1.1.TheStern-GerlachExperiment3z-axisABeamdirectionOvenTodetectorB -fieldCollimatingShapedSlitmagnet(polepieces)FIGURE1.1.The Stern-Gerlachexperimentproportional to the electron spin S,(1.1.1)μαs.where the precise proportionality factor turns out to be e /m .c (e <0 in thisbook) to an accuracy of about 0.2%Because the interaction energy of the magnetic moment with themagnetic field is just -μ·B, the z-component of the force experienced bythe atom is given byaB.d(1.1.2)F, =(μB)~μ:7zwhere we have ignored the components of B in directions other than thez-direction. Because the atom as a whole is very heavy, we expect that theclassical concept of trajectory can be legitimately applied, a point which canbe justified using the Heisenberg uncertainty principle to be derived later.With the arrangement of Figure 1.1, the μ, > 0 (S, < 0) atom experiences adownward force, while the μ,<0 (S, > O) atom experiences an upwardforce. The beam is then expected to get split according to the values of μz.In othcr words,thc SG (Stcrn-Gerlach)apparatus“measures"the z-component of μ or, equivalently, the z-component of S up to a proportionalityfactor.The atoms in the oven are randomly oriented; there is no preferreddirection for the orientation of μ. If the electron were like a classicalspinning object, we would expect all values of μ, to be realized between lμland - lμl. This would lead us to expect a continuous bundle of beamscoming out of the SG apparatus, as shown in Figure 1.2a. Instead, what we
Fundamental Concepts4ScreenScreen(b)(a)FIGURE 1.2.Beams from the SG apparatus: (a)is expected from classical physics,while (b)isactuallyobserved.experimentally ohserve is more like the situation in Figure 1.2b. In otherwords, the SG apparatus splits the original silver beam from the oven intotwo distinct components, a phenomenon referred to in the early days ofquantum theory as “space quantization." To the extent that μ can beidentified within a proportionality factor with the electron spin S, only twopossible values of the z-component of S are observed to be possible, S, upand S, down, which we call S, + and S, -. The two possible values of S,are multiples of some fundamental unit of angular momentum; numericallyit turns out that S, - h /2 and - h /2, whereh =1.0546×10-27erg-s=6.5822×10-16eV-s(1.1.3)This “quantization of the electron spin angular momentum is the firstimportant feature we deduce from the Stern-Gerlach experiment.Of course, there is nothing sacred about the up-down direction or thez-axis. We could just as well have applied an inhomogeneous field in ahorizontal direction, say in the x-direction, with the beam proceeding in they-direction. In this manner we could have separated the beam from the oveninto an S + component and an S - component.Sequential Stern-Gerlach ExperimentsLet us now consider a sequential Stern-Gerlach experiment. By thiswe mean that the atomic beam goes through two or more SG apparatuses insequence. The first arrangement we consider is relatively straightforward.We subject the beam coming out of the oven to the arrangement shown inFigure 1.3a, where SGz stands for an apparatus with the inhomogeneousmagnetic field in the z-direction, as usual. We then block the S, - compo-
51.1.The Stern-GerlachExperimentS,+ comp.Sz+ comp.OvenSG2SG2么-NoS,-compSz- comp.(a)Sz+ beam.Sx+ beam.SG2OvenSGxSx- beam.Sz-beam.(b)S.+beam.S.+beam.Sz+ beam.SG2OvenSGXSG2么Sz-beam.Sz- beam.Sx- beam.(c)FIGURE1.3.Sequential Stern-Gerlach experiments.nent coming out of the first SGz apparatus and let the remaining S, +component be subjected to another SGz apparatus. This time there is onlyone beam component coming out of the second apparatus-just the S, +component. This is perhaps not so surprising; after all if the atom spins areup, they are expected to remain so, short of any external field that rotatesthe spins between the first and the second SGz apparatuses.A little more interesting is the arrangement shown in Figure 1.3b.Here the first SG apparatus is the same as before but the second one (SGx)has an inhomogeneous magnetic field in the x-direction. The S, + beamthat enters the second apparatus (SGx) is now split into two componcnts, anS, + component and an S,- component, with equal intensities.How canwe explain this? Does it mean that 50% of the atoms in the S, + beamcoming out of the first apparatus (SG2) are made up of atoms characterizedby both S, + and S +, while the remaining 50% have both S, + and S, -?It turns out that such a picture runs into difficulty, as will be shown below.We now consider a third step, the arrangement shown in Figure1.3(c), which most dramatically illustrates the peculiarities of quantum-mechanical systems. This time we add to the arrangement of Figure 1.3byet a third apparatus, of the SG2 type. It is observed experimentally thattwo components emerge from the third apparatus, not one; the emergingbeams are seen to have both an S,+ component and an S,- componentThis is a complete surprise because after the atoms emerged from the first
