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

2. 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)

3. ∴ For acceptable ψ, boundary conditions
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!

4. Fig 9.2 Allowed energy levels for a particle
in a one-dimensional box
Normalized wavefunction:
n = 1, 2, 3, …
n ≠ 0 so:
is called the zero-point energy

5. Fig 9.3 First five normalized wavefunctions of PIB

6. Fig 9.4 First two normalized wavefunctions of PIB
with probability distributions

7. Real world PIB: a delocalized π electron
in a conjugated system
1 β-Carotene

8. Correspondence Principle:
Classical mechanics emerges from quantum
mechanics as high quantum numbers are reached
i.e., particle may be found anywhere as n → ∞

9. Fig 9.5 Probability of two wavefunctions
ψ1 and ψ3 are
orthogonal or
orthonormal
In Bra-ket notation:
〈1|3〉 = 0 when n ≠ n'

10. Fig 9.6 Two dimensional square well

11. 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

12. Fig 9.8 Contours for particle in 2-D square surface
Here, L1 = L2 = L
are said to be degenerate

13. Fig 9.9 Tunnelling of a particle through wall when V < ∞
Leakage by penetration through a classically forbidden region

14. Fig 9.13 Wavefunction of a heavy particle decays more
rapidly than that of a light particle
Light particles have
higher probability of
tunnelling

15. 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)

16. Fig 9.16 Tip of a Scanning Tunnelling Microscope (STM)
Pt-Rh or W
Ultrahigh
vacuum

17. 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.

18. Title : Stadium Corral
Media : Iron on Copper (111)