WEBVTT
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so welcome to these lectures where we are
reviewing our concepts that we have been utilizing
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all through this period for understanding
how you are able to implement the quantum
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computing aspects so in doing so we are going
back to the problems that we had given to
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you and based on their solutions and the understandings
we will be we are going ahead with our implementation
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problems so once again reviewing let us see
where we stand as of now in this reviewing
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concepts that we have been doing let us
look back to week two which we have been doing
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earlier and in this we have been analyzing
the problems that we have been doing there
01:02.000 --> 01:09.010
we have finished four of them earlier so let
us look at the fifth one where it was looking
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at the concept of the orbital now in the principle
of analyzing the orbital the question which
01:16.530 --> 01:22.420
was probing was to see how the node in orbital
is understood
01:22.420 --> 01:29.579
so the question is essentially asked to what
a node of a orbital represent and so it is
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a point or a plane where in the wave function
changes sign that was the solution given so
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let us revisit the concept of the orbital
and it's nodes so in order to do that let
01:42.189 --> 01:49.530
us look at the simplest of the orbitals which
is the s orbital where the s orbital wave
01:49.530 --> 01:56.469
function has the form where the radial part
corresponds to the distance the angular part
01:56.469 --> 02:02.170
corresponds this now the angular part for
the s orbital as we know is constant because
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there is no change along the a spherical
symmetrical
02:06.170 --> 02:13.540
so the angular part is going to be constant
so we get a spherical there are three possible
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representations that are possible for looking
at these orbitals one is to look at the probability
02:19.650 --> 02:26.540
along the different axis another one is to
look at it's contour along the two dimensions
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and finally it's to look at the probability
density in terms of dots or colors so
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generally speaking wave functions in orbitals
are defined by their quantum numbers l by
02:43.140 --> 02:51.500
their quantum numbers n l and n several orbital
is a wave function it is a region of space
02:51.500 --> 02:59.090
where the electronic exist and it has energy
shapes and orientations in space thus an orbital
02:59.090 --> 03:03.590
is defined as one electron wave function
03:03.590 --> 03:09.040
so here at the three s orbitals which are
all spherical symmetric but as you can see
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the sizes shapes and orientations of the orbital
depends on the quantum numbers so n determines
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the size l determines the shape and m so well
determines the orientation so given the nodal
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plane as we show here the the orientation
and the presence of the system is being shown
03:28.900 --> 03:36.570
here now the p x p y p z are the once which
have angular dependence so they are non symmetric
03:36.570 --> 03:41.210
along the nodal plane where as in terms of
the s orbital they are always going to be
03:41.210 --> 03:47.710
symmetric however nodes can exists irrespective
of the symmetry
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and other issues so in terms of s orbitals
the nodes exists in the radial part so one
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s has no nodes two s has one node and three
s has two nodes so typically in terms of
04:04.890 --> 04:09.920
these these are radial nodes which are represented
in terms of n minus one kind of nodes that
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you always get and so the first so the first
one one is doesnt have any node nodes in the
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p orbitals are of different kind because they
are angular nodes that passes through the
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nucleus and the orbital therefore becomes
a dumb bell shaped because it's no longer
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symmetric and in this case it's important
to note that the plus and the minus sign is
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shown that is shown for the p orbital refers
to the mathematical sign of the wave function
04:40.039 --> 04:42.039
and not anything to do with a electric charge
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because the where essentially looking at only
the electrons so there is no charge aspect
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associated with it it's the mathematical
sign of the wave function it's something to
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do with the symmetry of the system the nodes
in the orbitals for example here we show the
04:58.960 --> 05:03.669
three d orbitals there are two angular nodes
that passes through the nucleus the orbital
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is four leaf clover shaped and the d orbitals
are important for metals and so and so forth
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but generally speaking these as the some ways
of how to visualize these orbitals and to
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know that these nodes are the places where
