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Welcome back to the lectures on chemistry
and the course on introductory molecular spectroscopy
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the ah theory of spectroscopy is essentially
developed using quantum mechanics the quantum
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principles for molecular energy levels or
understood by the solution of the schrodinger
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equation and in this lecture and probably
the next one i shall give you a brief summary
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of the elementary quantum results that we
shall use for the whole course
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in the previous course to this called chemistry
one which is available from the n p tel website
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a c dot in or n p tel dot a c dot in i have
introduced basic principles of quantum ah
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models that is elementary models of quantum
mechanics by about ten to fifteen hours and
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therefore that may be you used as a reference
every now and then but here i shall give only
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this summary and the results that we shall
be using for vibrational and rotational and
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also magnetic resonance spectroscopies ok
so let me start with the the fundamental equation
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that be need to be concerned with the schrodinger
equation the time dependent the schrodinger
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equation is given by this formula by this
equation i h bar the partial derivative of
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psi with respect to time is equated to the
operation of the hamiltonian operator of the
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system under study acting on psi and psi here
which i shall use as a capital psi is a function
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of both the coordinates of the system and
is a function of time so that is what is denoted
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as psi however if h is independent of time
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then the equation i h bar dou psi by dou t
is equal to h psi can be ah reduced to a simpler
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time independent equation h psi is equal to
e psi this psi is different from capital psi
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which is used here and the relation is the
psi or r t is factored into a function which
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depends only on the position coordinates of
the system and a function which is only dependent
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and as a result if you substitute that into
this equation you will get the time independent
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for schrodinger equation h psi which is given
as the kinetic energy operator plus the potential
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energy operator acting on psi is equal to
e psi where the psi is the psi of the r that's
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the same the thing what you see here ok
quantum mechanics for spectroscopy uses both
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time dependent and time independent hamiltonians
the molecular energy levels in the absence
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of the electromagnetic field are obtained
by the solution of the schrodinger equation
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ah which is time independent in the present
of the radiation the excitation of a molecule
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is often studied using a time dependent hamiltonian
for which one needs to use time dependent
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schrodinger equation and use approximate procedure
approximation procedure such as perturbation
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theory to arrive at ah results for the intensity
of radiation the gets absorbed or emitted
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so spectroscopy uses both of these and in
different domains but for our problem we need
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to have the knowledge of the energy levels
of the molecular systems some picture has
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to be some mental picture has to be there
and therefore ah let me introduce the elementary
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mathematics and the model problem results
right away in this lecture ok
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first of all when you say the time independent
schrodinger equation is h psi is equal to
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e psi it means your whole series of wave functions
e n psi n where n can be an integer n is an
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integer n is an integer and it can be finite
or infinite for magnetic resonance systems
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that being function or finite for optical
resonance ah using harmonic oscillators electronic
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wave function and so on the m can be infinite
therefore a whole group of wave functions
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these are called eigen functions or used ok
and e n s are called eigen values these are
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what you will obtain if you measure the energy
of the molecular system one of these what
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you will get with some probability therefore
the wave function in general for a molecular
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system an arbitrary wave function of the system
is expressed as a linear combination one to
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whatever number lets write infinity here for
optical resonance what we have is a linear
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combination coefficient and the eigen functions
this is called the a linear superposition
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principle
in this model c n s have very specific ah
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properties associated with them and since
i mention that the psi ns are eigen functions
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they can be normalized and they are also orthogonal
to each other being eigen functions of the
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hamiltonian operator and the Hamiltonian operator
is a hermitian operator hermitian operators
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are associated with observable properties
experimentally observable properties so the
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wave functions are normalized the wave functions
can be normalized wave functions are orthogonal
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and they have the specific property namely
the integral psi n star psi n over all the
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coordinate system a space available to the
system a psi m is delta n m which is the kronecker
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delta delta n m is one if n is equal to m
and is equal to zero if n is not equal to
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m
with this property you can see that the wave
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function phi which is written as linear combination
of all the eigen functions the coefficient
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c n square or probabilities of finding the
system
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in the state psi n with an energy e n what
it means is that if you have identical copy
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of the system many many copies and you repeat
the experiment of measuring the energy of
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the system the energy would be possible eigen
values the outcome would be one of the eigen
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values and the state at which ah the system
will ah remain would be the eigen state of
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that eigen value therefore psi n would be
one of the measurable state of the system
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and e n is the energy associated with it what
is the probability that you look at the system
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you get one psi n or another or another that
probability is given by the linear combination
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square c n square and therefore since it is
a probability statement what it means is that
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all such probabilities should at to give you
unity that is the safe system will in one
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of the eigen states that when it is measured
so this is a fundamental principle and there
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are simple model problems to illustrate this
so let me just give you the three model problem
