Physical Chemistry Third Edition

15.3 The Particle in a Box and the Free Particle 673

whereκ^2 is the square of the magnitude of the wave vector. As in the one-dimensional

case the energy is not quantized and there is no zero-point energy.

Just as Eq. (15.3-38) represents a linear combination of waves moving in opposite

directions, a wave eigenfunction for a three-dimensional free particle can consist of a

superposition (linear combination) of waves moving in various directions with various

energies:

ψ(x,y,z)

∑

κx

∑

κy

∑

κz

Dκx,κy,κzeiκxxeiκyyeiκzz (15.3-42)

where Dκx,κy,κzrepresents a constant coefficient. If the particle has a definite energy but

not a definite direction, the sums over different values ofκx,κy, andκzare constrained

so that all wave vectors in the sum have the same magnitude.

The solutions to the Schrödinger equation for the particle in a box and the free

particle exhibit some of the most important features of quantum mechanics. We have

seen one case of quantized energy (the particle in a box) and one case of energy

that is not quantized (the free particle). We have seen that the Schrödinger equation

and its boundary conditions together dictate the nature of the wave functions and

energy eigenvalues that can occur. We have also seen that the principle of superpo-

sition applies: A valid wave function can be a linear combination of simpler wave

functions.

PROBLEMS

Section 15.3: The Particle in a Box and the Free Particle

15.8 Derive a formula for the kinetic energy of a particle with

de Broglie wavelength equal to 2a/nand show that this is

the same as the energy of a particle in a one-dimensional

box of lengthawith quantum numbern.

15.9 The particle in a one-dimensional box is sometimes used

as a model for the electrons in a conjugatedπ-bond

system (alternating double and single bonds).

a.Find the first three energy levels for aπelectron in

1,3-butadiene. Assume a carbon–carbon bond length

of 1. 39 × 10 −^10 m and assume that the box consists

of the three carbon–carbon bonds plus an additional

length of 1. 39 × 10 −^10 m at each end.

b.The molecule has fourπelectrons. Assume that two

are in the state corresponding ton1, and that two

are in the state corresponding ton2. Find the

frequency and wavelength of the light absorbed if an

electron makes a transition fromn2ton3.

15.10The particle in a one-dimensional box is sometimes used
as a model for the electrons in a conjugatedπ-bond
system (alternating double and single bonds).
a.Find the first three energy levels for aπelectron in
1,3,5-hexatriene. Assume a carbon–carbon bond

length of 1. 39 × 10 −^10 m and assume that the box

consists of the five carbon–carbon bonds plus an

additional length of 1. 39 × 10 −^10 m at each end.

b.The molecule has sixπelectrons. Assume that two are

in the state corresponding ton1, two are in the

state corresponding ton2, and that two are in the

state corresponding ton3. Find the frequency and

wavelength of the light absorbed if an electron makes

a transition fromn3ton4.

15.11 a.Sketch a graph of the product ofψ 1 andψ 2 , the first

two energy eigenfunctions of a particle in a

one-dimensional box. Argue from the graph that the

two functions areorthogonal, which means for real

functions that

∫∞

−∞

ψ 1 (x)ψ 2 (x)dx 0

b.Work out the integral and show thatψ 1 andψ 2 are

orthogonal. Remember thatψ0 outside of the box,

so that the integral extends only fromx0toxa.

15.12 a.Sketch a graph of the product ofψ 2 andψ 3 , two

energy eigenfunctions of a particle in a

one-dimensional box. Argue from the graph that the

two functions are orthogonal, which means for real

functions that