the sign changes are occurring or the symmetry
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aspects are changing as you go across in terms
of the orbitals and that's why these are important
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in terms of the nodes and how they are represented
in in each of these representations so with
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that let's now look at the case of the
schrodinger equation which in terms of more
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than multi electron system it is known
that it cannot be solved exactly and the critical
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problem for that remains in what form was
the question and the electron electron repulsion
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term has been pointed out as the main issue
so in order to know this let us first look
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at the helium atom picture and understand
that it has a one nucleus with two protons
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and it two electrons
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so there are this three body problem and so
the two electrons can exist around
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the nuclei two different directions which
gives raise to this condition where we have
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three coordinates to worry about
r one two r one and r two the state of the
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electron is given by the wave function which
depends on r one and r two and the potential
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energy is function of both r one r two as
well as r one two so the kinetic energy
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at the center mass of the system would be
also dependent on two of them but the potential
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energy part is the one which has the difficulty
because the electron electron repulsion part
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the is dependent on the r one two coordinate
which is the one which is the couple coordinate
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system in this case
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so if you write the schrodinger equation in
this form and the complete schrodinger equation
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it turns out that it's it depends on
three coordinates here r one r two and r one
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two because of the last term which is coupled
between r one and r two the coming from the
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potential energy the potential is
any more spherically symmetric and it depends
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on the angle between r one and r two and so
the schrodinger equation cannot be solved
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analytically so the way this has been looked
at is to so the way it has been looked at
07:51.900 --> 08:00.009
this has been done has been is to approximate
the problem and that will deal it later that
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generally the schrodinger equation has
the electron electron repulsion term which
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needs to be taken care if this has to be
solved so that was the part of the question
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which is asked
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the very next question however asks above
the electronic hamiltonian of the helium atom
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in atomic units which means that all the
aspects of the mass and those are all taken
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as unity in terms of the atomic units as
we know so it simplifies and becomes a form
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which only depends on the deferential
parts as well as the distances and so this
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is the actual form which i just showed you
earlier in atomic units in terms of the
08:44.540 --> 08:51.149
hamiltonian and that's the one which is been
shown here so so that there is nothing
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more to discuss in that so we come to the
next question which was asked which is
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the electronic wave function of the helium
atom in it's ground state
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now this is the under some approximation as
i have been and discussing when we looked
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at the hamiltonian before that we have to
we can only solve this under the condition
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that we assume that the r one two term
is not going to create a problem so let us
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look at how this is done and the approximate
model essentially is idea which is assuming
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that it's an independent electron approach
the electron electron repulsion term is going
09:27.760 --> 09:37.019
to be ignored and so then the product of the
two functions can be used as then the problem
09:37.019 --> 09:41.970
essentially depends only on r one and r two
the schrodinger equation is then separated
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into two equations and each one of the electrons
in the helium atom is independent of each
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other
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now this is an approximation as i have been
mentioning because there is an interaction
09:55.330 --> 10:02.290
between the two electrons but the interaction
is a ok as a first order level it can
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be ignore to at least solve it as two independent
