so welcome to these lectures where we are reviewing our concepts that we have been utilizing
all through this period for understanding how you are able to implement the quantum
computing aspects so in doing so we are going back to the problems that we had given to
you and based on their solutions and the understandings we will be we are going ahead with our implementation
problems so once again reviewing let us see where we stand as of now in this reviewing
concepts that we have been doing let us look back to week two which we have been doing
earlier and in this we have been analyzing the problems that we have been doing there
we have finished four of them earlier so let us look at the fifth one where it was looking
at the concept of the orbital now in the principle of analyzing the orbital the question which
was probing was to see how the node in orbital is understood
so the question is essentially asked to what a node of a orbital represent and so it is
a point or a plane where in the wave function changes sign that was the solution given so
let us revisit the concept of the orbital and it's nodes so in order to do that let
us look at the simplest of the orbitals which is the s orbital where the s orbital wave
function has the form where the radial part corresponds to the distance the angular part
corresponds this now the angular part for the s orbital as we know is constant because
there is no change along the a spherical symmetrical
so the angular part is going to be constant so we get a spherical there are three possible
representations that are possible for looking at these orbitals one is to look at the probability
along the different axis another one is to look at it's contour along the two dimensions
and finally it's to look at the probability density in terms of dots or colors so
generally speaking wave functions in orbitals are defined by their quantum numbers l by
their quantum numbers n l and n several orbital is a wave function it is a region of space
where the electronic exist and it has energy shapes and orientations in space thus an orbital
is defined as one electron wave function
so here at the three s orbitals which are all spherical symmetric but as you can see
the sizes shapes and orientations of the orbital depends on the quantum numbers so n determines
the size l determines the shape and m so well determines the orientation so given the nodal
plane as we show here the the orientation and the presence of the system is being shown
here now the p x p y p z are the once which have angular dependence so they are non symmetric
along the nodal plane where as in terms of the s orbital they are always going to be
symmetric however nodes can exists irrespective of the symmetry
and other issues so in terms of s orbitals the nodes exists in the radial part so one
s has no nodes two s has one node and three s has two nodes so typically in terms of
these these are radial nodes which are represented in terms of n minus one kind of nodes that
you always get and so the first so the first one one is doesnt have any node nodes in the
p orbitals are of different kind because they are angular nodes that passes through the
nucleus and the orbital therefore becomes a dumb bell shaped because it's no longer
symmetric and in this case it's important to note that the plus and the minus sign is
shown that is shown for the p orbital refers to the mathematical sign of the wave function
and not anything to do with a electric charge
because the where essentially looking at only the electrons so there is no charge aspect
associated with it it's the mathematical sign of the wave function it's something to
do with the symmetry of the system the nodes in the orbitals for example here we show the
three d orbitals there are two angular nodes that passes through the nucleus the orbital
is four leaf clover shaped and the d orbitals are important for metals and so and so forth
but generally speaking these as the some ways of how to visualize these orbitals and to
know that these nodes are the places where the sign changes are occurring or the symmetry
aspects are changing as you go across in terms of the orbitals and that's why these are important
in terms of the nodes and how they are represented in in each of these representations so with
that let's now look at the case of the schrodinger equation which in terms of more
than multi electron system it is known that it cannot be solved exactly and the critical
problem for that remains in what form was the question and the electron electron repulsion
term has been pointed out as the main issue so in order to know this let us first look
at the helium atom picture and understand that it has a one nucleus with two protons
and it two electrons
so there are this three body problem and so the two electrons can exist around
the nuclei two different directions which gives raise to this condition where we have
three coordinates to worry about r one two r one and r two the state of the
electron is given by the wave function which depends on r one and r two and the potential
energy is function of both r one r two as well as r one two so the kinetic energy
at the center mass of the system would be also dependent on two of them but the potential
energy part is the one which has the difficulty because the electron electron repulsion part
the is dependent on the r one two coordinate which is the one which is the couple coordinate
system in this case
so if you write the schrodinger equation in this form and the complete schrodinger equation
it turns out that it's it depends on three coordinates here r one r two and r one
two because of the last term which is coupled between r one and r two the coming from the
potential energy the potential is any more spherically symmetric and it depends
on the angle between r one and r two and so the schrodinger equation cannot be solved
analytically so the way this has been looked at is to so the way it has been looked at
