Physical Chemistry , 1st ed.

• Two-dimensional rotational motion, which describes motion in a circu-
  lar path


• Three-dimensional rotational motion, which describes motion on a
  spherical surface
  We will conclude this chapter with a discussion of the hydrogen atom. Recall
  that Bohr’s theory described the hydrogen atom, and correctly predicted its spec-
  trum. However, Bohr’s theory was based on some assumptions that, when applied,
  provided the right answer. Quantum mechanics has its postulates, and we will see
  that it, too, predicts the same spectrum for the hydrogen atom. In order to be a
  superior theory, quantum mechanics must not only do the same things as earlier
  theories but do more. In the next chapter, we will see how quantum mechanics is
  applied to systems larger than hydrogen (and most systems of interest are consid-
  erably larger than hydrogen!), thereby making it a better description of matter.

11.2 The Classical Harmonic Oscillator

The classical harmonic oscillatoris a repetitive motion that follows Hooke’s law.

For some mass m, Hooke’s law states that for a one-dimensional displacement

xfrom some equilibrium position, the force Facting against the displacement

(that is, the force that is acting to return the mass to the equilibrium point) is

proportional to the displacement:

Fkx (11.1)

where kis called the force constant.Note that both Fand xare vectors, and the

negative sign in the equation indicates that the force and displacement vectors

are opposite in direction. Since force has typical units of newtons or dynes and

displacement has units of distance, the force constant can have units like N/m

or, in other units that sometimes yield more manageable numbers, millidynes

per angstrom (mdyn/Å).

The potential energy, denoted V, of a Hooke’s-law harmonic oscillator is re-

lated to the force by a simple integral. The relationship and final result are

VFdx^12 kx^2

To simplify our presentation, we ignore the vector characteristic of the posi-

tion and focus on its magnitude,x. Since xis squared in the expression for V,

negative values ofxdon’t need to be treated in any special fashion. The re-

sulting working equation for the potential energy of a harmonic oscillator is

more simply written as

V^12 kx^2 (11.2)

The potential energy does not depend on the mass of the oscillator. A plot of

this potential energy is shown in Figure 11.1.*

Classically, the behavior of the ideal harmonic oscillator is well known. The

position of the oscillator versus time,x(t), is

x(t) x 0 sin 

m

k

t

where x 0 is the maximum amplitude of the oscillation,kand mare the force

constant and mass,tis time, and is somephase factor(which indicates the

316 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom

*An anharmonic oscillatoris one that does not follow Hooke’s law and, ultimately, does

not have a potential energy as defined in equation 11.2. Anharmonic oscillators are dis-

cussed in a later chapter.

x

V ^12 kx^2

Figure 11.1 A plot of the potential energy
diagram V(x) ^12 kx^2 for an ideal harmonic
oscillator.