electrons and that's the solution which has
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been shown here except that in this particular
case the spins of the two have been taken
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considered and alpha and beta represent the
plus and minus signs or the circular right
10:23.820 --> 10:30.510
circularly rotating versus left circularly
rotating electrons spin which is being combined
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here to the wave function to give it the total
wave function
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so whenever a total wave function is given
for a multi electron system the individual
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spins of the electrons are also to be considered
and that's what is been done here alpha beta
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so the spin of one alpha and the spin of the
other beta is been considered and the the
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possibility of one being or one kind and the
other and their linear combination is something
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which is been looked and since this
spin function due to polis exclusion principle
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is anti symmetric so when you change sign
so when the spin flips for one to the other
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the sign changes and therefore we have
this anti symmetric relationship between them
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so this sought out this model includes the
polis exclusion principle in terms of this
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spin
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now finally we also looked at some problems
which where just looking at if it was possible
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to understand how the energy transfers
was been looked at so it's a very simple question
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which ask as to if there only two energy levels
separated by say the ionization potential
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and hydrogen atom which is thirteen point
six electron volt a what should be the wavelength
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of the laser in nanometer that can match these
energy gap now this is been asked because
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it is sought of like unit transformation
and the simple equation in this case is to
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recognize that the energy can be written in
terms in many different terms and units and
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this is one of the ways of looking at it
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now this is another condition that this is
all under vacuum at the velocity that we are
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considering is the speed of light in vacuum
and therefore this is written in this term
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typically if it is mater is involved in this
then a refractive index term will come into
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the picture but as if now if you consider
the simplest possible case then the lambda
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can be simply calculated in this form which
turns out to be ninety one point eight nanometers
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and so this after this mathematics this is
what it comes out to now after having done
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all of this i also realize that it is time
perhaps that we look at how to solve the
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hydrogen atom to at least some level so that
all these understanding is put to in the
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right place so the hydrogen atom is the simplest
physical system containing interaction potential
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that is not just an isolated particle
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before coming to the hydrogen atom problem
we basically solve our schrodinger equation
13:11.350 --> 13:18.819
problems in terms of particle in a box or
rotating system rigid rotor or simple
13:18.819 --> 13:25.000
harmonic oscillator in doing so we basically
ensure that we are looking at potentials
13:25.000 --> 13:29.560
which are independent and they are not interacting
potentials here hydrogen atom is a complete
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problem in some sense and this also a good
problem because it can be solve completely
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using the schrodinger equation so that's the
reason why we can look at it and so it's a
13:41.779 --> 13:47.579
simple one proton one electron and the electrostatic
coulomb potential that holds them together
13:47.579 --> 13:53.139
so simple in a problem the potential energy
in this case is just e squared over four
13:53.139 --> 13:59.529
pi epsilon naught r at the attractive potential
between the charged of the proton and the
13:59.529 --> 14:05.880
electron is separated by distance r so it's
all columbic and this is the stationary state
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potential with node time dependence we could
just plug it in to the schrodingers equation
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a time dependent equation to get the form
as which is been shown here at the potential
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looks quite simple but it's a function
of radial part r and not of x or x y z
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so what can we do about that so what we have
to do is we have to recognize that we have
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to actually use the radial potential means
we are going to use use this in this form
