Quantum Physics

888 Chapter 27 Quantum Physicsmomentum of a particle of mass m and speed v is mv p, the de Broglie wavelengthof a particle isde Broglie’s hypothesis h p hmv[27.14]Further, de Broglie postulated that the frequencies of matter waves (waves associatedwith particles having nonzero rest energy) obey the Einstein relationshipfor photons, E hf, so thatFrequency of matter waves f E h[27.15]The dual nature of matter is quite apparent in Equations 27.14 and 27.15, becauseeach contains both particle concepts (mv and E ) and wave concepts ( and f ).The fact that these relationships had been established experimentally for photonsmade the de Broglie hypothesis that much easier to accept.The Davisson–Germer ExperimentDe Broglie’s proposal in 1923 that matter exhibits both wave and particle propertieswas first regarded as pure speculation. If particles such as electrons had wavelikeproperties, then, under the correct conditions, they should exhibit diffractioneffects. In 1927, three years after de Broglie published his work, C. J. Davisson(1881–1958) and L. H. Germer (1896–1971) of the United States succeeded inmeasuring the wavelength of electrons. Their important discovery provided thefirst experimental confirmation of the matter waves proposed by de Broglie.The intent of the initial Davisson–Germer experiment was not to confirm the deBroglie hypothesis. In fact, their discovery was made by accident (as is often thecase). The experiment involved the scattering of low-energy electrons (about 54 eV)from a nickel target in a vacuum. During one experiment, the nickel surface wasbadly oxidized because of an accidental break in the vacuum system. After the nickeltarget was heated in a flowing stream of hydrogen to remove the oxide coating, electronsscattered by it exhibited intensity maxima and minima at specific angles. Theexperimenters finally realized that the nickel had formed large crystalline regionsupon heating and that the regularly spaced planes of atoms in the crystalline regionsserved as a diffraction grating for electron matter waves. (See Section 27.5.)Shortly thereafter, Davisson and Germer performed more extensive diffractionmeasurements on electrons scattered from single-crystal targets. Their resultsshowed conclusively the wave nature of electrons and confirmed the de Broglierelation h/p. In the same year, G. P. Thomson (1892–1975) of Scotland alsoobserved electron diffraction patterns by passing electrons through very thin goldfoils. Diffraction patterns have since been observed for helium atoms, hydrogenatoms, and neutrons. Hence, the universal nature of matter waves has been establishedin various ways.Quick Quiz 27.3A nonrelativistic electron and a nonrelativistic proton are moving and have thesame de Broglie wavelength. Which of the following are also the same for the twoparticles?(a) speed (b) kinetic energy (c) momentum (d) frequencyQuick Quiz 27.4We have seen two wavelengths assigned to the electron: the Compton wavelengthand the de Broglie wavelength. Which is an actual physical wavelength associatedwith the electron? (a) the Compton wavelength (b) the de Broglie wavelength(c) both wavelengths (d) neither wavelength