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Home , Spin (physics)

1 Measurement Theory

For background to this section, reread Griffiths Ch. 4 on and Stern-Gerlach experiment.

We went through the structure of standard ”Copenhagen” interpretation of quan- tum last semester. Many elements of the theory are nonintuitive from point of view of classical , but we argued that classical intuition is useless or even misleading when applied on an atomic scale. The internal consistency of required the phenomenon of collapse of , wherein P measurement of Q in a physical system represented by ψ = n anψn yielded qn with 2 probability |an| (Qψˆ n ≡ qnψn), with the implication that wave function immedi- ately after that particular measurement was ψn. 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 Schr¨odingereqn, which is linear diffeq., superposition principle supposed to hold: if |ψi and |φi are allowed states of physical system, so is combination α|ψi + β|φi. The state vectors evolve according to S.’s eqn, ∂ ih¯ |ψ(t)i = H|ψ(t)i, (1) ∂t so since both |ψi and |φi are solns so is α|ψ(t)i + β|φ(t)i. Example 1: 2-slit expt. Go back to 2-slit expt. with 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 figure is probability distribution of particle positions x, e.g. dP A = |hx|Ai|2 (2) dx for state |Ai, etc. (Recall |xi is state with particle definitely at position x.) “Copen- hagen” QM says we don’t add probabilities in |Ai and |Bi to get probability in |Ci, but rather prob. amplitudes

1 So opr nreiswt lsia ae hr n nrybetween energy any where case, classical with energetics Compare magnetic inhomogeneous in place , con- e.g. illustration For field particles, particles. spin-1/2 different-spin neutral of sider separation spatial for device simple difference path on only depending pattern interference classical i.e., function wave as the Because slit from pattern? measured interference to rise give ψ terms these do Why xml :SenGrahapparatus Stern-Gerlach 2: Example ? Point A ( dP iue1: Figure x B ) C h ≡ ( /dx iersproiinpicpecuilt nesadn fti expt. this of understanding to crucial principle superposition linear : r .Rcl nryo pn12wt moment with spin-1/2 of energy Recall ). x | A | otisntonly not contains A i ssaewt lt1closed, 1 slit with state is i ∼ a siltr hrce ieawv mltd,roughly amplitude, wave a like character oscillatory has r 1 1 ,i i, r 1 | 2 C ( dP dx e i 1 = ikr = C 1 e , √ − !.S s w em n()aecnt. n w vary two 2nd consts., are (4) in terms two 1st So 2!). 1 = 2 ikr |h + ( x | 2 detector | A | B + A | h {|h i i x i| ssaewt lt2closed, 2 slit with state is U e + | ikr 2 x A : = | | and B ih A 2 “interference” e − i B i| − 2 ) ikr 2 ~µ , | |h x + · 1 i x ) B | + |h {z B ∼ dP dx x detector h i| | x r C B 1 2 1 | B i| but = r ~µ 1 2 2 ih nmgei edis field magnetic in |h | cos C A x nefrneterms interference i | | ssaewt ohsisopened. slits both with state is x C k i } ( } i| r 1 2

− detector r 2 (5) ) ± µB r 1 − sallowed. is e ikr r : 2 . i /r i (4) (3) (6) ( r i For quantum spin-1/2 particle, since spin is quantized to point either parallel or antiparallel to B, only allowed values are ±µB.

Now if B is inhomogeneous there is a classical on a ~µ equal to F = −∇U = −µ(±∇B). So assume incoming beam in figure is mixture of spins k and anti-k to field (B k xˆ), experience in opposite directions. Can separate spins, reverse ∇B, recombine as shown. Spin eigenstates for B k xˆ are     1 1  1   1  χ1 = √ ; χ2 = √ , (7) 2 1 2 −1 where labels 1 and 2 corresponding to paths followed by spins in figure. Note the spin quantization axis isz ˆ as usual although we’ve taken B k xˆ. What happens if particle with “spin up”,   1 χ =   (8) 0 is injected into field gradient? Can write as linear combination       1 1 1 1 1 χ =   =   +   (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 χ1 or χ2, and χ just represents√ our ignorance? No, for when we recombine, spin will be up again, χ = (1/ 2)(χ1 + χ2). If it were “really” χ1, say, it would leave as mixture of spin up and down.