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where we have to put back the x y z in this
form and we will be we need to use the help
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of symmetry of the problem to use our mathematical
approach the spherical symmetry potential
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in this case tells us that we can use spherical
polar coordinates just like what we have
14:57.759 --> 15:03.730
done for three d rigid rotor for instance
the only deference is here it's not a rigid
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rotor the r can also change
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so as before we can look at it and define
the volume element as r square sin theta d
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theta d phi d r and in the spherical polar
coordinates r is the length of the radius
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vector from origin to the point that we are
looking at in x y z dimension theta is the
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angle between the radius vector and the z
plus axis and phi is the angle between the
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projection of the radial vector on to the
x y plane and the plus x axis so phi is tan
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inverse y over x the equations on the preview
slide tells us how to express r theta phi
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in terms of x y z we can also express x y
z in terms of r theta phi so we can take advantage
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of that and we can write it out in the individual
terms and then we can rewrite the three d
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schrodinger equation in this form where
it is a function of x y z the energy in three
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dimensions and in spherical polar coordinates
we can rewrite this
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because we know how to convert cartesian coordinates
into spherical polar coordinates in this form
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as we show here now if we plug in our potential
v which we had discussed before and multiply
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both sides by r squared sin square theta we
get this form now this equation gives us the
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wave functions psi for the electron in the
hydrogen atom if we solve for psi in principle
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we know everything that is to know about the
hydrogen atom so that is the main part of
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about it because it's a one electron system
and that's what it is so when we solve the
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schrodinger equation in one dimension we found
out that one quantum number was necessary
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to describe our systems
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for example in the bohr atom the electron
moves in an orbit but we need only one parameter
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to specify it's position and the fixed orbit
so we need only one quantum number here in
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three dimensions and with three boundary conditions
we will find that we need three quantum numbers
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to describe our electron so this is what we
are getting to so we are really solve schrodinger
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equation for the electron in the hydrogen
atom however we talk about solving the hydrogen
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atom because our solution will provide us
with much more a what we need to know about
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the hydrogen thus because for one electron
system columbic interaction one you know particle
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then we have been essentially solved the whole
thing
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so in order to solving linear algebraic equations
as we have been taking about solving coupled
17:35.600 --> 17:41.280
algebraic equations for example x y together
and solving linear differential equations
17:41.280 --> 17:46.270
and solving coupled differential equations
example with derivatives mix together that's
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what we have in this case so we need to separate
the variables we have a couple linear differential
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equation to solve may be if you are clever
like when we were doing with the earlier calculations
17:59.640 --> 18:08.150
in some cases we can make the problem easier
a big improvement would be to uncouple the
18:08.150 --> 18:14.659
variables stated more mathematically when
we have an equation like the one that we are
18:14.659 --> 18:19.120
discussing we would like to see if we can
separate the variables or split the equation
18:19.120 --> 18:23.159
into different parts with only one part of
variable in each part
18:23.159 --> 18:28.780
so our problem will be much simplified if
we can write the wave function which is
18:28.780 --> 18:34.880
a function of r theta phi in terms of three
different functions r which is dependent only
18:34.880 --> 18:42.040
on r theta which is capital theta which is
dependent on the angle theta and capital phi
18:42.040 --> 18:47.820
which is dependent only on the phi so that
is what we are after so let us assume that
18:47.820 --> 18:53.110
it is possible to break it up into a these
parts and see what happens if i our assumption
18:53.110 --> 18:57.309
works then the orderly world of mathematics
will know that it will be right so we can
18:57.309 --> 19:02.390
write rewrite this six we are the original
expression in this form three different term
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founds where we have taken them
