38.4.1 Quantization of Energy

The fact that matter has wave-like properties is not merely a laboratory curiosity; the implications are profound. Foremost among them is that the energy of matter, like that of light, is quantized.

We’ll illustrate quantization with a simple system that physicists call “a particle in a box.” FIGURE 38.14a shows a particle of mass $m$ moving in one dimension as it bounces back and forth with speed $v$ between the ends of a box of length $L$ . The width of the box is irrelevant, so we’ll call this a one-dimensional box. We’ll assume that the collisions at the ends are perfectly elastic, so the particle’s energy—entirely kinetic—never changes. According to classical physics, there are no restrictions on the particle’s speed or energy.

The figure illustrates the movement of a particle in two parts inside a rectangular box of length L.

FIGURE 38.14 Full Alternative Text

But if matter has wave-like properties, perhaps we should consider the particle in a box to be a wave reflecting back and forth between the ends of the box, as shown in FIGURE 38.14b. These are the conditions that create standing waves. You learned in 《 SECTION 17.3 that a standing wave of length L must have one of the wavelengths given by

λ_{n} = \frac{2 L}{n} n = 1, 2, 3, \dots

(38.16)

If the confined particle has wave-like properties, it should satisfy both Equation 38.16 and the de Broglie relationship $λ = h / m v$ . That is, a particle in a box should obey the relationship

λ_{n} = \frac{h}{m v} = \frac{2 L}{n}

Thus the particle’s speed must be

v_{n} = n (\frac{h}{2 L m}) n = 1, 2, 3, \dots

(38.17)

In other words, the particle cannot bounce back and forth with just any speed. Rather, it can have only those specific speeds $v_{n}$ , given by Equation 38.17, for which the de Broglie wavelength creates a standing wave in the box.

Thus the particle’s energy, which is purely kinetic energy, is

E_{n} = \frac{1}{2} m v_{n}^{2} = n^{2} \frac{h^{2}}{8 m L^{2}} n = 1, 2, 3, \dots

(38.18)

De Broglie’s hypothesis about the wave-like properties of matter leads us to the remarkable conclusion that a particle confined in a box can have only certain energies. We say that its energy is quantized. The energy of the particle in the box can be $1 (h^{2} / 8 m L^{2})$ , or $4 (h^{2} / 8 m L^{2})$ , or $9 (h^{2} / 8 m L^{2})$ , but it cannot have an energy between these values.

The possible values of the particle’s energy are called energy levels, and the integer $n$ that characterizes the energy levels is called the quantum number. The quantum number can be found by counting the antinodes, just as you learned to do for standing waves on a string. The standing wave shown in Figure 38.14 is $n = 3$ , thus its energy is $E_{3}$ .

We can rewrite Equation 38.18 in the useful form

E_{n} = n^{2} E_{1}

(38.19)

where

E_{1} = \frac{h^{2}}{8 m L^{2}}

(38.20)

is the fundamental quantum of energy for a particle in a one-dimensional box. It is analogous to the fundamental frequency $f_{1}$ of a standing wave on a string.

EXAMPLE 38.7 The energy levels of a virus

A 30-nm-diameter virus is about the smallest imaginable macroscopic particle. What is the fundamental quantum of energy for this virus if confined in a one-dimensional cell of length $1.0 μ m ?$ The density of a virus is very close to that of water.

MODEL Model the virus as a particle in a box.

SOLVE The mass of a virus is $m = ρ V$ , where the volume is $\frac{4}{3} π r^{3}$ . A quick calculation shows that a 30-nm-diameter virus has mass $m = 1.4 \times 10^{- 20} kg$ . The confinement length is $L = 1.0 \times 10^{- 6} m$ . From Equation 38.20, the fundamental quantum of energy is

\begin{matrix} E_{1} & = & \frac{h^{2}}{8 m L^{2}} = \frac{(6.63 \times 10^{- 34} J s)^{2}}{8 (1.4 \times 10^{- 20} kg) (1.0 \times 10^{- 6} {m)}^{2}} \\ = & 3.9 \times 10^{- 36} J = 2.5 \times 10^{- 17} eV \end{matrix}

REVIEW This is such an incredibly small amount of energy that there is no hope of distinguishing between energies of $E_{1}$ or $4 E_{1}$ or $9 E_{1}$ . For any macroscopic particle, even one this tiny, the allowed energies will seem to be perfectly continuous. We will not observe the quantization.

EXAMPLE 38.8 The energy levels of an electron

As a very simple model of a hydrogen atom, consider an electron confined in a one-dimensional box of length 0.10 nm, about the size of an atom. What are the first three allowed energy levels?

MODEL Model the electron as a particle in a box.

SOLVE We can use Equation 38.20, with $m_{elec} = 9.11 \times 10^{- 31} kg$ and $L = 1.0 \times 10^{- 10} m,$ to find that the fundamental quantum of energy is $E_{1} = 6.0 \times 10^{- 18} J = 38 eV$ . Thus the first three allowed energies of an electron in a 0.10 nm box are

\begin{array}{l} E_{1} & = & 38 eV \\ E_{2} & = & 4 E_{1} = 152 eV \\ E_{3} & = & 9 E_{1} = 342 eV \end{array}

REVIEW You’ll soon see that the results are way off. This model of a hydrogen atom is too simple to capture essential details. Even so, this model correctly hints that atomic energy levels are in the eV range.

It is the confinement of the particle in a box that leads to standing matter waves and thus energy quantization. Our goal is to extend this idea to atoms. An atom is certainly more complicated than a one-dimensional box, but an electron is “confined” within an atom. Thus an electron in an atom must be some kind of three-dimensional standing wave and, like the particle in a box, must have quantized energies. De Broglie’s idea is steering us toward a new theory of matter.