hello this is the next session of this third
segment of the course on nano electronic device
fabrication and characterization and as i
- outlined last time the segment is concerned
about nano material systems and making an
attempt on to understand what concepts are
necessary to deal with nano systems now what
i will do this time is to recap what we did
in the previous session and then go on to
the basic concepts of quantum mechanics because
its necessary as i mentioned last time to
use quantum mechanics to deal with small scale
systems and within the quantum mechanical
aspects of the of the ah this segment we will
deal with the schrodinger equation the potential
barrier and tunneling ah concepts and also
deal with what happens when a particle is
in potential well and quantization of energy
that takes place under these circumstances
now what we get in the last class is really
to to try to first define nano systems and
then we went on to point out that quantum
mechanics is necessary to deal with ah such
systems and actually also statistical mechanics
because while quantum mechanics is necessary
to deal with a nano particle or let us say
a quantum particle like an electron a nano
system typically consists of thousands hundreds
and thousands of these nano particles so therefore
it is necessary to deal with statistical mechanics
aspects of such systems therefore what we
did last time was to point out that quantum
mechanics must be used to describe systems
when they are sufficiently small under how
small these should be depends on the strength
of the interactions involved among these particles
so you can have clusters of atoms quantum
dots electronic properties of nano wires and
thin films and so on all dealt with as quantum
systems
there is a concept that i mentioned last time
i will repeat it although we will will deal
with that in the next class namely the density
of states that is the number of energy states
per unit energy interval typically in electron
volts that determines how many processes in
the world of this small are determined or
how these prob process are determined
so the density of states is an important aspect
of dealing with nano systems so the very functional
form of the dependence of the density of states
on energy changes with the dimensionality
as you have mentioned last time you can have
nano systems with the zero dimensions one
dimensions two dimensions and three dimensions
therefore the density of the state depends
on the dimensionality of these systems now
as i said one has to deal with also this part
systems where you might have hundreds or thousands
of these quantum particles together in an
nano system while quantum mechanics dominates
the behavior of atoms statistical mechanics
is necessary and its pertinent to understanding
the behavior of ensembles of these atoms and
molecules therefore the science and technology
have nano systems is where there is a an intersection
of the behavior or complex systems with that
of quantum behavior
therefore quantum mechanics and statistical
mechanics have to be used together to deal
with these systems as i said a nano systems
they consist of tens of thousands of atoms
and each of them is a quantum mechanical system
and how quantum mechanical a system how microscopic
a system depends on the strength of interactions
as i already mentioned so what you should
do this time is to 
understand some basic concepts of quantum
mechanics all of you would be familiar right
from much earlier classes that you have taken
that energy is absorbed and emitted in packets
called quanta or a quantum
now this was the basic ah discovery of planks
more than hundred years ago and what he said
was energy is packed into in a quantum and
the energy in such a quantum is given by e
is equal to h nu where h is the plancks constant
whose value is six point six three into ten
to the power minus thirty four joule second
and nu is the frequency of the radiation involved
now again early in your learning you have
found or you learned about the bohrs theory
of atom and you have learned how by confining
a particle in this case an electron into a
region of space in this case around the nucleus
of the hydrogen atom such confinement leads
to energy quantization that is the electron
in such a system such a an environment can
only take discrete values unlike what would
happen in newtonian mechanics where there
is no such restriction and energy supposed
to be continuously variable for a particle
moving in under any any force any four force
field so that is the distinction between the
quantum concept and the classical concept
which is dealt with in the newtonian mechanics
now a primary aspect of the development of
quantum theory was the postulate of de broglie
luid de broglie who said that if you consider
a freely propagating particle then its position
can be predicted by associating with it a
wave of wavelength lambda given by h over
mv where v is the speed of the particle m
is its mass and h of course is the plancks
constant so what you have here is a relationship
between a particle in the newtonian sense
with the mass m and wave motion because lambda
the wavelength is associated with a wave so
these two are earlier considered to be incompatible
that is particle motion and wave motion and
de broglies concepts bring them brings them
together so lambda the value of lambda is
related to the question when is his when this
system to be considered quantum mechanical
and when is it classical so this can be illustrated
actually by considering an electron as an
example of a quantum particle or by applying
the de broglie rule to the electron whose
mass is nine point one into ten to the minus
thirty one kg and a charge of one point six
into ten to the minus nineteen coulombs
ok and if you subject it to a potential of
fifty kilo volts as typically happens in an
electron microscope example then it gains
energy and that energy is given by e is equal