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results the particle in a box one model the
ah particle on a ring the harmonic oscillator
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which is very important for vibrational spectroscopy
and then the hydrogen atom which ah is used
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as the basis for understanding molecular states
of all the molecules all the atoms on molecules
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with more than one electron hydrogen atom
is the simplest one electron model that could
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be solved and exactly and all these models
particles in the box or a ring harmonic oscillator
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they have exact solutions so let me recall
in the rest of this lecture and also the next
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lecture the primary results that we need to
know the eigen values the model the eigen
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vectors etcetera as i mentioned in the beginning
a more detail a more elaborate process of
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how we get these things is there in another
course chemistry one which is proceeded in
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this course therefore you may see the videos
ah at your convenience ah on on those specific
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topics if you need more information otherwise
you can also write them on the discussion
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board
now lets start with the simplest model namely
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the particle in a box model this is essentially
a potential free model inside the box and
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the potentials are infinity outside the box
and also the wall so what it means is that
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you study the hamiltonian you study the solutions
of the hamiltonian h psi is equal to e psi
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or h psi n is equal to e n psi n using the
hamiltonian which contains only the kinetic
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energy of the particle which in quantum mechanics
is given in one dimension by h square h bar
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square by two m d square by d x square h bar
is obviously h by two pi plancks constant
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please remember these things all the time
plancks constant and therefore you solve these
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equation for the potential model namely v
of x is zero for x in a certain region may
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be length of the box being l and v of x is
infinity otherwise meaning x less than or
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equal to zero x greater than or equal to l
ok
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so this is the box model and what are the
solutions the solutions are the wave function
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of psi n of x is normalized using the box
length by l sin n pi x by l and the n can
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take the values one two three etcetera all
the way to infinity and the energies e n are
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given by the formula h square n square by
eight m l square so the model of the mass
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of the particle and the length of the box
which are the parameters for this particle
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in a model determine the spread of the energies
on the discreetness in the energy because
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h square by eight m l square has the energy
unit or energy dimension ok and n square is
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the quantum number which tells you ah the
energies are one four sixty nine sixteen twenty
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five thirty six times this base unit and the
base unit of energy for that model depends
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on the length of the box that you chosen also
the mass of the particle that is being studied
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or the mass of the molecule that is being
studied the mass of the electron whatever
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that you do this is the solution this is the
simple one ah passes back you can immediately
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recall and of course you have the same relations
as i mentioned in the earlier namely the psi
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n star which being a real function is the
same as the psi n of x and psi m of x and
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d x the range for the particle is zero to
l the motion within the box and this is the
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d tau that you talked about this is delta
m n meaning that the wave functions are normalized
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when n is equal to m the probability of finding
the particle inside the box is unity for all
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states of the system and second is that the
wave functions are orthogonal to each other
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so this is the summary result and therefore
any particle wave function in principle any
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arbitrary wave function or any arbitrary state
of the system inside the box can be written
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as the sum n is equal to one to infinity c
n root two by l sin n pi x by l and for this
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state the average value of the energy turns
out to be sum over n one to infinity c n square
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the absolute square times h square n square
by eight m l square this is the average value
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for the energy but the average is obtained
only when you do many many measurements and
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take the average of all of this otherwise
every individual measurement will give you
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one of the energies and it will also fix the
state into that eigen state so this is a particle
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in a box model
the next important model that you have to
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remember for vibrational spectroscopy is the
harmonic oscillator now harmonic oscillator
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hamiltonian is the ah potential energy is
given by the potential energy which is parabolic
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or which is quadratic and the kinetic energy
is of course p square by two m she is written
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as minus h square by two m d square by d x
square plus half k x square and k is the harmonic
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force constant
expressing the stiffness of the harmonic emotion
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whether the emotion is very rigid or that
the motion is ah very flexible and the value
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of k will determine the harmonic the extent
to which the system can be harmonic
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so k is system dependent ok and every harmonic
oscillator has a fundamental frequency the
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classical frequency of the oscillator
is is known from elementary physics and that
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frequency nu is given by one by two pi square
root of k by m again you see that parameters
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which determine the system or m n k and the
frequency is expressed in terms of k by m
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the ratio k by m the ratio k by m the square
root this is of course classical what happens
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when you solve the schrodinger equation for
this system with h psi n of x is equal to
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e n psi n of x you get again infinity many
energies but ah one thing one one should remember
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is that n can be zero one two three etcetera
the integer n is equal to zero also allowed
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the solutions turn out to be for e n given
by this formula namely h bar omega times n
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plus half n plus half and omega is two pi
times nu the same nu that you have here therefore
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you see that the harmonic oscillator energy
is given in terms of h nu if you want to write
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that because there is a two pi in the denominator
that cancels this h nu into n plus half and
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the half ensures that n equal to zero is also
an allowed value which means even if the harmonic
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oscillator is in the lowest possible energy