19:03.779 --> 19:07.980
so with these assumption the partial derivatives
in the schrodinger equation become like this
19:07.980 --> 19:15.049
where the partial derivatives become full
derivatives because now they depend only one
19:15.049 --> 19:19.820
on each coordinate at one point of time to
separate the variables what we need to do
19:19.820 --> 19:25.850
is to plug in the wave function in to the
schrodinger equation and divide by them assuming
19:25.850 --> 19:32.950
that they are non zero and the result is
what is now shown here once we do this
19:32.950 --> 19:40.640
it immediately separates the phi variable
and the term one over phi d squared phi d
19:40.640 --> 19:42.970
phi is the function of phi only
19:42.970 --> 19:49.660
so let's put this over to the right side of
the equation and this gives us this firm
19:49.660 --> 19:57.090
where we have already separated out the phi
part so this equation has the form f of
19:57.090 --> 20:02.300
r theta which is equal to g of phi f is a
function of r and theta only where g is a
20:02.300 --> 20:08.779
function of phi only so how can this be the
that's the natural question which may sometimes
20:08.779 --> 20:14.050
arise the right hand side only has phi in
it but no r and theta where as the left hand
20:14.050 --> 20:19.980
side only has r and theta in it but no phi
and at left hand side and right hand side
20:19.980 --> 20:26.330
is equal this is only possible so only one
way when you have a constant independent of
20:26.330 --> 20:32.130
r theta and phi which is equivalent to
g of phi
20:32.130 --> 20:37.760
so does that mean everything is just a constant
that's not true it just means that the particular
20:37.760 --> 20:42.340
combination of terms in the left hand side
happens to add up to a constant which is the
20:42.340 --> 20:47.240
same as the constant given by the particular
combination of terms in the right hand side
20:47.240 --> 20:55.169
so when we use this idea that the left
hand side has this kind of a form then we
20:55.169 --> 21:00.580
can write it in this kind of an expression
where we have equated both the sides to a
21:00.580 --> 21:02.450
constant now this is really good because
21:02.450 --> 21:07.570
we have taken one nasty equation in r theta
phi and separated out into two equations one
21:07.570 --> 21:14.340
in r theta and the other one in phi only so
now the question is can we actually separate
21:14.340 --> 21:21.159
out the r and theta part also so in order
to do that let's examine that it turns
21:21.159 --> 21:27.440
out that although we dont want to do the detail
math here that the constant has to be a square
21:27.440 --> 21:33.750
of an integer so this particular constant
that we are taking about as to be particular
21:33.750 --> 21:39.690
constant like it has to be a square of an
integer and once we do that then these
21:39.690 --> 21:45.279
equations would have solutions otherwise there
are no solutions now this hunch we get from
21:45.279 --> 21:49.370
the fact that our left hand side is already
well defined and
21:49.370 --> 21:55.789
well known so we can write our right hand
side equation of this it turns out that we
21:55.789 --> 22:01.100
will not do the math here but the constant
has to be a square of an integer if not our
22:01.100 --> 22:07.840
differential equations have no solution and
we already know that the right hand side of
22:07.840 --> 22:13.230
the equation therefore can be written simply
as a part which is equivalent to a square
22:13.230 --> 22:21.730
of a a constant and it's a i am highlighting
the facts that the m actually depends on a
22:21.730 --> 22:27.810
l with the lower subscript so where did this
m so well components it's an integer and we
22:27.810 --> 22:34.380
just happen to give it that name the left
hand side of our big schrodinger equation
22:34.380 --> 22:37.159
also must be equal to m l squared now
22:37.159 --> 22:42.090
so if we set the left hand side equal to m
l square divided by sin square and rearrange
22:42.090 --> 22:47.450
this we get this kind of an expression
once again we have separated the variable
22:47.450 --> 22:52.830
the left hand side is a function of r only
and the right hand side is a function of theta
22:52.830 --> 22:59.240
only again the only way to satisfy this equation
is for the left hand side is to be a constant
22:59.240 --> 23:03.510
which is equal to a right hand side the solution
of these resulting differential equation will
23:03.510 --> 23:08.920
result in restrictions of this constant in
this case the constant must be equal to an
23:08.920 --> 23:14.090
integer times the next larger integer and
that's one of the tricks that we find out
23:14.090 --> 23:19.830
as to how it's happening and so what we find
is that we have taken our initial differential
23:19.830 --> 23:24.309
equation and split into three here are the
three pieces rewritten slightly differently
23:24.309 --> 23:33.360
the first two parts of the angular equations
where these double derivative of the phi function
23:33.360 --> 23:41.100
is a constant square this is the second part
is basically a part where the constant
23:41.100 --> 23:46.320
was l times l plus one which we have utilized
to write the second part which is our theta
23:46.320 --> 23:52.419
variable and finally the radial part is dependent
only on the r function and that's why we get
23:52.419 --> 23:58.070
the schrodinger equation has been separated
into three ordinary second order differential
23:58.070 --> 24:02.880
equations equations one two and three each
containing only one variable
24:02.880 --> 24:07.730
now the solutions need to be found for the
boundary condition and we no longer need to
24:07.730 --> 24:12.810
be dealing with partial differentials because
everything now falls into place by boundary
24:12.810 --> 24:16.929
conditions that the wave function amplitude
need to go to zero at infinity and that they
24:16.929 --> 24:23.740
need to be single valued and so on and so
forth so with this background then the different
24:23.740 --> 24:29.100
parts of the equations can be solved the radial
