Physical Chemistry Third Edition

15.3 The Particle in a Box and the Free Particle 667

The energy of a de Broglie wave is inversely proportional to the square of its

wavelength. The wavelength of the wave function is the value ofxsuch that the argu-

ment of the sine function in Eq. (15.3-10) equals 2π.

nπλ/a 2 π

or

λ 2 a/n (15.3-15)

which is analogous to Eq. (14.2-20). When the relationship between the wavelength

and the length of the box is used, the relation in Eq. (15.3-14) becomes the same as the

energy expression in Eq. (15.3-11).

Exercise 15.4

a.Show that the value of the wavelength corresponding toψnis equal to 2a/n.

b.Show that the same formula for the energy as in Eq. (15.3-11) is obtained by substituting the

result of part a into Eq. (15.3-14).

As the value ofnincreases, the energy increases, the wavelength decreases and the

number of nodes increases.It is an important general fact that a wave function with

more nodes corresponds to a higher energy.

If the potential energy inside the box is assigned a nonzero constant valueV 0 instead

of zero, the energy eigenfunction is unchanged and the energy eigenvalue is increased

byV 0.

Exercise 15.5

Carry out the solution of the time-independent Schrödinger equation for the particle in a one-

dimensional box with constant potentialV 0 in the box. Show that the energy eigenvalue is

En

h^2 n^2

8 ma^2

+V 0 (15.3-16)

but that the wave function is unchanged.

The result of the previous exercise is generally true. Adding a constant to the potential

energy adds the same constant to the energy eigenvalues but leaves the wave function

unchanged.

If a particle in a box is electrically charged, it can absorb or emit photons. Because

of the conservation of energy, the energy of a photon that is emitted or absorbed is

equal to the difference in energy of the initial and final states of the particle.

Exercise 15.6

a.Calculate the wavelength and frequency of the photon emitted if an electron in a one-

dimensional box of length 1.00 nm (1. 00 × 10 −^9 m) makes a transition fromn3ton 2

and the energy difference is entirely converted into the energy of the photon.

b.Obtain formulas for the wavelength and the frequency of the photon emitted if an electron

in a one-dimensional box of length 1. 000 × 10 −^9 m makes a transition fromn+1tonand

the energy difference is entirely converted into the energy of the photon.