Physical Chemistry Third Edition

15.3 The Particle in a Box and the Free Particle 663

which are generally shared by waves: (1)the wave function is single-valued, (2)the

wave function is continuous, and (3)the wave function is finite.These properties will

lead to boundary conditions that have important consequences, which we will illustrate

in the next section for the simplest case, the particle in a box.

PROBLEMS

Section 15.2: The Schrödinger Equation

15.5 The Schrödinger equation does not determine the
magnitude of the wave function.
a.Show that ifψis replaced byCψwhereCis any
constant, the time-independent Schrödinger equation is
still satisfied.
b.Show that ifΨis replaced byCΨwhereCis any
constant, the time-dependent Schrödinger equation is
still satisfied.

15.6 In classical mechanics, the addition of a constant to a
potential energy has no physical effect. The same is true of
quantum mechanics.
a.Show that if the potential energyVis replaced by
V+V 0 , whereV 0 is a constant, the same wave

function satisfies the time-independent Schrödinger

equation but the energy eigenvalue is nowE+V 0 ,

whereEis the original energy eigenvalue.

b.What is the effect on the solution of the time-dependent

Schrödinger equation of adding a constant to the

potential energy?

15.7 If the potential energyVin then-particle Hamiltonian of

Eq. (15.2-27) is equal to zero, show that the product wave

function

ψsystemψ 1 (x 1 ,y 1 ,z 1 )ψ 2 (x 2 ,y 2 ,z 2 )ψ 3 (x 3 ,y 3 ,z 3 )···

ψn(xn,yn,zn)

satisfies the time-independent Schrödinger equation of the

system.

15.3 The Particle in a Box and the Free Particle

In this section we solve the time-independent Schrödinger equation for the two simplest

model systems: the particle in a box and the free particle. This analysis will show how

the wave function and the values of the energy are determined by the Schrödinger

equation and the three conditions obeyed by the wave function.

The Particle in a One-Dimensional Box

The particle in a one-dimensional box is a model system that consists of a single

particle that can move parallel to thexaxis. The particle moves without friction, but is

confined to a finite segment of thexaxis, fromx0toxa. This interval is called a

one-dimensional box, but could also be called apotential well. This model system could

represent a particle sliding in a frictionless tube with closed ends or a bead sliding

on a frictionless wire between barriers. One chemical system that is approximately

represented by this model is a pi electron moving in a conjugated system of single and

double bonds. We will discuss this application in a later chapter.

We construct the Schrödinger equation for a particular system by inserting the appro-

priate potential energy function for the system into the equation. Since our particle

experiences no force inside the box, its potential energy is constant inside the box,

and we choose the value zero for this constant. In order to represent complete confine-

ment within the box we specify that the potential energy outside the box approaches a

positive infinite value.