position x one x two this is the position
as well as this motion associated would be
a velocity and so both position and momentum
can be specified and this is the classical
object point a system in this concept is a
collection of particles which interact among
themselves via internal forces and can also
interact with the outside world via external
forces
so the forces are of different kinds one are
one is the internal force and the other one
is the external force the state of a system
is a collection of this states of the particles
that comprise the system all properties of
a particle can be known to infinite precision
now this is the part which is very important
in classical mechanics and we have been used
to this principle so the conclusions from
classical look of how you wanted to state
is the trajectory is the state description
of newtonian physics evolution of the state
is the newtons second law the principle of
causality is the fact that two identical systems
with the same initial conditions subject to
the same measurement will yield the same result
now these are the three important aspects
of classical concept which give rise to or
which we are used to in terms of classical
mechanics so one of the important applications
of classical mechanics we know is always in
a astronomical objects like motion of stars
motion of planets all of this work quite well
solar system so we defined them in terms of
systems as well as individual objects so this
is all what we described in terms of classical
on the other hand in terms of quantum point
of view quantum particles can act both as
particles and waves so this is the first very
important aspect of quantum mechanics which
is it assumes to start with or it it has built
in wave particle duality
quantum state is a conglomeration of several
possible outcomes of measurement of physical
properties and quantum mechanics uses the
language of probability theory and therefore
a random chance is one of the important aspects
that we have to built in in quantum mechanics
another important point here to understand
is that the observer in this case cannot observe
a microscopic system without altering some
of its properties neither one can predict
how the state of the system will change in
terms of absolute precision and finally quantization
of energy is another property of this microscopic
particle which is different from classical
system because in terms of classical mechanics
energy can be continuous
so one of the most important points mentioned
in quantum way of looking at thing is the
inability to be certain in measurements and
that comes from the famous heisenberg uncertainty
principle which says that one cannot unambiguously
specify the values of particles position and
its momentum for a microscopic particle and
this is so different from the classical point
of view so this gives rise to the fact that
there is always some error in measurement
at any point of time the measurement is being
made so any measurement which is made will
have some error associated with it either
in space or in momentum and that product is
always going to be act best equal to h cross
which we know as h crosses h over two pi which
i clarify here ah with at best its going to
be h cross over two or more than that position
and momentum are therefore considered as incompatible
variables or in other words these are the
ones which cannot be measured with absolute
precision simultaneously
the heisenberg uncertainty principle strikes
at the very heart of classical physics the
particle trajectory because if we cannot ascertain
position and momentum simultaneously the entire
picture of trying to draw a particles motion
becomes difficult because neither the position
or the momentum can be certain to absolute
precision and that's the reason ah this becomes
one of the biggest problems for classical
physics or classical mechanics that is the
heisenberg uncertainty principle
so how do we apprehend this so in order to
make sure that quantum mechanics can still
be applied to macroscopic principle where
is something which is sort of like known as
the correspondence principle the idea being
that quantum mechanics when applied to macroscopic
systems it must reduce to the classical laws
or must reduce to the case which the classical
physics predicts and therefore the non classical
phenomena such as uncertainty and duality
must become undetectable niels bohr codified
this requirement into his correspondence principle
when he postulated his way of looking at quantum
mechanics
now bohrs theory is something which is still
considered as the old quantum mechanics what
we will now look at is ah how things have
evolved now ah for him it was necessary to
code this because he wanted the correspondent
principle of looking at quantum mechanics
or applying quantum mechanics to macroscopic
systems so that is one wave of looking at
it but there are now the system the ideas
have evolved to a wave that it need not be
coded it comes automatically the wave the
mathematics is done so in principle we have
the situation that when we are in the atoms
and molecule range in size ten to the power
minus ten or in that order then we are definitely
in this quantum mechanics domain and when
we are in macroscopic domains when we are
talking in terms of earth the trees buildings
planets for example then we are all in this
classical mechanics principle and there are
laws to connect them and there are physical
models which can lead us to the predictions
that we talk about in both these domains
so the wave particle duality which is the
other main aspect of quantum mechanics gives
rise to the very different behavior of microscopic
particles from that of classical particles