3 (Note for E& M purists: I’ve taken ∇B, B to point in same direction in figure, which could be arranged approximately, e.g. with a coil whose winding density varies along the axis direction. But of course to keep ∇ · B = 0 the system will generate some small transverse gradients as well. This doesn’t affect the argument, as you’ll be able to work out in prob. set. See also Griffiths, p. 181 et seq.)

Example 3: Ammonia molecule NH3

NH3 molecule is simple example of “2-level system”. Consists of triangle of H- and N- out of plane in minimum energy configuration. Since Hamilto- nian rotationally invariant, given fixed H-triangle there is no reason for N to be above rather than below, or v.v. In fact two states |topi and |boti must be degen- erate. Neither one is , however: rather than break symmetry, chooses symmetric mixture: 1 |0i = √ (|topi + |boti) (10) 2

and 1st is the asymmetric combination 1 |1i = √ (|topi − |boti) (11) 2 Here prob. ampl. for finding N atom at given position z relative to 3 H-atom plane plotted schematically for 4 states.

? N.B. |topi and |boti not energy eigenstates. If molecule is in, e.g. |topi at t = 0 it does not stay there, but “tunnels” into |boti with time. This phenomenon is , & very similar to 2-slit expt. Imagine we do x-ray scattering experiment off NH3 molecule as shown below.

4 If system is in state |topi, only one source for sph. If system is in ground state |0i, there are 2 scattered wave, so no interference in scattering pattern. waves which interfere, causing fringes on photographic plate. Example 4: Schr¨odinger’scat paradox We may be tempted to accept notion of molecule in superposition of 2 different configurations as mysteries of life at atomic scale, but harder to swallow similar im- plications at macroscopic scales. Famous Gedanken expt. propsed by Schr¨odinger: suppose at t = 0 box is filled with a) gun; b) atom in excited state c) cat; and d) device to detect when atom decays to grnd state and fire gun at cat. Atom is not in , therefore system is not, will evolve in time into admixture of state with excited atom and live cat |1, alivei and state with grnd. state atom and dead cat |0, deadi (Just as in NH3 case, where |topi evolves after some time into an admixture of |topi and |boti). 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 -sensitive TV cameras a and b along paths 1 and 2 corresponding to spin parallel and antiparallel tox ˆ.

If particle with spin alongx ˆ axis enters, it certainly is detected by camera a. If a particle with spin up (k zˆ) enters, according to rules, probability it’s detected by camera a is   µ ¶ 1 1 P = |hχ |χi|2 = | √1 √1   |2 = (12) a 1 2 2 0 2 Now when particle leaves the apparatus it is definitely k tox ˆ, not zˆ, since we know it went through arm 1, as only particles with spins k xˆ do. Act of measurement has changed spin state from χ to χ1. 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 |boti, there is certainly a scattering. If molecule is in state |topi certainly no scattering takes place. If the molecule is in ground state |0i, scattering is observed with 50% probability. If in given expt. no scattering is observed, molecule is in state |topi at end of observation. Thus starting from |0i, may happen that act of observation forced molecule into |topi, although no scattering takes place. Not just semantics: |topi is a higher energy state than |0i–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. Proposed “paradox” designed to prove not that qm was wrong, but that something was missing. Suppose particle in angular zero state at rest decays into two spin-1/2 particles, which must be in a spin , |ψi = (1/2)(| ↑↓i − | ↓↑i) to conserve ang. mom. Therefore as particles fly apart, no how far apart they are, each must be considered to be in mixed state of | ↑i & | ↓i! Now suppose one particle detected on Vulcan & found to be | ↑i (outcome had prob. 1/2). This collapses wave function instantaneously, such that when the particle is detected on Klingon home world it is in a state | ↓i with prob. 1. Measurement on 1st planet has instantaneously influenced measurement on 2nd =⇒ Copenhagen qm fundamentally nonlocal, apparently violates postulate of relativity! Copenhagen school response: in fact qm not acausal, doesn’t violate relativity, as no information or energy can be transferred as a result of collapse of wavefctn. Reason: observer on Vulcan can’t determine result of measurement beforehand.

