Quantum Physics

27.8 The Uncertainty Principle 893EXAMPLE 27.8 Locating an ElectronGoal Apply Heisenberg’s position–momentum uncertainty principle.Problem The speed of an electron is measured to be 5.00 10 3 m/s to an accuracy of 0.003 00%. Find the minimumuncertainty in determining the position of this electron.Strategy After computing the momentum and its uncertainty, substitute into Heisenberg’s uncertainty principle,Equation 27.16.SolutionCalculate the momentum of the electron:The uncertainty in p is 0.003 00% of this value:Now calculate the uncertainty in position using thisvalue of p x and Equation 27.17:p x m e v (9.11 10 31 kg)(5.00 10 3 m/s) 4.56 10 27 kg m/sp x 0.000 030 0p (0.000 030 0)(4.56 10 27 kg m/s) 1.37 10 31 kg m/sxp x x h4p x6.626 10 34 Js4(1.37 10 31 kgm/s) 0.384 103 m4: x h 0.384 mmRemarks Notice that this isn’t an exact calculation: the uncertainty in position can take any value, as long as it’sgreater than or equal to the value given by the uncertainty principle.Exercise 27.8Suppose an electron is found somewhere in an atom of diameter 1.25 10 10 m. Estimate the uncertainty in theelectron’s momentum (in one dimension).Answerp 4.22 10 25 kg m/sEXAMPLE 27.9GoalExcited States of AtomsApply the energy–time form of the uncertainty relation.Problem As we’ll see in the next chapter, electrons in atoms can be found in certain high states of energy calledexcited states for short periods of time. If the average time that an electron exists in one of these states is 1.00 10 8 s,what is the minimum uncertainty in energy of the excited state?StrategySubstitute values into Equation 27.17, the energy–time form of Heisenberg’s uncertainty relation.SolutionUse Equation 27.17 to obtain the minimum uncertaintyin the energy:E h4t 6.63 1034 Js4(1.00 10 8 s) 5.28 1027 J 3.30 10 8 eVRemarksThis is again an imprecise calculation, giving only a lower bound on the uncertainty.Exercise 27.9A muon may be considered to be an excited state of an electron, to which it decays in an average of 2.2 10 6 s.What’s the minimum uncertainty in the muon’s (rest) energy, according to the uncertainty principle?Answer2.40 10 29 J