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Measurement Theory

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Measurement Theory 1 Measurement Theory For background to this section, reread Gri±ths Ch. 4 on spin and Stern-Gerlach experiment. We went through the structure of standard "Copenhagen" interpretation of quan- tum mechanics last semester. Many elements of the theory are nonintuitive from point of view of classical physics, but we argued that classical intuition is useless or even misleading when applied on an atomic scale. The internal consistency of quantum mechanics required the phenomenon of collapse of wave function, wherein P measurement of Q in a physical system represented by à = n anÃn yielded qn with 2 probability janj (QÃ^ n ´ qnÃn), with the implication that wave function immedi- ately after that particular measurement was Ãn. Now we investigate the process of measurement more deeply, show that apparent inconsistencies arise when we try to apply ideas to macroscopic scale. 1.1 Linearity Because quantum mechanics supposed to be based on SchrÄodingereqn, which is linear di®eq., superposition principle supposed to hold: if jÃi and jÁi are allowed states of physical system, so is combination ®jÃi + ¯jÁi. The state vectors evolve according to S.'s eqn, @ ih¹ jÃ(t)i = HjÃ(t)i; (1) @t so since both jÃi and jÁi are solns so is ®jÃ(t)i + ¯jÁ(t)i. Example 1: 2-slit expt. Go back to 2-slit expt. with electron gun. Recall our explanation for interference fringes which appeared when both slits were opened had to do with the fact that probabilities don't add, probability amplitudes do. Curve plotted on the \screen" at the right in each ¯gure is probability distribution of particle positions x, e.g. dP A = jhxjAij2 (2) dx for state jAi, etc. (Recall jxi is state with particle de¯nitely at position x.) \Copen- hagen" QM says we don't add probabilities in jAi and jBi to get probability in jCi, but rather prob. amplitudes 1 detector detector detector Figure 1: jAi is state with slit 1 closed, jBi is state with slit 2 closed, jCi is state with both slits opened. 1 dP jCi = p (jAi + jBi); C = jhxjCij2 (3) 2 dx 2 2 So dPC=dx contains not only jhxjAij and jhxjBij but interference terms: dP C = fjhxjAij2 + jhxjBij2 dx + hxjAihBjxi + hxjBihAjxig (4) | {z } \interference" Why do these terms give rise to interference pattern? Because the wave function ikr ÃA(x) ´ hxjAi has oscillatory character like a wave amplitude, roughly e i =ri (ri measured from slit i; i = 1; 2!). So 1st two terms in (4) are consts., 2nd two vary as 1 1 1 1 ikr1 ¡ikr2 ikr2 ¡ikr1 » (e e + e e ) » cos k(r1 ¡ r2) (5) r1 r2 r1 r2 i.e., classical interference pattern depending only on path di®erence r1 ¡ r2. ?Point: linear superposition principle crucial to understanding of this expt. Example 2: Stern-Gerlach apparatus: simple device for spatial separation of di®erent-spin particles. For illustration con- sider neutral spin-1/2 particles, e.g. neutrons, place in inhomogeneous magnetic ¯eld B(r). Recall energy of spin-1/2 with moment ~¹ in magnetic ¯eld is U = ¡~¹ ¢ B (6) Compare energetics with classical case, where any energy between §¹B is allowed. 2 For quantum spin-1/2 particle, since spin is quantized to point either parallel or antiparallel to B, only allowed values are §¹B. Now if B is inhomogeneous there is a classical force on a magnetic moment ~¹ equal to F = ¡rU = ¡¹(§rB). So assume incoming beam in ¯gure is mixture of spins k and anti-k to ¯eld (B k x^), experience forces in opposite directions. Can separate spins, reverse rB, recombine as shown. Spin eigenstates for B k x^ are 0 1 0 1 1 1 @ 1 A @ 1 A Â1 = p ; Â2 = p ; (7) 2 1 2 ¡1 where labels 1 and 2 corresponding to paths followed by spins in ¯gure. Note the spin quantization axis isz ^ as usual although we've taken B k x^. What happens if particle with \spin up", 0 1 1  = @ A (8) 0 is injected into ¯eld gradient? Can write as linear combination 0 1 0 1 0 1 1 1 1 1 1  = @ A = @ A + @ A (9) 0 2 1 2 ¡1 This is strange: there is only 1 particle, but \part" of it must move along path 1 and the other \part" along path 2: there is a nonzero prob. amplitude for it to take each path. Could it be that the particle is \really" in Â1 or Â2, and  just representsp our ignorance? No, for when we recombine, spin will be up again,  = (1= 2)(Â1 + Â2). If it were \really" Â1, say, it would leave as mixture of spin up and down. 3 (Note for E& M purists: I've taken rB, B to point in