state with low quantum number input to that
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its still has an energy called ah energy equivalent
to half of the h bar omega and h is the plancks
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constant omega its ah natural angular frequency
what are the eigen functions for the system
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the eigen functions are given by a combination
of the gaussian e to the minus alpha x square
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by two and the harmonic polynomials h n root
of alpha x what is alpha alpha is square root
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of k m by h bar square again alpha is the
physical constant associated with the system
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the mass of the harmonic oscillator and the
force constant that the oscillator has so
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if the oscillator is very strong its force
constant is very high then alpha is very large
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square root of that of course k m and if the
oscillator is very heavy so is alpha
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so the parameters of the system the model
that you study are used to define the wave
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functions and the energies um the n n is a
normalization constant and in the case of
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harmonic oscillator x is taken all the way
from minus infinity to plus infinity therefore
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the wave functions when you talk about the
orthogonality these from minus infinity to
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plus infinity psi n star of x psi m of x d
x is delta n m which tells you that the wave
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functions also satisfy the same orthogonality
and normalization relation for the harmonic
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oscillator
what is important with the harmonic oscillator
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is something unique namely the energies are
h bar omega n plus half meaning that the energies
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are all equidistant so if you write half h
bar omega here for n equal to zero and if
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you write for n equal to one its three half
h bar omega and for n equal to two it is five
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h bar omega and so on three seven half h bar
omega and so on so energies are equidistant
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energy levels are equidistant for a particle
in a one dimensional box of course you remember
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is proportional to n square whatever is the
unit that you have therefore if you put n
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equal to one here and then if you put in equal
to two it is four n equal to two this is four
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times that fundamental unit this is one times
that fundamental unit the next one would be
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nine times one four so i believe the differences
the three three three so it should be somewhere
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here next one nine times and the third one
is sixteen times and so on so the energies
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are not ah separated from each other by the
same distance in the case of particle in a
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box in the case of harmonic oscillators they
are all ah separated by this same gap the
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difference between any pair of successive
energy levels is h bar omega which would be
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needed later when we study vibrational spectroscopy
of a harmonic oscillator the atomic molecule
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in the harmonic oscillator model you would
see that the ah spectrum for a simple diatomic
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molecule contains only one line and that one
line corresponds to the difference between
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the two nearby energy levels it is not possible
for a harmonic oscillator to be excited to
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levels for away from the starting levels there
are starting rules for that but the quantum
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mechanics of harmonic oscillator gives you
all these results later when we need them
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for a spectroscopy of a diatomic molecule
ok
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the next important model the real physical
model in addition to harmonic oscillator is
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the hydrogen atom the hydrogen atom with the
proton and then the electron separated by
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a distance or has they have a the proton electron
system has a potential energy given by minus
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z e square by four pi epsilon naught r where
epsilon the permittivity of the medium and
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the hydrogen atom is a two body system two
particle or a two body system and it can be
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separated into a center of mass coordinate
that is a system in which all the masses concentrated
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on the central mass and also a relative ah
particle system in which the relative particle
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has the mass mu and that is given by the proton
m p times the electron m e by m p plus m e
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and you know that m p is much much greater
than m e therefore mu is the reduced mass
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is approximately the same as me because this
will be m p and that cancels m p so its about
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m e and this is what we normally use in writing
down the one electron schrodinger equation
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for the hydrogen atom you do not worry about
the center of mass the nucleus is too heavy
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we don't ah consider the kinetic energy of
of the nucleus or kinetic energy of the whole
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whole atom system but only the one electron
system that we need to study and the hamiltonian
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for the one electron system is given by minus
h bar square by two m e the mass of the electron
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so the three dimensional system therefore
the kinetic energy contains three components
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the derivative with respect to the three coordinates
x e square plus dou square by dou y e square
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plus dou square by dou z e square and the
potential energy minus z e square by four
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pi epsilon naught r z is the charge on the
nucleus of course thats equal to plus one
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and so what you have is the z e square by
four pi epsilon naught r and these are the
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coordinates of the electron with respect to
the central mass where its approximately on
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the nucleus and so on therefore this the schrodinger
equation that we need to solve and obtain
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the solution h psi n which is now a function
of the three coordinates x e y e and z e is
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equal to e n times psi n of x e y e z e so
these solutions the psi n s and the e n s
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in the case of hydrogen atoms are slightly
more complex and what ha you see in the particle
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in the one dimensional box or the ha/harmonic
harmonic oscillator
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now its almost like ah thirty minutes ah since
we ah started this lecture so let me stop
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here and in the second part of this lecture
which is continuing to the quantum mechanics
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let me complete the exercise of writing down
the energy levels and the wave functions for
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the hydrogen atom and also the most important
aspect of the hydrogen atom that ah relevant
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to spectroscopy namely the concept of angular
momentum would be introduced in the second
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lecture the next lecture and this would be
a quick summary and a review of the basic
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quantum mechanics that you need for understanding
the rest of the course until then
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thank you very much