part of the equation is call the associated
24:29.100 --> 24:34.789
laguerre equation and the solutions r that
satisfy the appropriate boundary conditions
24:34.789 --> 24:39.240
are called associated laguerre polynomial
assuming that the ground state has l equals
24:39.240 --> 24:45.269
to zero and that requires m of l equals zero
the equation three which is the radial part
24:45.269 --> 24:51.679
becomes this takes this form where the derivative
of the r squared r d r yields two terms
24:51.679 --> 24:56.760
based on the product rule writing those terms
and inserting the spherical electrostatic
24:56.760 --> 25:00.010
potential we get the final form
25:00.010 --> 25:06.940
we can then try a solution which is of the
form a to the exponential minus r over a naught
25:06.940 --> 25:12.289
a is the nomination constant a zero is the
constant with the dimension of length on inserting
25:12.289 --> 25:17.510
the first and the second derivative of r we
are able to make the expression look much
25:17.510 --> 25:23.711
better and given the condition to satisfy
for any of the r is for each of the two expression
25:23.711 --> 25:28.289
is parenthesis has to be zero so we can set
the second parenthesis to be zero to solve
25:28.289 --> 25:32.730
for a zero which happens to be the bohr radius
as we find it here and then we can said the
25:32.730 --> 25:38.880
first parenthesis equal to zero and solve
for e which is what we get many often ignore
25:38.880 --> 25:44.710
the reduce mask and because it turns out to
be essentially the mass of the electron the
25:44.710 --> 25:50.340
difference of mass is so small where it is
the smaller mass which comes
25:50.340 --> 25:54.679
but both are equal to the bohr results and
which are backed by the spectral lines so
25:54.679 --> 26:00.299
the hydrogen atom radial wave functions of
the first few are now that we have seen
26:00.299 --> 26:05.330
them we can write write some of them from
the books we can see it in any of the books
26:05.330 --> 26:10.950
around and they subscripts specify the values
of ns of l n and l and these are the associated
26:10.950 --> 26:16.390
laguerre polynomials they are already normalized
similarly some of the associated laguerre
26:16.390 --> 26:22.200
polynomials are given here with respect to
the p l ms of l l and ms of l are constant
26:22.200 --> 26:26.309
which were used to separate the schrodinger
equation is spherical coordinates they were
26:26.309 --> 26:30.679
cleverly chosen and will become quantum numbers
eventually that's what the whole story is
26:30.679 --> 26:36.450
about and for general cases these has to
be normalized for usages solutions to the
26:36.450 --> 26:39.890
angular equations are given in this form as
we have to talk about
26:39.890 --> 26:45.200
the product of the solutions of the angular
and the azimuthal equations can also be found
26:45.200 --> 26:49.870
from the other part of the two expressions
which turn out to be as i mentioned before
26:49.870 --> 26:55.320
this spherical harmonics we with which we
started the whole problem and these group
26:55.320 --> 27:01.399
solutions can be given into two functions
and they can be normalize and looked at in
27:01.399 --> 27:07.860
this form theta and phi dependent spherical
harmonics the solution of the angular and
27:07.860 --> 27:13.770
azimuthal equations therefore give rise to
the entire full picture the radial wave function
27:13.770 --> 27:18.190
r and this spherical harmonics determine the
probability density of the various quantum
27:18.190 --> 27:26.070
states that total wave function psi r theta
phi depends on n l and m s of l the wave functions
27:26.070 --> 27:30.210
therefore become product function of the
two cases
27:30.210 --> 27:35.980
the radial part as well as the angular part
there are therefore two types of quantum numbers
27:35.980 --> 27:40.769
radial and angular they appropriate boundary
conditions needs to the following restrictions
27:40.769 --> 27:48.409
on the quantum numbers l and ms of l l can
take any values going from zero to l as long
27:48.409 --> 27:56.460
as it is less than n ms of l can take any
values from minus l through zero to plus l
27:56.460 --> 28:02.389
and mode of m s of l should be less than
equal to l so the predicted energy levels
28:02.389 --> 28:10.110
are only depended on the principle quantum
number n have with a lots of degeneracy
28:10.110 --> 28:13.559
e of n equal to minus e naught over n squared
28:13.559 --> 28:20.720
so basically there are n square d generate
when n is greater than zero for hydrogen atom
28:20.720 --> 28:24.870
n is always greater than zero and it's an
integer all quantum numbers can become very
28:24.870 --> 28:30.450
large for very highly excited state transition
to classical n can take any values from one
28:30.450 --> 28:36.210
to any integer all large values of possible
all quantum numbers can become very large
28:36.210 --> 28:42.130
for very highly excited states and which can
transit to the classical physical conditions
28:42.130 --> 28:46.950
the principle quantum number n is critical
in terms of defining the energy
28:46.950 --> 28:53.309
because only r radial part includes the potential
energy v of r the result of this quantized
28:53.309 --> 29:03.179
energy is e of n minus mu over two this
form which is equivalent to e naught over
29:03.179 --> 29:08.060
n squared which is just like what was predicted
in bohrs model the negative sign means that
29:08.060 --> 29:12.700
the energy indicates energy e indicates that
the electron and the proton are bound together
29:12.700 --> 29:17.321
as the energy only depends on n only there
will be lot of degeneracy due to the high
29:17.321 --> 29:22.180
symmetry of the potential at three d sphere
has the highest symmetry and that is possible
29:22.180 --> 29:25.559
in three d
29:25.559 --> 29:30.610
the orbital angular momentum quantum number
l and the spectroscopic notations can be
29:30.610 --> 29:35.910
also looked at in this same way in as we
are now doing this the letter names are given
29:35.910 --> 29:45.070