to half mv square and that is equal to q into
v where q is the charge of the electron and
v is the potential across which it has been
taken namely fifty kilo volts so if you do
the arithmetic you find that the velocity
of the electron under these conditions would
be about one point three into ten to the power
eight meters per second and therefore the
momentum which is m into v turns out to be
one point ten into ten to the minus twenty
two meter kg per second since lambda the de
broglie wavelength of the electron is the
ratio of the plancks constant of the momentum
then arithmetic tells us that that lambda
is of the order of five into ten to the power
of twelve meters or about five pico meters
now its important to keep in consider keep
in mind when we look at these numbers its
a very small distance very small (( )) five
into ten to the power minus twelve meters
is a very small length and therefore a very
small sizes for a particle but the size of
the electron there are different ways to estimate
it the size of the electron is of the order
of the ten to the minus sixteen meters therefore
what you have see is that the wavelength of
the wave associated with the electron that
has just gone through a potential difference
of ah fifty kilo volts is much larger than
the size of the electron itself the physical
size of the electron itself
in other words the de broglie wavelength associated
with an electron of this energy is significantly
greater than the size of the electron itself
therefore you can see that the application
of quantum mechanics in this case probably
should be significant or it must be necessary
because the de broglie wavelength associated
with the electron is so much larger than the
size of the electron itself again by extending
the argument suppose you think of a nano particle
and think of its size or rather the mass of
that to be of the order of let us say a pico
gram ten to the power of minus twelve gram
now if you try to calculate the wavelength
of the de broglie wave that is associated
with that you will see that it is a much smaller
number much smaller number than the de broglie
wavelength of the electron because the mass
is in the denominator therefore when the mass
is greater lambda becomes much smaller therefore
if you have a pico gram of material with about
the same energy then you will find that the
wavelength of that is orders of magnitude
smaller than the size of the particle itself
therefore what it illustrates is that when
the wavelength associated with a particle
or a physical object is much smaller than
its dimension then quantum mechanics does
not apply and in such a case it is valid to
treat the particle using newtons laws of motion
so that is a kind of a an order of magnitude
argument for where one has to deal with ah
quantum mechanics now what are the key concepts
some of the key concepts i should say because
i will deal with some of them today and some
of them tomorrow what are the key concepts
of quantum mechanics one is that the particle
behavior or motion can only be predicted in
terms of a probability that is quantum mechanics
shows how to make such probability predictions
of the motion of a particle that is subject
to quantum mechanical laws this is the contrast
with newtons laws where its not a question
of probability its a question of certainty
you can predict where for example the moon
will be in the orbit around the earth ok on
a given day thats how you are able to predict
the precised time and date of an eclipse for
example therefore that kind of certainty is
possible because quantum mechanical considerations
do not apply to large objects where the de
broglie wavelength is very much smaller than
the size of the object
now further the distribution of the particles
predicted through the tools of quantum mechanics
as i said you know we really talk about probabilities
and therefore we have to talk about distribution
of these probabilities so therefore the distribution
of this particles is wave like so there is
a concept which we will have no time to go
through in this short segment of the course
a the concept is a particle is to be associated
or described as a wave packet so its a wave
like distribution that represents a particle
in quantum mechanics now as i already said
the the de broglie wavelength associated with
the probability distribution of macroscopic
particles large objects is
so small that the quantum mechanical effects
are not a parent and certainly not observable
with any tool that we have today another important
part of the another important aspect of quantum
mechanics is the heisenberg uncertainty principle
again something that you would have come across
earlier in your schooling one statement of
the heisenberg uncertainty principle which
is actually in inequality is delta x into
delta p is greater than or equal to h bar
by two
now h bar which is really the plancks constant
divided by two pi has a special significance
in quantum mechanics therefore we shall generally
use h bar instead of h so what this equation
is saying is that there is a limit to the
precision with which one can simultaneously
determine delta x and delta p that is the
uncertainties in momentum and the position
of a particle that is you cannot have zero
for both you cannot have zero for delta x
and zero for delta p at the same time there
is a minimum uncertainty associated with both
and if you increase the uncertainty in rather
if you try to make the measurement of the
position precise then the uncertainty in the
position in the momentum goes up correspondingly
in such a way as to keep the inequality intact
so this is a direct result of what you said
earlier namely that in quantum mechanics one
describes particle motions through probabilities
and not certainties therefore the heisen hisenberg
uncertainty principle is an integral part
of the foundation of quantum mechanics
now again to illustrate this consider an electron
in hydrogen atom which is confined to a region
