30.4

De Broglie’s Explanation of Bohr’s Assumption about Angular Momentum

Of all the assumptions Bohr made in his model of the hydrogen atom, perhaps the most puzzling is the assumption about the angular momentum of the electron . Why should the angular momentum have only those values that are integer multiples of Planck’s constant divided by ? In 1923, ten years after Bohr’s work, de Broglie pointed out that his own theory for the wavelength of a moving particle could provide an answer to this question.
In de Broglie’s way of thinking, the electron in its circular Bohr orbit must be pictured as a particle wave. And like waves traveling on a string, particle waves can lead to standing waves under resonant conditions. Section 17.5 discusses these conditions for a string. Standing waves form when the total distance traveled by a wave down the string and back is one wavelength, two wavelengths, or any integer number of wavelengths. The total distance around a Bohr orbit of radius r is the circumference of the orbit or . By the same reasoning, then, the condition for standing particle waves for the electron in a Bohr orbit would be

where n is the number of whole wavelengths that fit into the circumference of the circle. But according to Equation 29.8 the de Broglie wavelength of the electron is , where p is the magnitude of the electron’s momentum. If the speed of the electron is much less than the speed of light, the momentum is p = mv, and the condition for standing particle waves becomes . A rearrangement of this result gives

which is just what Bohr assumed for the angular momentum of the electron. As an example, Figure 30-12 illustrates the standing particle wave on a Bohr orbit for which .

Figure 30-12 De Broglie suggested standing particle waves as an explanation for Bohr’s angular momentum assumption. Here, a standing particle wave is illustrated on a Bohr orbit where four de Broglie wavelengths fit into the circumference of the orbit.

De Broglie’s explanation of Bohr’s assumption about angular momentum emphasizes an important fact—namely, that particle waves play a central role in the structure of the atom. Moreover, the theoretical framework of quantum mechanics includes the Schrödinger equation for determining the wave function (Greek letter Psi) that represents a particle wave. The next section deals with the picture that quantum mechanics gives for atomic structure, a picture that supersedes the Bohr model. In any case, the Bohr expression for the energy levels (Equation 30.11) can be applied when a single electron orbits the nucleus, whereas the theoretical framework of quantum mechanics can be applied, in principle, to atoms that contain an arbitrary number of electrons.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.