same direction in ¯gure, which could be arranged approximately, e.g. with a coil whose winding density varies along the axis direction. But of course to keep r ¢ B = 0 the system will generate some small transverse gradients as well. This doesn't a®ect the argument, as you'll be able to work out in prob. set. See also Gri±ths, p. 181 et seq.) Example 3: Ammonia molecule NH3 NH3 molecule is simple example of \2-level system". Consists of triangle of H- atoms and N-atom out of plane in minimum energy con¯guration. Since Hamilto- nian rotationally invariant, given ¯xed H-triangle there is no reason for N to be above rather than below, or v.v. In fact two states jtopi and jboti must be degen- erate. Neither one is ground state, however: rather than break symmetry, nature chooses symmetric mixture: 1 j0i = p (jtopi + jboti) (10) 2 and 1st excited state is the asymmetric combination 1 j1i = p (jtopi ¡ jboti) (11) 2 Here prob. ampl. for ¯nding N atom at given position z relative to 3 H-atom plane plotted schematically for 4 states. ? N.B. jtopi and jboti not energy eigenstates. If molecule is in, e.g. jtopi at t = 0 it does not stay there, but \tunnels" into jboti with time. This phenomenon is observable, & very similar to 2-slit expt. Imagine we do x-ray scattering experiment o® NH3 molecule as shown below. 4 If system is in state jtopi, only one source for sph. If system is in ground state j0i, there are 2 scattered wave, so no interference in scattering pattern. waves which interfere, causing fringes on photographic plate. Example 4: SchrÄodinger'scat paradox We may be tempted to accept notion of molecule in superposition of 2 di®erent con¯gurations as mysteries of life at atomic scale, but harder to swallow similar im- plications at macroscopic scales. Famous Gedanken expt. propsed by SchrÄodinger: suppose at t = 0 box is ¯lled with a) gun; b) atom in excited state c) cat; and d) device to detect when atom decays to grnd state and ¯re gun at cat. Atom is not in stationary state, therefore system is not, will evolve in time into admixture of state with excited atom and live cat j1; alivei and state with grnd. state atom and dead cat j0; deadi (Just as in NH3 case, where jtopi evolves after some time into an admixture of jtopi and jboti). Therefore at later time t cat is neither alive nor dead, but some admixture of two? What happens when box is opened? Then you \measure" system, determine if cat is alive or dead{collapse wave function. Observation itself is responsible for killing cat or keeping it alive. Seems absurd|leave as question for now. 5 1.2 Measurement and collapse of wave function Go back to Stern-Gerlach apparatus, and see what happens if we try to determine path particle takes. We'll put special neutron-sensitive TV cameras a and b along paths 1 and 2 corresponding to spin parallel and antiparallel tox ^. If particle with spin alongx ^ axis enters, it certainly is detected by camera a. If a particle with spin up (k z^) enters, according to rules, probability it's detected by camera a is 0 1 µ ¶ 1 1 P = jh jÂij2 = j p1 p1 @ A j2 = (12) a 1 2 2 0 2 Now when particle leaves the apparatus it is de¯nitely k tox ^, not z^, since we know it went through arm 1, as only particles with spins k x^ do. Act of measurement has changed spin state from  to Â1. Slightly more subtle: suppose we had only put camera b in arm 2, and it didn't register anything. If the camera is perfect this means with probability 1 the particle was in arm 1 and wave function is collapsed anyway, even though it was never \directly" observed. In case of NH3 molecule, imagine we can create a beam of x-rays so tight that we can determine whether the N atom is above or below triangle of H-atoms, as 6 shown: If molecule is in state jboti, there is certainly a scattering. If molecule is in state jtopi certainly no scattering takes place. If the molecule is in ground state j0i, scattering is observed with 50% probability. If in given expt. no scattering is observed, molecule is in state jtopi at end of observation. Thus starting from j0i, may happen that act of observation forced molecule into jtopi, although no scattering takes place. Not just semantics: jtopi is a higher energy state than j0i{where did extra energy come from? Einstein-Podolsky-Rosen \Paradox" EPR (1935) suggested that Copenhagen qm was an incomplete theory, because events could only be predicted in probabilistic sense.
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