Atkins’ Physical ChemistryEighth Edition

Chapter 9Quantum Theory:

Techniques and Applications

Copyright © 2006 by Peter Atkins and Julio de Paula

Peter Atkins • Julio de Paula

Chap 9Chap 9Quantum Theory: Techniques and Quantum Theory: Techniques and

ApplicationsApplications

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 conditionsmust 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

L

xn1/2

L2

n sin(x)Ψ π

2

22

nmL8

hnE

Normalized wavefunction:

n = 1, 2, 3, …

2

2

1mL8

hE n ≠ 0 so:

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 electronin 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

or

orthonormal

0dΨΨ 3*1 τ

In Bra-ket notation:

⟨1|3 = 0 when n ≠ n'⟩

'nn

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

Lxn

Lxn

L2

n,n21

21sinsin)y,x(Ψ ππ

2

222

21nn

mL8

h)nn(E

21

Here, L1 = L2 = L

2,11,2 Ψ and Ψ

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 havehigher probability of

tunnelling

TunnelingTunneling

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 WPt-Rh or W

Ultrahigh

vacuum

Title : The Making of the Circular Corral Title : The Making of the Circular Corral Media : Iron on Copper (111) Media : Iron on Copper (111)

We can predict what goes on in the corral by solving the classic eigenvalueproblem in quantum mechanics -- a particle in a hard-wall circular box.

Title : Stadium Corral

Media : Iron on Copper (111)