11

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T

HE PREVIOUS CHAPTER INTRODUCED the basic postulates of quan-

tum mechanics, illustrated key points, and applied the postulates to a

simple ideal system, the particle-in-a-box. Although it is an ideally defined

model system, the particle-in-a-box ideas are applicable to compounds having

carbon-carbon double bonds like ethylene, and also to systems that have mul-

tiple conjugated double bonds, like butadiene, 1,3,5-hexatriene, and some dye

molecules. The electrons in these systems do not act as perfect particles-in-a-

box, but the model does a credible job of describing the energies in these mol-

ecules, certainly better than classical mechanics could describe them. Consider

what quantum mechanics has provided so far: a simple, approximate, yet ap-

plicable description of electrons in some bonds. This is more than anything

classical mechanics provided.

Other model systems can be solved mathematically and exactly using the

time-independent Schrödinger equation. In such systems, the Schrödinger

equation is solved analytically;that is, by deriving a specific expression that

yields exact answers (like the expressions for the wavefunctions and energies

of the particle-in-a-box). Only for a few systems can the Schrödinger equation

be solved analytically, and we will consider most of those. For all other systems,

the Schrödinger equation must be solved numerically, by inserting numbers or

expressions and seeing what answers come out. Quantum mechanics provides

the tools for doing that, so don’t let the rarity of analytic solutions shake the

knowledge that quantum mechanics is the best theory for understanding the

behavior of electrons and, therefore, atoms and molecules and chemistry in

general.

11.1 Synopsis

We will consider the following systems, the behavior of which have exact, an-

alytic solutions for in the Schrödinger equation:

• The harmonic oscillator, wherein a mass moves back and forth in a
  Hooke’s-law type of motion and whose potential energy is proportional
  to the square of the displacement

11.1 Synopsis

11.2 The Classical Harmonic

Oscillator

11.3 The Quantum-Mechanical

Harmonic Oscillator

11.4 The Harmonic Oscillator

Wavefunctions

11.5 The Reduced Mass

11.6 Two-Dimensional Rotations

11.7 Three-Dimensional Rotations

11.8 Other Observables in

Rotating Systems

11.9 The Hydrogen Atom: A

Central Force Problem

11.10 The Hydrogen Atom: The
Quantum-Mechanical
Solution

11.11 The Hydrogen Atom
Wavefunctions

11.12 Summary

Quantum Mechanics:

Model Systems and the

Hydrogen Atom