let us first consider the first bohr orbit
the size of that orbit is one angstrom approximately
that is a bohr radius multiplied by two therefore
there is an uncertainty in the position of
the electron around the nucleus of the order
of one angstrom therefore delta x is of the
order of one angstrom using the uncertainty
principle then we see that the minimum delta
p is h divided by four pi delta x using the
above inequality and that transferred to be
five point three into ten to the minus twenty
five kilogram meter per second and therefore
the uncertainty in the velocity which is delta
p divided by m is of the order of five point
eight into ten to the five meter per second
that actually gives us an order of magnitude
of the velocity of the electron as it moves
around the nucleus under its electrostatic
force so what is actually says is that a particle
that is confined to a small volume has a large
momentum now what we have shown here is delta
p we have calculated delta p here as about
ten to the minus twenty five kilogram metre
per second
if we use that momentum to calculate the energy
of the electron through the formula e is equal
to p square by two m to get an order of magnitude
what you will find is that the only the arithmetic
view what you will find is that the energy
of the electron in such a circumstance where
it is confined is easily find to be of the
order of several electron volts which we know
to be true from our bohr theory therefore
what this shows is that using the uncertainty
principle its possible to calculate the order
of magnitude of energies that quantum mechanical
particles would have under certain known circumstances
now now rev recall that the uncertainty principle
is really the product of delta x and delta
p and the product has the product is greater
than or equal to the plancks constant recall
that the units of the plancks constant is
energy multiplied by time joule second therefore
this suggests i have used the wrong word here
i said this reveals i should say this suggests
another form of the uncertainty relationship
which we could write as delta e into delta
t greater than or equal to the plancks constant
delta e representing joules and delta t representing
second together farming the units for the
planks constant
now consider an electron that undergoes a
transition across a band gap of four electron
volts in some semiconductor you know semiconductors
have band gaps we will written though that
later consider an electron that undergoes
a an electronic transition across the band
gap of a semiconductor whose band gap is four
electron volts they therefore delta e in this
case can be taken to be four electron volts
so if you dont know for example whether the
electron is in the upper band or the lower
band therefore data e is equal to four electron
volts using the uncertainty relationship we
can see that therefore delta t which is h
divided by delta e according to the uncertainty
relationship turns out to be of the order
of ten to the power minus fifteen seconds
or one fempto second therefore what this says
is that the lifetime of an electronic transition
across a band gap of four electron volts is
of the order of one femtosecond that is the
uncertainty in the energy of a particle observed
for a very short time one fem femtosecond
is very great therefore what another formula
the uncertainty princi[ple]- relationship
is that if you try to measure the energy of
a quantum particle within a very short period
of time
if you try to make an observation if its energy
make a measurement of its energy over a very
short period of time then the result that
you get of the energy is uncertained by a
large proportion again that means that you
cannot determine the energy of a quantum particle
precisely unless you make the measurement
over very long periods of time thats what
it means now coming back to the probabilistic
behaviour of quantum [bechadic/mechanic] mechanic
quantum particles that is the probabilistic
behaviour that quantum mechanics asserts for
particles in the ah quantum mechanical regime
the values of the probability amplitude are
now postulated to be given by a wave function
represented typically by the greek letter
psi which is a function of the position r
and time t so this is analogous to the newtonian
concept of how you can define the particle
a particle by its position as a function of
time so you follow the motion of a particle
by saying where it is going to be as a function
of time
so this is an analogous concept where we have
a probability amplitude will come back to
why its called the amplitude momentarily that
is given by psi of r and t and r and t define
where the quantum mechanical particle is in
time and space an important aspect of quantum
mechanics again as a departure from classical
mechanics is that psi can be a complex function
because what you have said earlier is that
psi is the probability amplitude again by
a certain interpretation of quantum mechanics
mechanics due to max born the meaning of the
meaning of psi lies in the probabilistic interpretation
of quantum mechanics which means that the
modulus of psi squared the wave function squared
is the quantity that represents a probability
of finding the position finding the particle
at r and t at at a position r at a time t
now as i said the ah wave function is a complex
quantity it can be complex therefore psi square
modulus of psi square is really the product
of the complex conjugate of psi and the real
psi therefore what you have here is the real
quantity so the even though the probability
amplitude can be complex psi square the probability
of finding the particle at a given position
at a given time is a real quantity
now its possible that this psi r and t is
actually just psi r where it is independent
of t that is the probability of finding the
particle is independent of the time for a
given position r in that case what you have
is a stationary system in that case what you