6Fundamental Conceptsapparatus, we made sure that the S, - component was completely blocked.How is it possible that the S, - component which, we thought, wc climinatedearlier reappears? The model in which the atoms entering the third appara-tus are visualized to have both S, + and S, + is clearly unsatisfactoryThis cxample is often used to illustrate that in quantum mechanicswc cannot determine both S,and S,simultaneously.Moreprecisely,we cansay that the selection of the S, + beam by the second apparatus (SGx)completely destroys any previous information about S,.It is amusing to compare this situation with that of a spinning top inclassical mechanics, where the angular momentumL=Iw(1.1.4)can be measured by determining the components of the angular-velocityvector w.By observinghow fast the objectis spinning in whichdirectionwecan determine Wx, wy, and w, simultaneously. The moment of inertia I iscomputableifweknowthemassdensityandthegeometricshapeofthespinning top, so there is no difficulty in specifying both L, and L in thisclassicalsituationIt is to be clearly understood that the limitation we have encounteredin determining S, and S, is not due to the incompetence of the experi-mentalist. By improving the experimental techniques we cannot make theS, - component out of the third apparatus in Figure 1.3c disappear. Thepeculiarities of quantum mechanics are imposed upon us by the experimentitself. The limitation is, in fact, inherent in microscopic phenomena.Analogy with Polarization of LightBecause this situation looks so novel, some analogy with a familiarclassical situation may be helpful here. To this end we now digress toconsider the polarization of light waves.Consider a monochromatic light wave propagating in the z-direction.A linearly polarized (or plane polarized) light with a polarization vector inthe x-direction, which we call for short an x-polarized light, has a space-tirnedependent electric field oscillating in the x-directionE= E,xcos(kz - wt).(1.1.5)Likewise, we may consider a y-polarized light, also propagating in thez-direction,(1.1.6)E = E.ycos(kz - wt).Polarized light beams of type (1.1.5) or (1.1.6) can be obtained by letting anunpolarized light beam go through a Polaroid filter. We call a filter thatselects only beams polarized in the x-direction an x-filter. An x-filter, ofcourse, becomes a y-filter when rotated by 9o° about the propagation (z)
1.1.The Stern-Gerlach ExperimentNo beamNo lightxfilteryfilter(a)100%x'filteryfilterxfilter(45°diagonal)(b)FIGURE 1.4.Light beams subjccted to Polaroid filters.direction. It is well known that when we let a light heam go through anx-filter and subsequently let it impinge on a y-filter, no light beam comesout provided, of course, we are dealing with 100% eficient Polaroids; seeFigure1.4aThe situation is even more interesting if we insert between the x-filterand the y-filter yet another Polaroid that selects only a beam polarized inthe direction-which we call the x'-direction-that makes an angle of 45owith the x-direction in the xy plane; see Figure 1.4b. This time, there is alight beam coming out of the y-filter despite the fact that right after thebeam went through the x-filter it did not have any polarization componentin the y-direction. In other words, once the x'-filter intervenes and selectsthe x'-polarized beam, it is immaterial whether the beam was previouslyx-polarized. The selection of the x'-polarized beam by the sccond Polaroiddestroys any previous information on light polarization. Notice that thissituation is quite analogous to the situation that we encountered earlier withthe SG arrangement of Figure 1.3b, provided that the following correspon-denceis madeS, ± atoms x-, y-polarized light(1.1.7)S, ± atoms αx'-, y'-polarized light,where the x'- and the y'-axes are defined as in Figure 1.5.Let us examine how we can quantitatively describe the behavior of45°-polarized beams (x'- and y'-polarized beams) within the framework of