In the years following Bohr's development of a model for the hydrogen atom, the dual nature of radiant energy became a familiar concept. Depending on the experimental circumstances, radiation appears to have either a wavelike or a particle-like (photon) character. Louis de Broglie (1892-1987), who was working on his Ph.D. thesis in physics at the Sorbonne in Paris, boldly extended this idea. If radiant energy could, under appropriate conditions, behave as though it were a stream of particles, could matter, under appropriate conditions, possibly show the properties of a wave? Suppose that the electron orbiting the nucleus of a hydrogen atom could be thought of as a wave, with a characteristic wavelength. De Broglie suggested that the electron in its movement about the nucleus has associated with it a particular wavelength. He went on to propose that the characteristic wavelength of the electron or of any other particle depends on its mass, *m,* and velocity, *v:*

[6.7]

(*h* is Planck's constant.) The quantity *mv* for any object is called its **momentum.** De Broglie used the term **matter waves** to describe the wave characteristics of material particles.

Because de Broglie's hypothesis is applicable to all matter, any object of mass *m* and velocity *v* would give rise to a characteristic matter wave. However, Equation 6.7 indicates that the wavelength associated with an object of ordinary size, such as a golf ball, is so tiny as to be completely out of the range of any possible observation. This is not so for an electron because its mass is so small.

What is the characteristic wavelength of an electron with a velocity of 5.97 10^{6} m/s? (The mass of the electron is 9.11 10^{–28} g.)

**SOLUTION** The wavelength of the electron is given by Equation 6.7. Using the value of Planck's constant, *h *= 6.63 10^{–34} J-s, and recalling that 1 J = 1 kg-m^{2}/s^{2}, we have:

By comparing this value with the wavelengths of electromagnetic radiations shown in Figure 6.4, we see that the characteristic wavelength is about the same as that of X- rays.

At what velocity must a neutron be moving in order for it to exhibit a wavelength of 500 pm? The mass of a neutron is given in the table on the back inside cover of the text. ** Answer:** 7.92 10

Within a few years after de Broglie published his theory, the wave properties of the electron were demonstrated experimentally. Electrons were diffracted by crystals, just as X rays are diffracted.

The technique of electron diffraction has been highly developed. In the electron microscope the wave characteristics of electrons are used to obtain pictures of tiny objects. This microscope is an important tool for studying surface phenomena at very high magnifications. Figure 6.16 is a photograph of an electron microscope image. Such pictures are powerful demonstrations that tiny particles of matter can indeed behave as waves.

The discovery of the wave properties of matter raised some new and interesting questions about classical physics. Consider, for example, a ball rolling down a ramp. By using classical physics, we can calculate *exactly* its position, its direction of motion, and its speed of motion at any time. Can we do the same for an electron that exhibits wave properties? A wave extends in space, and its location is not precisely defined. We might therefore anticipate that it is not possible to determine exactly where an electron is located at a specific time.

The German physicist Werner Heisenberg (Figure 6.17) concluded that the dual nature of matter places a fundamental limitation on how precisely we can know both the location and the momentum of any object. The limitation becomes important only when we deal with matter at the subatomic level, that is, with masses as small as that of an electron. Heisenberg's principle is called the **uncertainty principle.** When applied to the electrons in an atom, this principle states that it is inherently impossible for us to know simultaneously both the exact momentum of the electron and its exact location in space. Thus, it is not appropriate to imagine the electrons as moving in well-defined circular orbits about the nucleus.

De Broglie's hypothesis and Heisenberg's uncertainty principle set the stage for a new and more broadly applicable theory of atomic structure. In this new approach, any attempt to define precisely the instantaneous location and momentum of the electrons is abandoned. The wave nature of the electron is recognized, and the electron's behavior is described in terms appropriate to waves.