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.The magnetic field B is a function of r, the position of the center of mass of theatom (the variation of B over the size of the atom is completely negligible).The classical equations of motion are obtained from the Poisson brackets= , H = m , (1.5)r[ ] PB/P= H ]PB = " ( µ · B ), (1.6)p ,[= µ H = g( µ × B ).(1.7),[]PBThe last equation follows from [Sx , Sy]PB = S and its cyclic permutations.Notezthat the internal variables S have vanishing Poisson brackets with the center ofmass variables r and p.Equation (1.7) implies that µ precesses around the direction of B.This di-rection cannot be constant in space, since this would violate Maxwell s equation" · B = 0.One can however approximately solve (1.7) if , the mean value ofB in the magnet gap, is much larger than the variation of B in that gap andif, moreover, the duration of passage of the atom through the magnet is muchlonger than its precession time 2À /g B.If these conditions hold, the atom willprecess many times around the direction of , so that we can neglect, on atime average, the components of µ orthogonal to.Let us write = e B,1where e is a unit vector and B is a constant.It then follows from (1.7) that1µ · e1 is a constant, and we can, on a time average, replace µ by µ1 e1 , where:=µ µ · e1.(1.8)1(The symbol := means is defined as .) From Eq.(1.6) we obtainde1 · B'( p) = µ1 , (1.9)dtwhere B' : = · " · B depends only on the construction of the magnet.(e ) ( e )1 1The force (1.6) acts during a time L/v, where v is the longitudinal velocityof the atoms, and L is the length of the magnet.The transverse momentumimparted to the atoms by this force is µ1 B'L/v, and their deflection angle1is µ B'L/2E, where E = mv2.All these terms, except µ , are determined1 12by the macroscopic experimental setup (the oven, the velocity selector, themagnet, etc.) and are fixed for a given experiment.The surprising result foundby Gerlach and Stern15 was that µ could take only two values, ± µ.1This result is extremely surprising from the point of view of classical physics,because Gerlach and Stern could have chosen different orientations for theirmagnet, for example e and e , making angles of ±120° with e1 , as shown in23Fig.1.6.They would have measured then 16 Introduction to Quantum Physics·µ = µ e2 or = µ · e3 , (1.10)2 µ3respectively.As the laws of physics cannot be affected by merely rotating themagnet, they would have found, likewise, µ2 = ± µ or µ3 = ±µ.This creates,however, an apparent contradiction when we add Eqs.(1.8) and (1.10):µ1 + µ2 + µ3 = µ · e1 + e + e3) a" 0.(1.11)( 2Obviously, µ1 , and µ3 cannot all be equal to ±µ , and also sum up to zero.µ2Of course, it is impossible to measure in this way the values of µ1 and µ2andµ3 of the same atom the magnet can have only one of the three positions.There is no need to invoke quantum uncertainties here.This is a purelyclassical impossibility, inherent in the experiment described by Fig.1.6.(WhatFig.1.6.Three possible orientations for the Stern-Gerlach magnet, making 120°angles with each other.The three unit vectors e1 , e and e sum up to zero.23quantum theory tells us is that this is not a defect of this particular experimentalmethod for measuring a magnetic moment: No experiment whatsoever candetermine µ1 and µ and µ3 simultaneously.) Yet, even if the three experimental2setups sketched in Fig.1.6 are incompatible, it is certainly possible16 to measureµ , or µ , instead of µ1.Thus, if we attribute to the word measurement its2 3ordinary meaning, namely the acquisition of knowledge about some objectivepreexisting reality, we reach a contradiction.The contradiction is fundamental.Once we associate discrete valueswith the components of a vector which can be continuously rotated,the meaning of these discrete values cannot be that of objective vectorcomponents, which would be independent of the measurement process.16You may feel uneasy with this counterfactual reasoning.While we are free to imagine thepossible outcomes of unperformed experiments, Eq.(1.11) goes farther: it involves, simultane-ously, the results of three incompatible experiments.At most one of the mathematical symbolswritten on the paper can acquire an actual meaning.The two others then exist only in ourimagination.Is that equation legitimate? Can we draw from it reliable conclusions? Moreover,Eq.(1.11) assumes that, in these three possible but incompatible experiments, the magneticmoment of the silver atom has the same orientation.That is, our freedom of choice for theorientation of the magnet does not affect the silver atoms that evaporate from the oven.If youthink that this is obvious, wait until after you have read Chapter 6. What is a measurement? 17A measurement is not a passive acquisition of knowledge.It is an active pro-cess, making use of extremely complex equipment, usually involving irreversibleamplification mechanisms
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