Quantum Physics

27.6 The Dual Nature of Light and Matter 889EXAMPLE 27.7GoalThe Electron versus the BaseballApply the de Broglie hypothesis to a quantum and a classical object.Problem (a) Compare the de Broglie wavelength for an electron (m e 9.11 10 31 kg) moving at a speed of1.00 10 7 m/s with that of a baseball of mass 0.145 kg pitched at 45.0 m/s. (b) Compare these wavelengths with thatof an electron traveling at 0.999c.Strategy This is a matter of substitution into Equation 27.14 for the de Broglie wavelength. In part (b), the relativisticmomentum must be used.Solution(a) Compare the de Broglie wavelengths of theelectron and the baseball.Substitute data for the electron into Equation 27.14:Repeat the calculation with the baseball data:(b) Find the wavelength for an electron traveling at0.999c.Replace the momentum in Equation 27.14 with the relativisticmomentum:Substitute:e hm e v 7.28 10 11 mb hm b v e 6.63 10 34 Js(9.11 10 31 kg)(1.00 10 7 m/s)6.63 1034 Js(0.145 kg)(45.0 m/s) hm e v/√1 v 2 /c h √1 v 2 /c 22 m e v1.02 10 34 m(6.63 10 34 Js)√1 (0.999c) 2 /c 2(9.11 10 31 kg)(0.9993.00 10 8 m/s) 1.09 10 13 me Remarks The electron wavelength corresponds to that of x-rays in the electromagnetic spectrum. The baseball,by contrast, has a wavelength much smaller than any aperture through which the baseball could possibly pass,so we couldn’t observe any of its diffraction effects. It is generally true that the wave properties of large-scale objectscan’t be observed. Notice that even at extreme relativistic speeds, the electron wavelength is still far larger than thebaseball’s.Exercise 27.7Find the de Broglie wavelength of a proton (m p 1.67 10 27 kg) moving with a speed of 1.00 10 7 m/s.Answer3.97 10 14 mApplication: The Electron MicroscopeA practical device that relies on the wave characteristics of electrons is the electronmicroscope. A transmission electron microscope, used for viewing flat, thin samples,is shown in Figure 27.17 (page 890). In many respects, it is similar to an opticalmicroscope, but the electron microscope has a much greater resolving powerbecause it can accelerate electrons to very high kinetic energies, giving them veryshort wavelengths. No microscope can resolve details that are significantly smallerthan the wavelength of the radiation used to illuminate the object. Typically, thewavelengths of electrons are about 100 times smaller than those of the visible lightused in optical microscopes. (Radiation of the same wavelength as the electrons inan electron microscope is in the x-ray region of the spectrum.)APPLICATIONElectron Microscopes