this has been done has been is to approximate the problem and that will deal it later that
generally the schrodinger equation has the electron electron repulsion term which
needs to be taken care if this has to be solved so that was the part of the question
which is asked
the very next question however asks above the electronic hamiltonian of the helium atom
in atomic units which means that all the aspects of the mass and those are all taken
as unity in terms of the atomic units as we know so it simplifies and becomes a form
which only depends on the deferential parts as well as the distances and so this
is the actual form which i just showed you earlier in atomic units in terms of the
hamiltonian and that's the one which is been shown here so so that there is nothing
more to discuss in that so we come to the next question which was asked which is
the electronic wave function of the helium atom in it's ground state
now this is the under some approximation as i have been and discussing when we looked
at the hamiltonian before that we have to we can only solve this under the condition
that we assume that the r one two term is not going to create a problem so let us
look at how this is done and the approximate model essentially is idea which is assuming
that it's an independent electron approach the electron electron repulsion term is going
to be ignored and so then the product of the two functions can be used as then the problem
essentially depends only on r one and r two the schrodinger equation is then separated
into two equations and each one of the electrons in the helium atom is independent of each
other
now this is an approximation as i have been mentioning because there is an interaction
between the two electrons but the interaction is a ok as a first order level it can
be ignore to at least solve it as two independent electrons and that's the solution which has
been shown here except that in this particular case the spins of the two have been taken
considered and alpha and beta represent the plus and minus signs or the circular right
circularly rotating versus left circularly rotating electrons spin which is being combined
here to the wave function to give it the total wave function
so whenever a total wave function is given for a multi electron system the individual
spins of the electrons are also to be considered and that's what is been done here alpha beta
so the spin of one alpha and the spin of the other beta is been considered and the the
possibility of one being or one kind and the other and their linear combination is something
which is been looked and since this spin function due to polis exclusion principle
is anti symmetric so when you change sign so when the spin flips for one to the other
the sign changes and therefore we have this anti symmetric relationship between them
so this sought out this model includes the polis exclusion principle in terms of this
spin
now finally we also looked at some problems which where just looking at if it was possible
to understand how the energy transfers was been looked at so it's a very simple question
which ask as to if there only two energy levels separated by say the ionization potential
and hydrogen atom which is thirteen point six electron volt a what should be the wavelength
of the laser in nanometer that can match these energy gap now this is been asked because
it is sought of like unit transformation and the simple equation in this case is to
recognize that the energy can be written in terms in many different terms and units and
this is one of the ways of looking at it
now this is another condition that this is all under vacuum at the velocity that we are
considering is the speed of light in vacuum and therefore this is written in this term
typically if it is mater is involved in this then a refractive index term will come into
the picture but as if now if you consider the simplest possible case then the lambda
can be simply calculated in this form which turns out to be ninety one point eight nanometers
and so this after this mathematics this is what it comes out to now after having done
all of this i also realize that it is time perhaps that we look at how to solve the
hydrogen atom to at least some level so that all these understanding is put to in the
right place so the hydrogen atom is the simplest physical system containing interaction potential
that is not just an isolated particle
before coming to the hydrogen atom problem we basically solve our schrodinger equation
problems in terms of particle in a box or rotating system rigid rotor or simple
harmonic oscillator in doing so we basically ensure that we are looking at potentials
which are independent and they are not interacting potentials here hydrogen atom is a complete
problem in some sense and this also a good problem because it can be solve completely
using the schrodinger equation so that's the reason why we can look at it and so it's a
simple one proton one electron and the electrostatic coulomb potential that holds them together
so simple in a problem the potential energy in this case is just e squared over four
pi epsilon naught r at the attractive potential between the charged of the proton and the
electron is separated by distance r so it's all columbic and this is the stationary state
potential with node time dependence we could just plug it in to the schrodingers equation
a time dependent equation to get the form as which is been shown here at the potential
looks quite simple but it's a function of radial part r and not of x or x y z
so what can we do about that so what we have to do is we have to recognize that we have
to actually use the radial potential means we are going to use use this in this form
where we have to put back the x y z in this form and we will be we need to use the help
of symmetry of the problem to use our mathematical approach the spherical symmetry potential
in this case tells us that we can use spherical polar coordinates just like what we have