so in a sense in some experiments we have
a situation where the behavior of classical
wave is shown by particles so the particle
shows resembles the behavior of classical
wave that is its not localized in space in
other experiments the particle behaves like
a classical particle localized in space so
in some sense it is showing a duality between
the particle and the wave like picture corpuscular
theories of light treat light as though it
were composed of particles but cannot explain
diffraction and interference
on the other hand maxwells theory of electromagnetic
radiation can explain diffraction and interference
phenomena and that was the reason why corpuscular
theory was abandoned earlier so this is from
earlier times in quantum mechanics we are
apprehensive of the fact that both can coexist
and that is the reason why the property of
light used to have both wave like in terms
of maxwells phenomena or particle like as
for as the corpuscular theory so in terms
of wave particle duality we have specific
examples where waves as particles exist max
planck work on black body radiation in which
he assumed that the molecules of the cavity
walls described using simple oscillator model
can only exchange energy in quantized units
so that was one place where max planck introduced
the concept when he used to study the black
body radiation so this is the case when particle
nature was used in nineteen o five einstein
proposed that energy of an electromagnetic
field is not spread out overs the entire wave
front but localized in individual packets
or quanta each quantum of frequency new travel
through space with speed of light carrying
discrete amounts of energy and momentum which
are known as photons which are needs to explain
the photoelectric effect and this was later
on confirmed through x ray experiments of
compton so these are the two cases where for
example we have waves to be treated as particles
black body radiation and photo electric effect
similarly for particles as wave we have double
slit experiments where instead of using a
light source if one uses an electro gun the
electrons are diffracted by slit and then
interfere in the region between the diaphragm
and the detector to give rise to the same
kind of pattern which looks like the case
of interfering waves ok so the principle of
diffraction and interference are shown by
electrons and therefore electron particles
behave as wave under these conditions so little
bit more on blackbody radiation because this
was one of the first important points which
gave rise to the idea of quantum mechanics
if this kind of radiation has been known since
centuries since whenever a material is radiated
this kind of radiation has been known for
centuries since when a material is heated
it radiates heat and its color depends on
its temperature so for example heating elements
of a stove turned from dark red to bright
red when the temperature goes from five fifty
degree centigrade to seven hundred degree
centigrade and it can even further go in to
orange and yellow and finally white when the
temperature goes really hot so depending on
the temperature the radiation and its color
changes so the emission spectrum as we call
it as a result of this change depends on the
material heating as well as on the material
property
blackbody is defined as a material that is
constantly exchanging heat with its surroundings
to remain at a constant temperature it observes
and emits radiation the problem is it can
reflect incoming radiations which makes a
theoretical description much more difficult
depending on the environment and so on and
so forth however a blackbody is a perfect
absorber incoming radiations are totally absorbed
and none is reflected so that is the ideal
definition of blackbody another example of
blackbody can be something like a cavity which
is someone shown here such that a metal box
with a small hold drilled in it the incoming
radiation enters the hole keep bouncing around
inside the box with negligible change of escaping
again through the hole that means it gets
absorbed completely the hole is the perfect
absorber for example blackbody radiation emission
does not depend on the material the box is
made of this is generally universal in nature
so there are many ways blackbody and blackbody
radiations can be designed and can be checked
and it has been therefore several this concept
has been around for very long however the
difficulty this principle faced and it was
labeled as one of the outstanding problems
at beginning of the twentieth century was
to described the blackbody emission spectra
it was difficult because classical physical
principles could not quite explain what was
going on here so when attempts to fit the
low and the high wave length part of this
spectrum was tried simultaneously the theory
always failed the theory worked quite well
for the high temperature zone but it always
had a problem when it was trying to fit both
low and high wave length part of the spectrum
that's because as per classical physics emission
spectrum is a super position of electromagnetic
waves of different frequencies and the frequencies
allowed where the standing waves inside the
cavity again the equipartition of energy would
give rise to the fact that every standing
wave carries k t of energy boltzmann constant
and temperature t of the energy the flaw is
in this approach that when the wave length
is approaching zero the number of standing
waves keep on increasing leading to an energy
which goes to infinity and that's why almost
all the laws sort of show the theory in terms
of classical physics showed that the energy
keeps on increasing as the wave length goes
lower
this came to be known as the ultra of highlight
catastrophe and that led to the failure of
