Physical Chemistry Third Edition

15.3 The Particle in a Box and the Free Particle 669

For example, when a photon is absorbed or emitted by a molecule, the initial and

final molecule states correspond to energy eigenfunctions.

2. The wave function is some function other than a product of an energy eigenfunction
   and a time-dependent factor. The wave function is not proportional to a factor that
   obeys the time-independent Schrödinger equation, but it must still obey the time-
   dependent Schrödinger equation.

EXAMPLE15.3

Show that the following linear combination obeys the time-dependent Schrödinger equation:

Ψ(x,t)

∑∞

n 1

Anψn(x)e−iEnt/h ̄ (15.3-20)

whereA 1 ,A 2 ,...are a set of constants and whereψ 1 ,ψ 2 ,...represent energy eigenfunctions

of the system.

Solution

ĤΨĤ

∑∞

n 1

Anψne−iEnt/h ̄

∑∞

n 1

AnEnψne−iEnt/ ̄h

ih

(

∂Ψ

∂t

)

ih ̄

(

∂

∂t

)∑∞

n 1

Anψne−iEnt/h ̄

∑∞

n 1

AnEnψne−iEnt/ ̄h

It is apparent thatΨdoes not obey the time-independent Schrödinger equation, sinceĤΨ

does not equal a single constant timesΨ.

Equation (15.3-20) expresses theprinciple of superposition, which means that an

arbitrary wave function can be represented by a linear combination of energy eigen-

functions.

The Particle in a Three-Dimensional Box

The principal application of this model system is to represent the translational motion

of a gas molecule, and we will discuss this application in Chapters 22 and 26. We

assume our three-dimensional box is rectangular and we place the box with its lower

left rear corner at the origin of coordinates and its walls perpendicular to the coordinate

axes, as depicted in Figure 15.5. We denote the length of the box in thexdirection by

a, the length in theydirection byb, and the length in thezdirection byc.

Particle

z 5 c

z

x

y y 5 b

x 5 a

Figure 15.5 A Particle in a Three-

Dimensional Box.

If the particle is completely confined in the box the potential energy is equal to

a constant inside the box and is infinite outside the box. We set the constant equal to

zero. The resulting time-independent Schrödinger equation can be solved by separation

of variables, as described in Appendix F. The energy eigenfunction (coordinate wave

function) is a product of three factors, each one of which is the same as a wave function

for a particle in a one-dimensional box of lengtha,b,orc:

ψnxnynz(x,y,z)Csin

(n

xπx

a

)

sin

(nyπy

b

)

sin

(n

zπz

c

)

(15.3-21)