Classical%20Model%20of%20Rigid%20Rotor

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Classical Model of Rigid Rotor
A particle rotating around a fixed point, as
shown below, has angular momentum and rotational
kinetic energy (rigid rotor)
The classical kinetic energy is given by If
the particle is rotating about a fixed point at
radius r with a frequency ? (s-1 or Hz), the
velocity of the particle is given by
where ? is the angular frequency (rad s-1 or rad
Hz). The rotational kinetic energy can be now
expressed as
Also
where
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Consider a classical rigid rotor corresponding to
a diatomic molecule. Here we consider only
rotation restricted to a 2-D plane where the two
masses (i.e., the nuclei) rotate about their
center of mass.
The rotational kinetic energy for diatomic
molecule in terms of angular momentum
Note that there is no potential energy involved
in free rotation.
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Momentum Summary
Classical
QM
Linear
Momentum
Energy
Rotational (Angular)
Momentum
Energy
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Angular Momentum
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Angular Momentum
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Angular Momentum
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Angular Momentum
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Two-Dimensional Rotational Motion
Polar Coordinates
y
r
f
x
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Two-Dimensional Rotational Motion
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Two-Dimensional Rigid Rotor
Assume r is rigid, ie. it is constant
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Two-Dimensional Rigid Rotor
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Solution of equation
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Energy and Momentum
As the system is rotating about the z-axis
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Two-Dimensional Rigid Rotor
m
18.0
12.5
E
8.0
4.5
2.0
0.5
Only 1 quantum number is require to determine the
state of the system.
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Spherical coordinates
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Spherical polar coordinate
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Hamiltonian in spherical polar coordinate
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Rigid Rotor in Quantum Mechanics
Transition from the above classical expression to
quantum mechanics can be carried out by replacing
the total angular momentum by the corresponding
operator
Wave functions must contain both ? and F
dependence
are called spherical harmonics
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Schrondinger equation
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Two equations
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Solution of second equation
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Solution of First equation
Associated Legendre Polynomial
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Associated Legendre Polynomial
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For l0, m0
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First spherical harmonics
Spherical Harmonic, Y0,0                        
                       
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l 1, m0
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l 1, m0
? cos2?
0 1
30 3/4
45 1/2
60 1/4
90 0
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l2, m0
? cos2? 3cos2?-1
0 1 2
30 3/4 (9/4-1)5/4
45 1/2 (3/2-1)1/2
60 1/4 (3/4-1)-1/4
90 0 -1
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l 1, m1
Complex Value??
If ?1 and ?2 are degenerateeigenfunctions, their
linear combinations are also an eigenfunction
with the same eigenvalue.
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l1, m1
Along x-axis
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Three-Dimensional Rigid Rotor States
l
m
3
2
6.0
1
0
-1
-2
-3
E
2
3.0
1
0
-1
-2
1
1.0
0
-1
0
0.5
Only 2 quantum numbers are required to determine
the state of the system.
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Rotational Spectroscopy
J Rotational quantum number
Rotational Constant
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Rotational Spectroscopy
Wavenumber (cm-1)
Rotational Constant
Line spacing
v
Dv
Frequency (v)
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Bond length

• To a good approximation, the microwave spectrum
  of H35Cl consists of a series of equally spaced
  lines, separated by 6.261011 Hz. Calculate the
  bond length of H35Cl.