done for three d rigid rotor for instance the only deference is here it's not a rigid
rotor the r can also change
so as before we can look at it and define the volume element as r square sin theta d
theta d phi d r and in the spherical polar coordinates r is the length of the radius
vector from origin to the point that we are looking at in x y z dimension theta is the
angle between the radius vector and the z plus axis and phi is the angle between the
projection of the radial vector on to the x y plane and the plus x axis so phi is tan
inverse y over x the equations on the preview slide tells us how to express r theta phi
in terms of x y z we can also express x y z in terms of r theta phi so we can take advantage
of that and we can write it out in the individual terms and then we can rewrite the three d
schrodinger equation in this form where it is a function of x y z the energy in three
dimensions and in spherical polar coordinates we can rewrite this
because we know how to convert cartesian coordinates into spherical polar coordinates in this form
as we show here now if we plug in our potential v which we had discussed before and multiply
both sides by r squared sin square theta we get this form now this equation gives us the
wave functions psi for the electron in the hydrogen atom if we solve for psi in principle
we know everything that is to know about the hydrogen atom so that is the main part of
about it because it's a one electron system and that's what it is so when we solve the
schrodinger equation in one dimension we found out that one quantum number was necessary
to describe our systems
for example in the bohr atom the electron moves in an orbit but we need only one parameter
to specify it's position and the fixed orbit so we need only one quantum number here in
three dimensions and with three boundary conditions we will find that we need three quantum numbers
to describe our electron so this is what we are getting to so we are really solve schrodinger
equation for the electron in the hydrogen atom however we talk about solving the hydrogen
atom because our solution will provide us with much more a what we need to know about
the hydrogen thus because for one electron system columbic interaction one you know particle
then we have been essentially solved the whole thing
so in order to solving linear algebraic equations as we have been taking about solving coupled
algebraic equations for example x y together and solving linear differential equations
and solving coupled differential equations example with derivatives mix together that's
what we have in this case so we need to separate the variables we have a couple linear differential
equation to solve may be if you are clever like when we were doing with the earlier calculations
in some cases we can make the problem easier a big improvement would be to uncouple the
variables stated more mathematically when we have an equation like the one that we are
discussing we would like to see if we can separate the variables or split the equation
into different parts with only one part of variable in each part
so our problem will be much simplified if we can write the wave function which is
a function of r theta phi in terms of three different functions r which is dependent only
on r theta which is capital theta which is dependent on the angle theta and capital phi
which is dependent only on the phi so that is what we are after so let us assume that
it is possible to break it up into a these parts and see what happens if i our assumption
works then the orderly world of mathematics will know that it will be right so we can
write rewrite this six we are the original expression in this form three different term
founds where we have taken them
so with these assumption the partial derivatives in the schrodinger equation become like this
where the partial derivatives become full derivatives because now they depend only one
on each coordinate at one point of time to separate the variables what we need to do
is to plug in the wave function in to the schrodinger equation and divide by them assuming
that they are non zero and the result is what is now shown here once we do this
it immediately separates the phi variable and the term one over phi d squared phi d
phi is the function of phi only
so let's put this over to the right side of the equation and this gives us this firm
where we have already separated out the phi part so this equation has the form f of
r theta which is equal to g of phi f is a function of r and theta only where g is a
function of phi only so how can this be the that's the natural question which may sometimes
arise the right hand side only has phi in it but no r and theta where as the left hand
side only has r and theta in it but no phi and at left hand side and right hand side
is equal this is only possible so only one way when you have a constant independent of
r theta and phi which is equivalent to g of phi
so does that mean everything is just a constant that's not true it just means that the particular
combination of terms in the left hand side happens to add up to a constant which is the
same as the constant given by the particular combination of terms in the right hand side
so when we use this idea that the left hand side has this kind of a form then we
can write it in this kind of an expression where we have equated both the sides to a
constant now this is really good because
we have taken one nasty equation in r theta phi and separated out into two equations one
in r theta and the other one in phi only so now the question is can we actually separate
out the r and theta part also so in order to do that let's examine that it turns
out that although we dont want to do the detail math here that the constant has to be a square
of an integer so this particular constant that we are taking about as to be particular
constant like it has to be a square of an integer and once we do that then these