get is psi is equal to psi r that is the probability
of finding the particle at a position r is
independent of time such a system is a stationary
system 
now continuing the concept of the wave function
what i wrote in the previous slide is the
wave function where it is a function of r
which is a three dimensional wave vector three
dimensional position vector with you know
cartesian coordinates xy and z but to simplify
things
we can consider the wave function as a function
of x only in one dimension and if you write
psi is equal to psi x without the t there
then you have a stationary wave function it
is a simple fact that such a particle has
to be found somewhere that is you would have
the particle somewhere in the universe therefore
if x for example or varies all over the place
that is you go from minus infinity to to plus
plus infinity for example and you integrate
this probability function psi r psi star into
psi r if we integrate this then the integral
should be equal to unity because that says
the probability of finding the particle somewhere
is exactly equal to one in one dimensions
this becomes minus infinity to plus infinity
integral of psi x psi star x is equal to one
so this is how one defines the certainty of
finding the particle somewhere within the
domain of interest 
another basic aspect of quantum mechanics
is the schrodinger equation the schrodinger
equation is the equivalent of the newtons
law of quantum mechanics and it applies to
wave functions or the probability amplitude
of a quantum particle now just like newtons
laws the schrodinger equation is a postulate
so you one cannot derive schrodinger equation
although how it takes on the forum it does
has its roots in the development of co classical
mechanics in the later stages now in one dimension
this schrodinger equation has this form where
the first term is a second derivative in space
so minus h bar h bar square by two m delta
square psi delta x square and at the second
function or the second term has the potential
function vxt vxt is the time dependent potential
to which this particle is subject so it'ls
moving under this potential and the right
hand side has the time derivative of the wave
function multiplied by i and h bar so this
itself the presence of i the square root of
minus one once again asserts that this is
a quantum mechanical system where you can
have complex functions now this is a so called
time dependent schrodinger equation because
on the left hand side you have a derivative
with respect to space on the right hand side
you have a derivative with respect to time
therefore it is the time dependent schrodinger
equation
however its possible to deal with and actually
more common to deal with the time independent
schrodinger equation if the potential vxt
does not dependent on time then as i said
you would have a particle in a stationary
state if the potential is independent of time
then the solution to schrodinger equation
which describes the motion of the particle
they are time independent therefore you have
a stationary state in such a case one can
write psi as the product of a time dependent
part and a time independent part and a time
dependent part so we write psi xt as psi of
x into phi of t because now the potential
function vxt is just vx independent of a time
then the schrodinger equation becomes ok if
we go back to how the schrodinger equation
was in the previous side and we substitute
psi xt as psi x into phi t then it is the
ah schrodinger equation then becomes ok as
a as you shown here you get a separation of
the spatial part on the left hand side and
the time dependent part on the right hand
side so the left hand side is independent
of time and the right hand side is independent
of x now they are equal if they are to be
equal then they both must be a constant
so on the left hand side you have a time independent
part on the right hand side you have a time
dependent part both of them are being equal
then they are equal to a constant with respect
to time and space so this is the energy of
the system why is called the energy and so
on is a part as i said of hamiltonian mechanics
and so on we have no time to go into that
but this turns out to be the energy of the
system 
so we take the right hand side and therefore
we write we can write the time dependent part
equal to e the the constant that is the energy
of the system
so this is a simple first order time dependent
differential equation and the solution is
such that therefore psi if you solve this
is simple equation psi then becomes psi of
x into e to the minus i into e by h bar into
t where e because we have a quantum system
e is equal to h bar omega ok because this
is what plank postulated earlier so e is equal
to h bar omega therefore one has a simple
sinusoidal function for the ah wave function
more here now psi xt therefore is the product
of psi x the space dependent part multiplied
by exponential that is the time dependent
part notice that this is a complex function
therefore the complex conjugate gives us a
change of sign in the exponent therefore the
product of psi and pi star the complex conjugate
becomes a real function psi xt squared actually
psi x squared ok so what you see here is that
this probability is there for independent
of time that is what we have now is the stationary
state the probability of finding a particle
in a given position is independent of time
we have a stationary state so this is the
simple consequence of being able to separate
the variables because the potential is independent
of time 
now again we equate the left hand side in
the schrodinger equation which was the space
dependent part to e and when you do that what
you get this equation is the equation at on
this line the time independent schrodinger
equation minus h bar square divided by two
m second derivative of psi plus v into psi
is equal to e into psi so this is the famous
time independent schrodinger equation
