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An Introduction into Molecular Physics
for the Atmospheric Sciences
Björn-Martin Sinnhuber
University of Bremen
Summer Semester 2004
www.iup.physik.uni-bremen.de/~bms
Sinnhuber, Molecular Physics, 2004
X Contents of the Lecture
• Introduction
– Quantum Mechanical Basics
– Electromagnetic Radiation
– Absorption and Emission
• Molecular Rotations
Microwave Spectroscopy
• Molecular Vibrations
Infrared Spectroscopy
• Electronic Transitions in Atoms and Molecules
UV-visible Spectroscopy
Sinnhuber, Molecular Physics, 2004
X Literature
• C.N. Banwell and E.M McCash,
Fundamentals of Molecular Spectroscopy
• P.W. Atkins and R.S. Friedman,
Molecular Quantum Mechanics
• P.W. Atkins, Physical Chemistry
• G. Herzberg,
Molecular Spectra and Molecular Structure
[classical textbook]
Sinnhuber, Molecular Physics, 2004
X Diatomic Molecule
H
Cl
Example: HCl
Sinnhuber, Molecular Physics, 2004
X The Water Molecule
O
H
0.09578 nm
104.48°
Sinnhuber, Molecular Physics, 2004
H
X Introduction: Planck‘s Quantum
Light is both particle- and wave-like
Light quantum has Energy:
E = hν
Sinnhuber, Molecular Physics, 2004
X Introduction: Atomic Spectra
Spectrum of atomic hydrogen
Sinnhuber, Molecular Physics, 2004
X Introduction: Atomic Spectra
Balmer series of atomic hydrogen
 1 1 
~
ν = RH  2 − 2 
2 n 
wavenumber
~
ν =ν / c
n = 3, 4,K
(wavenumber)
−1
~
ν = λ λ = c /ν
Rydberg constant:
5
RH = 1.097 ×10 cm
−1
Sinnhuber, Molecular Physics, 2004
X Introduction: Atomic Spectra
~
ν = Eu − El
µe
En = − 2 2 2
8h ε 0 n
4
n = 1, 2, K
1 µ = 1 me + 1 m p
Sinnhuber, Molecular Physics, 2004
X Introduction: Molecular Spectrum of CO
Sinnhuber, Molecular Physics, 2004
X Introduction: Basic Quantum Mechanics
Schrödinger Equation:
Operators:
x→x
Hψ = Eψ
h ∂
p→
i ∂x
Sinnhuber, Molecular Physics, 2004
X Introduction: Basic Quantum Mechanics
H = T +V
2
T = p / 2m
2
2
h d
H =−
+ V ( x)
2
2m dx
Sinnhuber, Molecular Physics, 2004
X Introduction: Free Particle
solve
Hψ = Eψ
2
with
Solution:
2
h d
H =−
2
2m dx
ψ ( x) = C cos kx + D sin kx
with kh = (2mE )
12
Sinnhuber, Molecular Physics, 2004
X Introduction: Momentum of Free Particle
2
E = p / 2m
2
2
E = k h / 2m
p = kh
Sinnhuber, Molecular Physics, 2004
X Introduction: Particle in a Box
V ( x) = 0 for 0 < x < L
V ( x) = ∞ else
Sinnhuber, Molecular Physics, 2004
X Introduction: Particle in a Box
Solution inside Box as for free particle:
ψ ( x) = C cos kx + D sin kx kh = (2mE )
12
Outside Box:
ψ =0
Sinnhuber, Molecular Physics, 2004
X Introduction: Particle in a Box
ψ ( x) = C cos kx + D sin kx kh = (2mE )
12
Boundary conditions:
ψ ( 0) = 0 ⇒ C = 0
ψ ( L) = 0 ⇒ D sin kL = 0
nπ
⇒k =
L
with n = 1, 2, K
Sinnhuber, Molecular Physics, 2004
X Introduction: Particle in a Box
2
2
E = k h / 2m
with
we get:
nhπ
nh
En =
=
2
2
2mL
8mL
2
2
2
2
2
with n = 1, 2,K
Energy is quantized! (Only discrete levels allowed.)
Sinnhuber, Molecular Physics, 2004
X Introduction: Particle in a Box
nhπ
En =
2
2mL
2
Sinnhuber, Molecular Physics, 2004
2
2
n = 1, 2,K
X Introduction: Particle in a Box
ψ ( x) = sin kx
nπ
k=
L
Sinnhuber, Molecular Physics, 2004
n = 1, 2,K
X Introduction: Particle in a Box
Sinnhuber, Molecular Physics, 2004
X Introduction: Harmonic Oscillator
2
2
h d
1 2
H =−
+ kx
2
2m dx
2
Sinnhuber, Molecular Physics, 2004
X Introduction: Harmonic Oscillator
Energy levels:
Ev = (v + )hω v = 0,1, 2,K
1
2
12
k
ω = 
m
Sinnhuber, Molecular Physics, 2004
X Introduction: Harmonic Oscillator
ψ v = N v H v ( y )e
−1 2 y 2
,
14
h 

y = , α = 
α
 mk 
x
2
12
1


with N v =  v 1 2 
 2 v!π α 
Normalization constant
H v ( y ) Hermite polynomals
Sinnhuber, Molecular Physics, 2004
X Introduction: Hermite Polynomals
H 0 ( y) = 1
H1 ( y ) = 2 y
2
H 2 ( y) = 4 y − 2
3
H 3 ( y ) = 8 y − 12 y
M
H v +1 ( y ) = 2 yH v − 2vH v −1
Sinnhuber, Molecular Physics, 2004
X Introduction: Harmonic Oscillator
Sinnhuber, Molecular Physics, 2004
X Introduction: Harmonic Oscillator
Sinnhuber, Molecular Physics, 2004