equations would have solutions otherwise there are no solutions now this hunch we get from
the fact that our left hand side is already well defined and
well known so we can write our right hand side equation of this it turns out that we
will not do the math here but the constant has to be a square of an integer if not our
differential equations have no solution and we already know that the right hand side of
the equation therefore can be written simply as a part which is equivalent to a square
of a a constant and it's a i am highlighting the facts that the m actually depends on a
l with the lower subscript so where did this m so well components it's an integer and we
just happen to give it that name the left hand side of our big schrodinger equation
also must be equal to m l squared now
so if we set the left hand side equal to m l square divided by sin square and rearrange
this we get this kind of an expression once again we have separated the variable
the left hand side is a function of r only and the right hand side is a function of theta
only again the only way to satisfy this equation is for the left hand side is to be a constant
which is equal to a right hand side the solution of these resulting differential equation will
result in restrictions of this constant in this case the constant must be equal to an
integer times the next larger integer and that's one of the tricks that we find out
as to how it's happening and so what we find is that we have taken our initial differential
equation and split into three here are the three pieces rewritten slightly differently
the first two parts of the angular equations where these double derivative of the phi function
is a constant square this is the second part is basically a part where the constant
was l times l plus one which we have utilized to write the second part which is our theta
variable and finally the radial part is dependent only on the r function and that's why we get
the schrodinger equation has been separated into three ordinary second order differential
equations equations one two and three each containing only one variable
now the solutions need to be found for the boundary condition and we no longer need to
be dealing with partial differentials because everything now falls into place by boundary
conditions that the wave function amplitude need to go to zero at infinity and that they
need to be single valued and so on and so forth so with this background then the different
parts of the equations can be solved the radial part of the equation is call the associated
laguerre equation and the solutions r that satisfy the appropriate boundary conditions
are called associated laguerre polynomial assuming that the ground state has l equals
to zero and that requires m of l equals zero the equation three which is the radial part
becomes this takes this form where the derivative of the r squared r d r yields two terms
based on the product rule writing those terms and inserting the spherical electrostatic
potential we get the final form
we can then try a solution which is of the form a to the exponential minus r over a naught
a is the nomination constant a zero is the constant with the dimension of length on inserting
the first and the second derivative of r we are able to make the expression look much
better and given the condition to satisfy for any of the r is for each of the two expression
is parenthesis has to be zero so we can set the second parenthesis to be zero to solve
for a zero which happens to be the bohr radius as we find it here and then we can said the
first parenthesis equal to zero and solve for e which is what we get many often ignore
the reduce mask and because it turns out to be essentially the mass of the electron the
difference of mass is so small where it is the smaller mass which comes
but both are equal to the bohr results and which are backed by the spectral lines so
the hydrogen atom radial wave functions of the first few are now that we have seen
them we can write write some of them from the books we can see it in any of the books
around and they subscripts specify the values of ns of l n and l and these are the associated
laguerre polynomials they are already normalized similarly some of the associated laguerre
polynomials are given here with respect to the p l ms of l l and ms of l are constant
which were used to separate the schrodinger equation is spherical coordinates they were
cleverly chosen and will become quantum numbers eventually that's what the whole story is
about and for general cases these has to be normalized for usages solutions to the
angular equations are given in this form as we have to talk about
the product of the solutions of the angular and the azimuthal equations can also be found
from the other part of the two expressions which turn out to be as i mentioned before
this spherical harmonics we with which we started the whole problem and these group
solutions can be given into two functions and they can be normalize and looked at in
this form theta and phi dependent spherical harmonics the solution of the angular and
azimuthal equations therefore give rise to the entire full picture the radial wave function
r and this spherical harmonics determine the probability density of the various quantum
states that total wave function psi r theta phi depends on n l and m s of l the wave functions
therefore become product function of the two cases
the radial part as well as the angular part there are therefore two types of quantum numbers
radial and angular they appropriate boundary conditions needs to the following restrictions
on the quantum numbers l and ms of l l can take any values going from zero to l as long
as it is less than n ms of l can take any values from minus l through zero to plus l
and mode of m s of l should be less than equal to l so the predicted energy levels
are only depended on the principle quantum number n have with a lots of degeneracy
e of n equal to minus e naught over n squared