for the various l values when the reference
is two n electron so the ls of value of zero
29:45.070 --> 29:55.720
is letter notation is s one is p two is d
three is f four is g five is h the electronic
29:55.720 --> 30:02.590
states are refer to by their n and l a state
with n equal to two and l equal to one is
30:02.590 --> 30:11.571
called a two piece state the boundary conditions
required that n is always greater than l when
30:11.571 --> 30:19.820
refer to the hydrogen atom we therefore have
s p d kind of the shells and the spectroscopic
30:19.820 --> 30:24.850
notation involves the capital access where
as the orbitals are always represented
30:24.850 --> 30:26.279
by the small letters
30:26.279 --> 30:30.159
the spectroscopic notations for the atomic
shells and the sub shells are as we have been
30:30.159 --> 30:36.790
showing here has been given in terms of k
l m n o p where as the shell symbol has
30:36.790 --> 30:43.419
been given as s p d f g and h so in summary
the quantum numbers which we have used at
30:43.419 --> 30:50.100
the three main once which is a principle quantum
number orbital angular quantum number magnetic
30:50.100 --> 30:55.540
quantum number ms of l the boundary condition
of the wave functions to go to zero at x goes
30:55.540 --> 31:04.179
to infinity result in n values can go anywhere
can take any integer one to n l can go from
31:04.179 --> 31:10.889
zero to n minus one ms of l can go from minus
l through zero to plus l and the restrictions
31:10.889 --> 31:15.710
for the quantum numbers are such that n has
to be greater than zero l has to be less than
31:15.710 --> 31:24.620
zero and less than maximum is n minus one
m s of l mode is less than equal to l
31:24.620 --> 31:31.330
with this summary let us sought out finish
on the idea of the hydrogen atom because
31:31.330 --> 31:38.399
anything more than this would require the
solution to the relativistic part of the schrodinger
31:38.399 --> 31:45.289
equation which we are not doing the idea of
the spin was introduced by poly in a different
31:45.289 --> 31:52.460
manner of symmetry and which we will be looking
at later the final question in the problem
31:52.460 --> 32:00.120
that we are looking at in week two was
concerning the approximations that are
32:00.120 --> 32:03.940
very importantly done are very useful and
32:03.940 --> 32:10.419
so this is what the statement was being discussed
the common and contrasting factors for the
32:10.419 --> 32:16.179
born oppenheimer approximation and the frank
condon principle are tested in terms of
32:16.179 --> 32:21.399
true false which is which is what is being
talked about born oppenheimer is for molecules
32:21.399 --> 32:23.590
while frank condon is for atoms
32:23.590 --> 32:31.880
now the fact of these are the fact is not
true because as we so the more important
32:31.880 --> 32:38.620
part to realize up as a questioning of this
is to revisit the idea of born oppenheimer
32:38.620 --> 32:43.490
approximation and the frank condon principle
which we will just now do but let us read
32:43.490 --> 32:47.830
through the problem which have been looked
at which is that born oppenheimer is for generating
32:47.830 --> 32:52.899
molecular orbitals whereas frank condon is
for spectroscopic transitions both relay which
32:52.899 --> 32:57.860
is a true statement both relay on the fact
that the nuclear mass is extremely large compared
32:57.860 --> 33:03.000
to that of the electrons which is also true
and both are gross approximations and mostly
33:03.000 --> 33:09.130
fail is a wrong statement because amazingly
this these two approximations work wonderfully
33:09.130 --> 33:15.289
truly for most of the time and so the basic
point of these two approximations are at in
33:15.289 --> 33:16.970
quantum chemistry and molecular physics
33:16.970 --> 33:21.390
the born oppenheimer approximation is the
assumption that the motion of the atomic nuclei
33:21.390 --> 33:27.610
and electrons in a molecule or an atom can
be separated based on the fact that mass difference
33:27.610 --> 33:34.710
between the two is extremely large which
means that the motion of the electron is extremely
33:34.710 --> 33:39.780
fast as compare to that of the molecule and
so the separation of variables that which
33:39.780 --> 33:46.330
we just discussed in terms of the hydrogen
atom is in fact true due to the born oppenheimer
33:46.330 --> 33:52.179
approximation the frank on principle on the
other hand is the transition condition for
33:52.179 --> 33:58.409
any spectroscopic rule and as we have been
discussing the both where discussed in this
33:58.409 --> 34:05.270
particular lecture in both saying token and
this is a rule in spectroscopy and condon
34:05.270 --> 34:10.840
chemistry that explains the intensity of vibronic
transitions vibronic transitions are simultaneously
34:10.840 --> 34:15.240
change in the electronic and vibrational energy
levels for molecule due to the absorption
34:15.240 --> 34:21.659
emission of a photon or the absorption of
energy and there in it can only occur through
34:21.659 --> 34:23.140
vertical transition
34:23.140 --> 34:29.020
that is because once again the nuclear coordinate
motion is extremely sluggish as compare to
34:29.020 --> 34:34.139
the motion of the electrons and that's the
basic idea behind all of these studies that
34:34.139 --> 34:39.419
we have been discussing until now and they
form the basis of most of the implementation
34:39.419 --> 34:43.829
aspects in quantum computing and quantum information
processing that we are actually dealing with
34:43.829 --> 34:51.010
in this entire course and therefore i tell
it was extremely important that we revisit
34:51.010 --> 34:56.659
and understand these basic concepts once more
to make sure that whatever we have discussing
34:56.659 --> 35:04.230
currently in terms of implementations do not
become faded because of the logical understanding
35:04.230 --> 35:10.329
and these things are not clear with this
let us end the lecture today and i look
35:10.329 --> 35:16.680
forward to meeting you in the next round
35:16.680 --> 35:18.920
thank you