Download Chapter 9 - Lecture 1

Peter Atkins • Julio de Paula
Atkins’ Physical Chemistry
Eighth Edition
Chapter 9
Quantum Theory:
Techniques and Applications
Copyright © 2006 by Peter Atkins and Julio de Paula
Chap 9
Quantum Theory: Techniques and
Applications
Objectives:
Solve the Schrodinger equation for:
•
•
•
Translational motion (Particle in a box)
Vibrational motion (Harmonic and anharmonic
oscillator)
Rotational motion (Particle on a ring & on a sphere)
Fig 9.1 Particle in a one-dimensional box
• Particle is not free
• ∴ For acceptable ψ,
boundary conditions
must be set:
• ψ must vanish
at x = 0 and x = L
• Implies quantization!
Fig 9.2 Allowed energy levels for a particle
in a one-dimensional box
Normalized wavefunction:
Ψn (x) 
En 
n 2h 2
2
8mL
n ≠ 0 so:
 

nπx
2 1/2
sin
L
L
n = 1, 2, 3, …
E1 
h2
8mL2
is called the zero-point energy
Fig 9.3 First five normalized wavefunctions of PIB
0
L
Fig 9.4 First two normalized wavefunctions of PIB
with probability distributions
Real world PIB: a delocalized π electron
in a conjugated system
1 β-Carotene
Correspondence Principle:
• Classical mechanics emerges from quantum
mechanics as high quantum numbers are reached
• i.e., particle may be found anywhere as n → ∞
Fig 9.5 Probability of two wavefunctions
ψ1 and ψ3 are
orthogonal

Ψ1* Ψ3dτ  0
or
In Bra-ket notation:
orthonormal
n n'
〈1|3〉 = 0 when n ≠ n'
Fig 9.6 Two dimensional square well
Fig 9.7 Contours for particle in 2-D rectangular well
n1 = n2 =1
n1 = 1, n2 =2
n1 = 2, n2 =1
n1 = 2, n2 =2
Fig 9.8 Contours for particle in 2-D square surface
Here, L1 = L2 = L
Ψn1,n2 (x, y) 
2
L
n1πx
sin L
En1n2  (n12  n22 )
n2 πx
sin L
h2
8mL2
Ψ1,2 and Ψ 2,1
are said to be degenerate
Fig 9.9 Tunnelling of a particle through wall when V < ∞
Leakage by penetration through a classically forbidden region
Fig 9.13 Wavefunction of a heavy particle decays more
rapidly than that of a light particle
• Light particles have
higher probability of
tunnelling
Tunneling
Chemical effects of tunneling:
• Isotope-dependence of reactions rates
• Transfer of a proton in an acid-base reaction
• Mechanism of enzyme-catalyzed reactions
• Electron transfer in redox reactions
• Scanning tunneling microscopy (STM)
Fig 9.16 Tip of a Scanning Tunnelling Microscope (STM)
Pt-Rh or W
Ultrahigh
vacuum
Title : The Making of the Circular Corral
Media : Iron on Copper (111)
We can predict what goes on in the corral by solving the classic eigenvalue
problem in quantum mechanics -- a particle in a hard-wall circular box.
Title : Stadium Corral
Media : Iron on Copper (111)