Quantum Physics

28.6 Quantum Mechanics and the Hydrogen Atom 913rABFigure 28.10 (a) Standing-wavepattern for an electron wave in astable orbit of hydrogen. There arethree full wavelengths in this orbit.(b) Standing-wave pattern for avibrating stretched string fixed at itsends. This pattern also has three fullwavelengths.l(a)(b)For more than a decade following Bohr’s publication, no one was able to explain whythe angular momentum of the electron was restricted to these discrete values. Finally,de Broglie gave a direct physical way of interpreting this condition. He assumed thatan electron orbit would be stable (allowed) only if it contained an integral number ofelectron wavelengths. Figure 28.10a demonstrates this point when three completewavelengths are contained in one circumference of the orbit. Similar patterns can bedrawn for orbits containing one wavelength, two wavelengths, four wavelengths, fivewavelengths, and so forth. These waves are analogous to standing waves on a string,discussed in Chapter 14. There, we found that strings have preferred (resonant) frequenciesof vibration. Figure 28.10b shows a standing-wave pattern containing threewavelengths for a string fixed at each end. Now imagine that the vibrating string is removedfrom its supports at A and B and bent into a circular shape that brings thosepoints together. The end result is a pattern such as the one shown in Figure 28.10a.In general, the condition for a de Broglie standing wave in an electron orbit isthat the circumference must contain an integral number of electron wavelengths.We can express this condition as2r n n 1, 2, 3, . . .Because the de Broglie wavelength of an electron is h/m e v, we can write thepreceding equation as 2r nh/m e v, orm e vr nThis is the same as the quantization of angular momentum condition imposed byBohr in his original theory of hydrogen.The electron orbit shown in Figure 28.10a contains three complete wavelengthsand corresponds to the case in which the principal quantum number n 3. Theorbit with one complete wavelength in its circumference corresponds to the firstBohr orbit, n 1; the orbit with two complete wavelengths corresponds to the secondBohr orbit, n 2; and so forth.By applying the wave theory of matter to electrons in atoms, de Broglie was ableto explain the appearance of integers in the Bohr theory as a natural consequenceof standing-wave patterns. This was the first convincing argument that the wave natureof matter was at the heart of the behavior of atomic systems. Although theanalysis provided by de Broglie was a promising first step, gigantic strides weremade subsequently with the development of Schrödinger’s wave equation and itsapplication to atomic systems.28.6 QUANTUM MECHANICSAND THE HYDROGEN ATOMOne of the first great achievements of quantum mechanics was the solution of thewave equation for the hydrogen atom. The details of the solution are far beyondthe level of this course, but we’ll describe its properties and implications foratomic structure.