this can be written for shorthand as h of
psi x equal to e into psi x where h which
is the so called hamiltonian operator is the
second derivative differential operator represented
by minus h bar square over two m derivative
of second derivative of with respect to space
plus vx so this hamiltonian is therefore set
to operate on the wave function and the result
of that operation of this second derivative
operator on the wave function is for us to
get the wave function back with the energy
as its factor so the solution to the schrodinger
equation therefore yields what are known as
eigenfunctions that is functions that give
us exact solutions to the hamiltonian operator
and eigen energies that is the exact energies
that such a quantum mechanical system would
have so what one typically does in quantum
mechanics is to solve the time independent
schrodinger equation for a known potential
for example you may have a constant potential
you may have a potential such as in a the
ah hydrogen atom where it is the central potential
you could have a simple harmonic oscillator
where you have a corresponding potential for
v and so on
therefore the primary object of quantum mechanics
is to solve the schrodinger equation for a
given potential and sometimes of course you
actually approximate a potential in order
to be able to solve a physical problem as
often happens now let us consider a particle
in a constant potential that is vx is equal
to v it is not space dependent then the schrodinger
equation is simplified ok you have minus h
bar square about two m delta square delta
x square phi x plus v into psi x is equal
to e into psi x it is rearranged this way
where you have the second derivative that
is equal to the wave function multiplied by
a constant e is a constant to be found v is
the potential and these are m and h are constants
for a given particle the solution for this
is straight forward as long as for example
in this case the solution is a into e to the
ikx where k is given by k square is equal
to two m into e minus v divided by h power
square so this is a simple sinusoidal function
so what we have a solution for the motion
of the particle is through the wave function
psi x which is the sinusoidal function that
is the motion of a particle quantum mechanical
particle in a constant potential that is time
independent position independent is just a
sinusoidal wave now k as you as you see here
is related to the potential and the mass of
the particle now k into x because it is in
the exponent kx is dimensionless therefore
k must have a dimension that is the inverse
of the dimension of x and its position therefore
this is the so called reciprocal vector x
is the real vector real space so this k belongs
to the reciprocal space which you would have
come across for example in x ray diffraction
now looking at the form of the equation for
k square k is real or imaginary depending
on whether e is greater than equal to v is
greater than v or e is less than v so the
motion of the particle therefore depends on
the relative value of the energy that the
particle may have with respect to the potential
constant potential in which it is moving 
a simple case of the constant potential is
where v is equal to zero in that case e is
you can see here from this equation if v is
equal to zero
then e is equal to h bar square k square over
two m so in such a case you have a simple
expression for energy and if energy is also
written as p square over two m where p is
the momentum we can then see that p the momentum
can be written as h bar k so there is a direct
relationship between the momentum of the particle
and the wave vector of the particle or the
reciprocal vector of the particle now recall
that in the earlier case where we have a time
independent schrodinger equation psi xt is
psi into exponential e to the power of minus
i into e by h bar t ok e by h bar is just
omega so what we have is psi x into t xt is
equal to osi x into e to the power of minus
i omega t so we bring that over here to write
the general solution for the schrodinger equation
for a particle with ah moving under zero potentials
so called free particle
so you have the space dependent part which
is a e to the ikx a sinusoidal function with
respect to space and the time dependent part
is also sinusoidal therefore the wave function
of a free particle has the form of a wave
both in time and space now note that p is
equal to h bar k and k is equal to actually
two pi a by lambda and therefore what you
get is p is equal to h bar h divided by lambda
which is the de broglie relationship so everything
is consistent 
we can proceed next to dealing with the very
important concept of quantum mechanical tunneling
concept of quantum mechanical tunneling which
is an illustration of the solution of the
problem of a particle with energy e that is
trying to surmount a barrier of potential
v v being greater than e so that is represented
in this diagram so a particle is moving towards
a barrier potential barrier whose height so
to speak is greater than the energy that the
particle processes
the barrier is erected at x equal to zero
now in classical mechanics this particle let
us say an electron with energy e that faces
a potential barrier of v greater than e would
simply bounce of the barrier that is think
of this is a projectile it comes here finds
the barrier cannot surmount the barrier so
it just gets back goes back considering that
you have an elastic collision now what this
quantum mechanics tell us about what happens
in such a case 
now quantum mechanics imposes boundary conditions
that lead to to solutions different from the
classical one
now what are the boundary conditions that
is remember what you are dealing with in quantum
mechanics are solutions in the form of the
probability amplitude namely the wave function
in such a case what should really happen with
respect to the solution what should be what
are the conditions that the solutions should
