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. Fig 9.20 Potential energy of a harmonic oscillator
Restoring force:
F = -kx
where k ≡ force constant
Potential energy:
Schrodinger equation:

4. Fig 9.21 Energy levels of a harmonic oscillator
Solving
Gives:
where:
Zero point energy

5. Form of the harmonic oscillator wavefunctions:
Ψ(x) = N · (polynomial in x) · (Gaussian function)
Fig Graph of Gaussian function:

6. Form of the harmonic oscillator wavefunctions:
Ψ(x) = N · (polynomial in x) · (Gaussian function)
Precisely:
where: and

7. When the Schrodinger equation for the harmonic oscillator
is solved, the solutions contain this set of polynomials, named the Hermite polynomials.

8. Fig 9.23 Normalized Ψ and Ψ2 for the v = 0 state
of a harmonic oscillator

9. Fig 9.24 Normalized Ψ and Ψ2 for the v=1 state
of a harmonic oscillator

10. Interpretation of Harmonic Oscillator Wavefunctions
Gaussian function goes strongly to zero as
displacement increases. i.e., ψ → 0 at large x.
The exponent y2 ∝ x2 (mk)1/2 so ψ → 0 more rapidly
with high masses and stiff springs.
3) Wavefunctions spread more a v increases.

11. Fig 9.25 Normalized Ψ’s for the v=0 – v=4 states
of a harmonic oscillator

12. Oscillator may be in classically forbidden regions at low v
Fig Probability distributions for the v = 0 – v = 4 states
of a harmonic oscillator
classical
turning points
at high v
Oscillator may be in classically
forbidden regions at low v
i.e. tunnelling!

13. Correspondence Principle:
Classical mechanics emerges from quantum
mechanics as high quantum numbers are reached
i.e., particle may be found anywhere as v → ∞
As v → ∞ the oscillator becomes macroscopic (e.g.
a pendulum)