bei48482_FM

En n 1, 2, 3,... (3.18)

Each permitted energy is called an energy level,and the integer nthat specifies an

energy level Enis called its quantum number.

We can draw three general conclusions from Eq. (3.18). These conclusions apply

to anyparticle confined to a certain region of space (even if the region does not have

a well-defined boundary), for instance an atomic electron held captive by the attraction

of the positively charged nucleus.

1 A trapped particle cannot have an arbitrary energy, as a free particle can. The fact

of its confinement leads to restrictions on its wave function that allow the particle to

have only certain specific energies and no others. Exactly what these energies are de-

pends on the mass of the particle and on the details of how it is trapped.

2 A trapped particle cannot have zero energy. Since the de Broglie wavelength of the

particle is hm, a speed of 0 means an infinite wavelength. But there is no

way to reconcile an infinite wavelength with a trapped particle, so such a particle must

have at least some kinetic energy. The exclusion of E0 for a trapped particle, like

the limitation of Eto a set of discrete values, is a result with no counterpart in classi-

cal physics, where all non-negative energies, including zero, are allowed.

3 Because Planck’s constant is so small—only 6.63  10 ^34 J s—quantization of en-

ergy is conspicuous only when mand Lare also small. This is why we are not aware

of energy quantization in our own experience. Two examples will make this clear.

Example 3.4

An electron is in a box 0.10 nm across, which is the order of magnitude of atomic dimensions.

Find its permitted energies.

Solution

Here m9.1  10 ^31 kg and L0.10 nm 1.0  10 ^10 m, so that the permitted electron

energies are

En6.0  10 ^18 n^2 J

 38 n^2 eV

The minimum energy the electron can have is 38 eV, corresponding to n1. The sequence of

energy levels continues with E 2 152 eV, E 3 342 eV, E 4 608 eV, and so on (Fig. 3.11). If

such a box existed, the quantization of a trapped electron’s energy would be a prominent feature

of the system. (And indeed energy quantization is prominent in the case of an atomic electron.)

Example 3.5

A 10-g marble is in a box 10 cm across. Find its permitted energies.

Solution

With m10 g 1.0  10 ^2 kg and L10 cm1.0  10 ^1 m,

En

5.5  10 ^64 n^2 J

(n^2 )(6.63  10 ^34 J s)^2



(8)(1.0  10 ^2 kg)(1.0  10 ^1 m)^2

(n^2 )(6.63  10 ^34 J s)^2



(8)(9.1  10 ^31 kg)(1.0  10 ^10 m)^2

n^2 h^2



8 mL^2

Particle in a box

Wave Properties of Particles 107

Figure 3.11Energy levels of an

electron confined to a box

0.1 nm wide.

n = 2

700

600

500

400

300

200

100

0

n = 1

n = 3

n = 4

Energy, eV

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