29.5

The De Broglie Wavelength and the Wave Nature of Matter

CONCEPTS AT A GLANCE As a graduate student in 1923, Louis de Broglie (1892–1987) made the astounding suggestion that since light waves could exhibit particle-like behavior, particles of matter should exhibit wave-like behavior. De Broglie proposed that all moving matter has a wavelength associated with it, just as a wave does. The Concept-at-a-Glance chart in Figure 29-12, which is a continuation of the chart in Figure 29-3, shows that the notions of energy, momentum, and wavelength are applicable to particles as well as to waves.

Figure 29-12 CONCEPTS AT A GLANCE Both a moving particle and a wave possess energy, momentum, and a wavelength. This photograph shows a highly magnified view of a Drosophila fruit fly, made with a scanning electron microscope. This microscope uses electrons instead of light. The resolution of the fine detail is exceptional because the wavelength of an electron can be made much smaller than that of visible light. (David Scharf/SPL/Photo Researchers, Inc.)

De Broglie made the explicit proposal that the wavelength of a particle is given by the same relation (Equation 29.6) that applies to a photon:

| (29.8)

where h is Planck’s constant and p is the magnitude of the relativistic momentum of the particle. Today, is known as the de Broglie wavelength of the particle.
Confirmation of de Broglie’s suggestion came in 1927 from the experiments of the American physicists Clinton J. Davisson (1881–1958) and Lester H. Germer (1896–1971) and, independently, those ofthe English physicist George P. Thomson (1892–1975). Davisson and Germer directed a beam of electrons onto a crystal of nickel and observed that the electrons exhibited a diffraction behavior, analogous to that seen when X-rays are diffracted by a crystal (see Section 27.9 for a discussion of X-ray diffraction). The wavelength of the electrons revealed by the diffraction pattern matched that predicted by de Broglie’s hypothesis, . More recently, Young’s double-slit experiment has been performed with electrons and reveals the effects of wave interference illustrated in Figure 29-1.
Particles other than electrons can also exhibit wave-like properties. For instance, neutrons are sometimes used in diffraction studies of crystal structure. Figure 29-13 compares the neutron diffraction pattern and the X-ray diffraction pattern caused by a crystal of rock salt (NaCl).

Figure 29-13 (a) The neutron diffraction pattern and (b) the X-ray diffraction pattern for a crystal of sodium chloride (NaCl). (a, From Wollan, Shull, and Marney, Phys. Rev. 73 (5): 527, 1948; b, Courtesy Edwin Jones, University of South Carolina)

Although all moving particles have a de Broglie wavelength, the effects of this wavelength are observable only for particles whose masses are very small, on the order of the mass of an electron or a neutron, for instance. Example 5 illustrates why.

| Example 5 | | | The De Broglie Wavelength of an Electron and of a Baseball

Determine the de Broglie wavelength for (a) an electron () moving at a speed of and (b) a baseball () moving at a speed of 13 m/s. Reasoning In each case, the de Broglie wavelength is given by Equation 29.8 as Planck’s constant divided by the magnitude of the momentum. Since the speeds are small compared to the speed of light, we can ignore relativistic effects and express the magnitude of the momentum as the product of the mass and the speed.
Solution

a. | Since the magnitude p of the momentum is the product of the mass m of the particle and its speed v, we have . Using this expression in Equation 29.8 for the de Broglie wavelength, we obtain

A de Broglie wavelength of is about the size of the interatomic spacing in a solid, such as the nickel crystal used by Davisson and Germer, and, therefore, leads to the observed diffraction effects.
b. | A calculation similar to that in part (a) shows that the de Broglie wavelength of the baseball is . This wavelength is incredibly small, even by comparison with the size of an atom (10−10 m) or a nucleus (10−14 m). Thus, the ratio of this wavelength to the width W of an ordinary opening, such as a window, is so small that the diffraction of a baseball passing through the window cannot be observed.

The de Broglie equation for particle wavelength provides no hint as to what kind of wave is associated with a particle of matter. To gain some insight into the nature of this wave, we turn our attention to Figure 29-14. Part a shows the fringe pattern on the screen when electrons are used in a version of Young’s double-slit experiment. The bright fringes occur in places where particle waves coming from each slit interfere constructively, while the dark fringes occur in places where the waves interfere destructively.

Figure 29-14 In this electron version of Young’s double-slit experiment, the characteristic fringe pattern becomes recognizable only after a sufficient number of electrons have struck the screen. (From A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa Am. J. Phys. 57 (2): 117, Feb. 1989.)

When an electron passes through the double-slit arrangement and strikes a spot on the screen, the screen glows at that spot, and parts b, c, and d of Figure 29-14 illustrate how the spots accumulate in time. As more and more electrons strike the screen, the spots eventually form the fringe pattern that is evident in part d. Bright fringes occur where there is a high probability of electrons striking the screen, and dark fringes occur where there is a low probability. Here lies the key to understanding particle waves. Particle waves are waves of probability, waves whose magnitude at a point in space gives an indication of the probability that the particle will be found at that point. At the place where the screen is located, the pattern of probabilities conveyed by the particle waves causes the fringe pattern to emerge. The fact that no fringe pattern is apparent in part b of the figure does not mean that there are no probability waves present; it just means that too few electrons have struck the screen for the pattern to be recognizable.
The pattern of probabilities that leads to the fringes in Figure 29-14 is analogous to the pattern of light intensities that is responsible for the fringes in Young’s original experiment with light waves (see Figure 27-4). Section 24.4 discusses the fact that the intensity of the light is proportional to either the square of the electric field strength or the square of the magnetic field strength of the wave. In an analogous fashion in the case of particle waves, the probability is proportional to the square of the magnitude (Greek letter Psi) of the wave. is referred to as the wave function of the particle.
In 1925 the Austrian physicist Erwin Schrödinger (1887–1961) and the German physicist Werner Heisenberg (1901–1976) independently developed theoretical frameworks for determining the wave function. In so doing, they established a new branch of physics called quantum mechanics. The word “quantum” refers to the fact that in the world of the atom, where particle waves must be considered, the particle energy is quantized, so only certain energies are allowed. To understand the structure of the atom and the phenomena related to it, quantum mechanics is essential, and the Schrödinger equation for calculating the wave function is now widely used. In the next chapter, we will explore the structure of the atom based on the ideas of quantum mechanics.

Check Your Understanding 3

A beam of electrons passes through a single slit, and a beam of protons passes through a second, but identical, slit. The electrons and the protons have the same speed. Which one of the follow ing correctly describes the beam that experiences the greatest amount of diffraction? (a) The electrons, because they have the smaller momentum and, hence, the smaller de Broglie wavelength. (b) The electrons, because they have the smaller momentum and, hence, the larger de Broglie wavelength. (c) The protons, because they have the smaller momentum and, hence, the smaller de Broglie wavelength. (d) The protons, because they have the larger momentum and, hence, the smaller de Broglie wavelength, (e) Both beams experience the same am ount of diffraction, because the electrons and protons have the sam e de Broglie wavelength. (The answer is given at the end of the book.)
Background: The amount of diffraction depends on the ratio of the de Broglie wavelength to the width of the slit (see Section 27.5), and the de Broglie wavelength of a particle depends on its momentum.
For similar questions (including calculational counterparts), consult Self-Assessment Test 29.2, which is described at the end of Section 29.6.

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