BioPHYSICAL chemistry

and no sharp turns are allowed;

that is, the wavefunction must be

smooth and continuous.

6 The solution to Schrödinger’s

equation will always be a series

of relatedsolutions because the

equation is an eigenfunction. The

solutions to all eigenfunctions

have common properties. Each of

the solutions is orthogonal, mean-

ing that for each of the wave-

functions the solution is unique

and, when the wavefunctions for

the hydrogen atom are found, they

will each represent a different pos-

sible orbital for the electrons.

7 Quantum mechanics does not

replace classical mechanics but

complements it when consider-

ing small energies and particles.

According to the Correspond-

ence Principle, the results from

quantum mechanics always must

agree with classical mechanics.

Thus, the results from quantum

mechanics must agree with the

classical mechanics in the appro-

priate limits. For example, quantum

mechanics must still conserve

energy of the observed states.

General approach for solving Schrödinger’s equation

There is a general strategy that will be

used to solve Schrödinger’s equation

for the different applications. The

first step is to determine the depend-

ence of the potential energy upon the

distance. We start with the classical

expression of the potential and then

substitute the appropriate operators,

which for distance are simply the

190 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

Not allowed

Discontinuous

Not allowed

Derivative

Not continuous

Not allowed

Multiple values

Not allowed

Infinite

x

 (^) 
ψ Allowed
ψ
ψ
ψ
ψ
Figure 9.8The wavefunction ψ(x) must be a smooth,
continuous, and single-valued function. Solutions are
unacceptable if they have a discontinuity or more than one
value for a given value of x.