Particle on a Circular Path Classical
 Motion of a particle in 1-D described by linear momentum
⃑
⃑⃑
 Motion of particle on a circular path described by angular momentum
⃑⃑ ⃑ ⃑
 If ⃑ and are ⃑⃑ ⊥ so
Dynamic properties
Linear motion
Linear displacement (x)
Mass (m)
Speed (v)
Momentum (p)
Rotational motion
Angular displacement, (θ, radians)
Moment of inertia (I)
)
for particle on a ring (
Angular speed (, rad.s-1)
Angular momentum (L)
Relations
for particle on a ring (
Quantization of Rotation
 Classically, kinetic energy of particle constrained
to a ring:
 De Broglie relation, wave behavior of particle,
)
 Wave function must be single valued
 Require constructive interference
 Require integral number of wavelengths in full circle
 From standing wave in a ring solution
 In terms of angular momentum:
 Wave behavior of particle & cyclic boundary condition → quantization
(
)
 Quantized energy states
(
)
Quantum Solution for Particle on a Ring
 Particle free to move on the ring so, V ( x,y) = 0 on the ring
 Schrödinger Equation for ring system (in spherical coordinates)
(
)
 This Equation have the following Solutions
√
,
√
(
(
√
√

)
)
(
(|
| )
(|
| )
)
means states with given |ml | are 2-fold DEGENERATE (except for ml= 0)
 Energy Level Diagram & Standing Wave in a Ring
4h 2
E2 =
8m 2 r 2
h2
E1 =
8m 2 r 2
E0 = 0
C
S
 The particle on a ring model can be applied to electrons moving freely (π electrons)
in a molecule
 Consider the benzene molecule as an example of the free particle move on a ring, in
which the six  electrons, only, are free to move, while the  electrons are frozen in
bonds with atoms
Benzene
 Benzene has an absorption band at 200 nm for the 1st π→ π* transition. As a simple
approximation, consider benzene as being a particle on a ring of radius 1.4 A. and
consider the 6 π electrons to occupy the levels calculated using the particle on ring
model
No. of e’s = 6
r = 1.4 Å
E=ELUMO – EHOMO
(
E2
2
E1
1
E0
0
LUMO
HOMO
)
 Comment on the results
 Calculate
for naphthalene and anthracene considering the radius = no. of
rings * 1.4 A
Rigid Rotor
 Diatomic molecule approximated as “Rigid Rotor”
 = Two masses (atoms) rotating around the center of mass separated by a fixed
distance (= bond length, r)
 To describe rotation around center of mass →reduced mass
 Reduced mass
 Moment of inertia
 Schrödinger equation of rotating diatomic
 Solutions = spherical harmonics
(
)
 Rotational energy levels given by:
(
(
)
 Energy levels given by:
(
)
(
)
)
 Degeneracy =
 Selection Rules = Requirements for transitions to occur

 Require molecule has permanent electric dipole moment
 need fluctuating charge to interact with light
 Allowed transitions:
(
 Spectrum = series of lines separated by 2B...
 Units of B
 B has units of energy (J)
)
 It can has units of frequency (Hz)
 and it can be take unit of wave number cm-1
 We can determine the bond length from the line spacing in the spectrum and moment
of inertia (r= bond length of diatomic)
 Examples:
 The microwave spectrum of 1H35Cl consists of a series of equally spaced lines
separated by 6.26×1011Hz. Calculate the bond length of H-Cl.
 If the wave number of the rotational transition l =0  1 of 1H81Br is 16.93 cm-1.
a- Calculate the rotational constant B (Hz)
b- Calculate the bond length of HBr (Å)
c- Calculate the energy of l =5  6 transition (J)
d- If we deuterate HBr without affecting the bond length, what will happen to the
position of the absorption peak?