satisfy the boundary conditions are psi is
continuous across the barrier
that is if you take the barrier if you take
the wave function just to the left of the
barrier and a wave function just to the right
of the barrier they should be continuity in
other words you are thinking about continuous
function as in the case of analysis mathematical
analysis so i want analytical continuity for
the wave function for values of x just less
than zero and just greater than zero and also
the derivative of the wave function should
also be continuous across the barrier that
is once again the derivative of x derivative
of psi with respect to x to the left of the
barrier should be the same as a derivative
just to the right of the barrier so we are
talking amount continuous psi as well as the
continuous continuity in this first derivative
of spy psi so these are the conditions that
a quantum mechanical solution should satisfy
now why these conditions why these conditions
is that suppose suppose the wave function
was discontinuous at the barrier at at x equal
to zero suppose the first derivative of the
wave function was was discontinuous at x equal
to zero zero that is there is sudden jump
in psi across this barrier and there is a
sudden jump in the derivative of psi across
this barrier what that implies is that you
would have to provide you know since since
this since the derivative for example can
be infinite across the barrier because the
second that is the second derivative becomes
infinite if the first derivative discontinuous
and the first derivative becomes infinite
if the wave function is discontinuous at this
boundary now such discontinuities are singularities
really mean that there must be some large
amount of energy infinite amount of energy
that would be responsible for such discontinuities
in real physical situations thats not the
case therefore the boundary conditions for
the solution of the problem would be typically
that psi is continuous across the barrier
and the first derivative of psi is continuous
across the barrier now one thing i forgot
to say along the way is that the schrodinger
equation as we have written here is a second
order differential equation now because it
is a second order differential equation the
solution to that or the integration of the
equation to get solutions would involve two
constants so you have to determine two constants
to have complete solution for the second order
differential equation and those two constants
would then be be determined by these boundary
conditions that
we just defined that is there are two boundary
conditions that the solution must satisfy
and there are two constants should be determined
therefore its possible using these conditions
to determine the solutions to the schrodinger
equation uniquely now let us look at the case
of the barrier height being greater than e
then k is equal to square root of two m into
e minus v divide by h square now e is less
than v therefore what we now have is an imaginary
quantity so k is now an imaginary quantity
i into square root of all this therefore the
solution to the schrodinger equation which
if you remember this spatial part of it is
just e to the ikx that solution now has a
into psi e to the ikx now you have a an imaginary
quantity multiplied by i so that becomes a
real quantity therefore what you have is the
solution for x as a into psi e to the minus
kappa x where kappa is the quantity in the
square root its a real quantity
so what you have is not a sinusoidal variation
of ah sinusoidal wave function for the ah
solution but you have an exponentially declining
because k is positive what you have is an
exponentially decaying function so when you
have a particle going across a barrier of
this art then what you have is a case where
over here where e is less than v you have
sinusoidal variation for the solution but
across the barrier because of continuity conditions
it falls exponentially down on the other side
so you have a an exponential decay of the
wave function on the other side of the barrier
that is x greater than zero now remember that
in classical mechanics the chance of this
particle being on the side of the barrier
is zero the fact that the wave function is
non zero this side even if it is exponential
declining it means that there is a finite
probability the wave function is essentially
square root of probability as we said because
probability is given by psi squared therefore
a non zero value of the wave function on this
side of the barrier means that there is a
non zero probability of the particle being
here
so this phenomenon where in quantum mechanics
a particle with what would be an insufficient
energy in classical mechanics is able to surmount
a barrier that is larger than its energy this
is called tunneling which is a very very important
part of quantum mechanics there are actually
devices that operate on the concept of tunneling
and there are very important measurement instruments
you would have heard about the scanning tunneling
microscope the stn which actually opened the
era of nano sciences nano technology so that
depends on the concept of tunneling
so this arises from the schrodinger equation
and the boundary conditions that are necessary
to be imposed on realistic physical systems
that obey quantum mechanics so we will stop
here and we will continue next time after
reviewing these concepts but before we leave
for the day what i would like to say is that
we have tried to over here introduce the basic
concepts of quantum mechanics some of the
basic concepts i should say because we are
not come to a couple of others and use that
to tell us how the motion of a particle in
a quantum mechanical system is to be calculated
the schrodinger equation and a simple example
of the schrodinger equation almost to the
first example that gives us a an unusual result
a characteristic result of tunneling which
is a very important part of quantum mechanics
thank you