the classical theories one of the most celebrated
work in this the work of rayleigh jeans was
considered as state of the art using well
tested theories which have very good agreement
with many experimental results in many other
circumstances and even in this case wave length
part was well explained only when the wavelengths
went to lower numbers it failed does there
was a need of a new theory so when planck
looked at this problem he was also an expert
in thermo dynamics he assumed that the radiation
of the cavity was emitted and observed by
some sort of oscillators contained in the
walls and this concept is something which
he introduce he use the boltzmann statistical
methods to arrive at the formula which is
the plancks radiation law
now planck made two modifications to the classical
theory in coming to this form the oscillators
of electromagnetic origin can only have certain
discrete energies determined by e equal to
h nu n h nu where n is an integer nu is the
frequency and h is call the plancks constant
which was given by a number which was very
small six point six times ten to power minus
thirty four however with the units of joule
second so when there the oscillators can absorb
or emit energy in discrete multiples of the
fundamental quantum of energy given by delta
e equals h nu and that was one of the most
important points of the plancks radiation
law which showed that energy was in packets
so this give rise to the principle of quantization
blackboard emission spectrum explain by introducing
quantization of energy transfers resolved
the ultraviolet catastrophe so that was one
of the most important contributions at that
time in this filed by max planck even at low
wave lengths and at high frequencies this
particular theory worked at small wave length
the energy h nu needed to fill up the oscillator
states the probability of the occupied oscillators
decrease rapidly that is faster than the rate
found in the rayleigh jeans formula so no
ultra violet catastrophe happened
so at small lambda the energy h nu needs to
fill up the oscillators straights increase
their probability to be occupied then decreases
rapidly for example faster than the rate in
the rayleigh jeans formula so no ultra violet
catastrophe happens however it became very
highly disputed planck himself looked for
a few years find whether h could be made to
go to zero the planck constant without any
success so that was the basic principle behind
quantization where even planck who discovered
it had difficulty in ah apprehended so as
a result of plancks law the energy which is
the classical one continuous any energy could
be there became discrete the particle could
have only specific quantized energies and
this was in conjunction with idea that there
was some sort of a oscillating particle inside
this entire picture which was being shown
to think in terms of blackbody radiation
after this einstein in nineteen o five found
out that the same principle of quantization
was necessary when he was studying the photoelectric
effect what he found that his law required
that the radiation field be quantized his
observations where that when the light fell
on the emitter and photo electrons where being
generated which showed as a potential in the
circuit there was a threshold frequency until
which there were no emitted electrons however
once a threshold was reached many emitted
electrons came out so essentially there was
a threshold in terms of the energy which was
necessary for the electrons to come out showing
that it had a particle property rather than
a continuous property
so this threshold frequency or the threshold
energy gave rise to the idea of quantization
again this led to the other part which was
creating a problem in terms of classical mechanics
to look at an atom in terms of classical mechanics
and electron in orbit around an atomic nucleolus
should emit electromagnetic radiation or photons
continuously as per maxwell equation because
its continuously accelerating in curved part
the resulting laws in energy implies that
the electron should spiral into the nucleolus
in a very short time that is an atom cannot
exist so this was one of the issues which
the bohr model was able to handle because
once this concept of classical nature of radiation
being emitted at any energy was possible to
be looked at differently he was able to come
up with the first quantized atom picture and
that immediately explained the atomic spectra
of the hydrogen atom the line spectra and
all that so his theory in nineteen thirteen
was the one where these energies of these
electron around the nucleolus was considered
as quantized
so that the transition of an electron could
only happen at discrete intervals and so only
discrete line spectra which was always observed
was immediately explainable because of this
principle so the atomic spectra of the hydrogen
atom line spectra became understood from this
picture introduced by bohr and so that is
the first ah bohrs theory on that's the first
theory on the hydrogen atom in terms of quantized
aspect which was given so will get more into
quantum mechanics after this lecture in the
next one because we have just come to the
point where the first bohrs model of quantization
and the old quantum mechanics as we call it
is justified by using the principle of packets
or discrete energies which was obvious from
the previous experiments and ideas which were
necessary to explain this kind of observations
that we are observing
so both max planck and einstein quantization
principle of very small energy packets which
were given based on the plancks constant were
being utilized to give rise to this principle
of quantum mechanics more on this in the next
lecture
