Concise Physical Chemistry

c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

258 WAVE MECHANICS OF SIMPLE SYSTEMS

FIGURE 16.6 Degenerate energy levels in a one-dimensional box. Degeneracy increases

withnin the sequence 1, 4, 9,...

by Born’s principle. Normalization yieldsA=

√

2

l

(see Problems) for the particle

in a one-dimensional box. Normalization constants, sometimes rather messy ones,

appear as premultipliers of wave functions.

Further elaboration of the method [see many textbooks, including Levine (2000)

and Barrow (1996)] yields solutions for the harmonic oscillator (one dimensional

vibrator) and the rigid rotor. These systems have the same kind of energy level

spectrum (Section 16.5) as the particle in a box except that the spacings are different.

The spacing is in equal steps for vibration of the harmonic oscillator but not for

rotation.

16.7 THE HYDROGEN ATOM

The kinetic energy operator of an electron moving in the vicinity of a proton (H+

nucleus) is the same as a particle in a box:

Tˆ= h ̄

2 me

∇^2

Many texts replace the mass of the electronmebyμdesignating the reduced mass

of the electron and proton rotating about their center of gravity. The effect of this

correction is very small for the proton–electron pair, and we shall ignore it until we

reach comparable problems of molecular rotation where more nearly equal masses

are involved.

The potential energy is not zero in this problem; rather, it is

V=

e^2

r

according to the electrostatic attraction between the nucleus and the electron. This

leads to

∇^2 (r,θ,φ)=

2 me

̄h^2

(E(r,θ,φ)−V(r,θ,φ))(r,θ,φ)= 0

where we have expressed(r,θ,φ)in spherical polar coordinatesr,θ,andφ.The

problem is simpler this way because of its spherical symmetry. Routine algebraic

methods exist for conversion of problems in Cartesian coordinates to spherical polar

coordinates (Barrante 1998).