Quantum Physics

892 Chapter 27 Quantum PhysicsFigure 27.18 A thought experimentfor viewing an electron with apowerful microscope. (a) The electronis viewed before colliding with thephoton. (b) The electron recoils(is disturbed) as the result of thecollision with the photon.IncidentphotonBeforecollisionScatteredphotonAftercollisionElectronRecoilingelectron(a)(b)microscope into your eye, as shown in Figure 27.18b. When it strikes the electron,however, the photon transfers some unknown amount of its momentum to the electron.Thus, in the process of locating the electron very accurately (that is, by makingx very small), the light that enables you to succeed in your measurement changesthe electron’s momentum to some undeterminable extent (making p x very large).The incoming photon has momentum h/. As a result of the collision, the photontransfers part or all of its momentum along the x-axis to the electron. Therefore,the uncertainty in the electron’s momentum after the collision is as great asthe momentum of the incoming photon: p x h/. Further, because the photonalso has wave properties, we expect to be able to determine the electron’s positionto within one wavelength of the light being used to view it, so x . Multiplyingthese two uncertainties givesx p x hThe value h represents the minimum in the product of the uncertainties. Becausethe uncertainty can always be greater than this minimum, we havex p x hApart from the numerical factor 1/4 introduced by Heisenberg’s more preciseanalysis, this inequality agrees with Equation 27.16.Another form of the uncertainty relationship sets a limit on the accuracy withwhich the energy E of a system can be measured in a finite time interval t :E t h[27.17]It can be inferred from this relationship that the energy of a particle cannot bemeasured with complete precision in a very short interval of time. Thus, when anelectron is viewed as a particle, the uncertainty principle tells us that (a) its positionand velocity cannot both be known precisely at the same time and (b) itsenergy can be uncertain for a period given by t h/(4 E ).h4Applying Physics 27.4A common, but erroneous, description of the absolutezero of temperature is “that temperature at which allmolecular motion ceases.” How can the uncertaintyprinciple be used to argue against this description?Motion at Absolute ZeroExplanation Imagine a particular molecule in a piece ofmaterial. The molecule is confined within the material,so there is a fixed uncertainty x in its position alongone axis, corresponding to the size of that piece of material.If all molecular motion ceased at absolute zero, thegiven molecule’s velocity, in particular, would be exactlyzero, so its uncertainty in velocity would be v 0,meaning its uncertainty in momentum would also bezero, since p mv. The product of zero uncertainty inmomentum and a nonzero uncertainty in position iszero, violating the uncertainty principle. So according tothe uncertainty principle, there must be some molecularmotion even at absolute zero.