so basically there are n square d generate when n is greater than zero for hydrogen atom
n is always greater than zero and it's an integer all quantum numbers can become very
large for very highly excited state transition to classical n can take any values from one
to any integer all large values of possible all quantum numbers can become very large
for very highly excited states and which can transit to the classical physical conditions
the principle quantum number n is critical in terms of defining the energy
because only r radial part includes the potential energy v of r the result of this quantized
energy is e of n minus mu over two this form which is equivalent to e naught over
n squared which is just like what was predicted in bohrs model the negative sign means that
the energy indicates energy e indicates that the electron and the proton are bound together
as the energy only depends on n only there will be lot of degeneracy due to the high
symmetry of the potential at three d sphere has the highest symmetry and that is possible
in three d
the orbital angular momentum quantum number l and the spectroscopic notations can be
also looked at in this same way in as we are now doing this the letter names are given
for the various l values when the reference is two n electron so the ls of value of zero
is letter notation is s one is p two is d three is f four is g five is h the electronic
states are refer to by their n and l a state with n equal to two and l equal to one is
called a two piece state the boundary conditions required that n is always greater than l when
refer to the hydrogen atom we therefore have s p d kind of the shells and the spectroscopic
notation involves the capital access where as the orbitals are always represented
by the small letters
the spectroscopic notations for the atomic shells and the sub shells are as we have been
showing here has been given in terms of k l m n o p where as the shell symbol has
been given as s p d f g and h so in summary the quantum numbers which we have used at
the three main once which is a principle quantum number orbital angular quantum number magnetic
quantum number ms of l the boundary condition of the wave functions to go to zero at x goes
to infinity result in n values can go anywhere can take any integer one to n l can go from
zero to n minus one ms of l can go from minus l through zero to plus l and the restrictions
for the quantum numbers are such that n has to be greater than zero l has to be less than
zero and less than maximum is n minus one m s of l mode is less than equal to l
with this summary let us sought out finish on the idea of the hydrogen atom because
anything more than this would require the solution to the relativistic part of the schrodinger
equation which we are not doing the idea of the spin was introduced by poly in a different
manner of symmetry and which we will be looking at later the final question in the problem
that we are looking at in week two was concerning the approximations that are
very importantly done are very useful and
so this is what the statement was being discussed the common and contrasting factors for the
born oppenheimer approximation and the frank condon principle are tested in terms of
true false which is which is what is being talked about born oppenheimer is for molecules
while frank condon is for atoms
now the fact of these are the fact is not true because as we so the more important
part to realize up as a questioning of this is to revisit the idea of born oppenheimer
approximation and the frank condon principle which we will just now do but let us read
through the problem which have been looked at which is that born oppenheimer is for generating
molecular orbitals whereas frank condon is for spectroscopic transitions both relay which
is a true statement both relay on the fact that the nuclear mass is extremely large compared
to that of the electrons which is also true and both are gross approximations and mostly
fail is a wrong statement because amazingly this these two approximations work wonderfully
truly for most of the time and so the basic point of these two approximations are at in
quantum chemistry and molecular physics
the born oppenheimer approximation is the assumption that the motion of the atomic nuclei
and electrons in a molecule or an atom can be separated based on the fact that mass difference
between the two is extremely large which means that the motion of the electron is extremely
fast as compare to that of the molecule and so the separation of variables that which
we just discussed in terms of the hydrogen atom is in fact true due to the born oppenheimer
approximation the frank on principle on the other hand is the transition condition for
any spectroscopic rule and as we have been discussing the both where discussed in this
particular lecture in both saying token and this is a rule in spectroscopy and condon
chemistry that explains the intensity of vibronic transitions vibronic transitions are simultaneously
change in the electronic and vibrational energy levels for molecule due to the absorption
emission of a photon or the absorption of energy and there in it can only occur through
vertical transition
that is because once again the nuclear coordinate motion is extremely sluggish as compare to
the motion of the electrons and that's the basic idea behind all of these studies that
we have been discussing until now and they form the basis of most of the implementation
aspects in quantum computing and quantum information processing that we are actually dealing with
in this entire course and therefore i tell it was extremely important that we revisit
and understand these basic concepts once more to make sure that whatever we have discussing
currently in terms of implementations do not become faded because of the logical understanding
and these things are not clear with this let us end the lecture today and i look
forward to meeting you in the next round
thank you