7 1.3 Role of Observer

All examples: ψ changes in 2 ways. 1) Deterministic evolution according to H|ψi = ih∂¯ |ψi/∂t between observations, and 2) abrupt changes in state upon observation. 2nd change is not deterministic, but probabilistic. Clearly strange, discontinuous things happen in the observation process in Copen- hagen description. Why not account explicitly for role played by observer, try to describe both observer and experiment in deterministic way? von Neumann sug- gested idealized model system: measuring z component of spin. Before

measurement (frame 1), observer & expt. well-separated, can therefore describe to- tal state by specifying state of physicist & state of spin individually (|ψi = |Ai|ai). At later time t1(frame 2) physicist gets up close & personal with spin, must have |ψ(t1)i = Uˆ(t1)|ψi, where Uˆ is op. (State is no longer direct prod- uct!) She now moves away & records observation in notebook at t2 (frame 3). 0 0 State has evolved to Uˆ(t2)|ψi = |A i|a i, i.e. the act of observation has potentially altered both spin and physicist (again well-separated).

Suppose the spin initially in state of definite Sz, e.g., |ai = | ↑i. Suppose further: observer able to make measurement and leaves spin in up state (|a0i = | ↑i). Then final state is

Uˆ(t2)|ψi = |A+i| ↑i (13) where A+ represents the way in which the observer’s knowledge that the spin is up has altered her. So far, no problem. Now assume that spin is initially in linear superposition of up & down. Initial state vector then α|Ai| ↑i + β|Ai| ↓i (14)

By linearity principle, this must evolve at later time t2 to

|ψ(t2)i = Uˆ(t2)(α|Ai| ↑i + β|Ai| ↓i)

8 = Uˆ(t2)α|Ai| ↑i + Uˆ(t2)β|Ai| ↓i (15)

= α|A+i| ↑i + β|A−i| ↓i (16)

So after measurement physicist is in linear combination of states |A+i and |A−i, i.e. the notebook doesn’t contain a definite entry on the spin state. This is absurd, so seems to be impossible to arrange for purely deterministic quantum mechanics without concept of (see below, however). No need to require direct observation by a person (a bit anthropocentric!), sufficient to require collapse whenever microscopic system induces change in macroscopic object which can be described by quantum mechanics, e.g. detector of some kind. Hmmm... someone needs to read the detector, though...

1.4 “Resolution” of measurement paradoxes

Success of qm forces us, reluctantly, to believe that an NH3 molecule can be put in a superposition of different states, but it’s harder to swallow that a cat can be in such a lin. comb. Saw there is a logical inconsistency as well if we allow measuring device (“physicist” of sec. 1.3) can be put in superposition–how can measurement be completed? Some ways out: (no generally accepted answers!) 1. Copenhagen approach (most common). Relax requirement that every element of physical theory correspond to element of “reality”. Only goal of physical theory should be to systematize our knowledge, increase it, and make predic- tions which agree with experiments. Wave function ψ serves as device used in computation of probabilities of events to be recorded in macroscopic notebooks or macroscopic brains, and this is all we can hope to know. Collapse of wave function no problem: once measurement on microscopic sys- tem is made, we have new knowledge & this alters all probabilities for all subsequent measurements. Prescription which describes all microscopic qm: after measurement, start computing events with new wave function. Question of whether N atom in NH3 problem is “really” in two places at once in ground state is ill-posed–“at once” is an experience-laden term which is irrelevant to question of what happens in a measurement, which qm tells us, albeit proba- bilistically. Measurement process at macroscopic level: Schr¨odinger’scat, Wigner’s friend. Macroscopic system itself not really allowed to be in lin. comb. of distinct

9 states. We were sloppy when we described the measurement process, which actually occurs 1st time microscopic system interacts with macroscopic object. In case of S’s cat this was when atomic decay triggered gun. EPR “paradox” no problem: nonlocal influences do exist in nature, but are of a sort where no information is transferred, consistent with relativity. 2. Wigner approach • Linearity an approximation only valid at microscopic level–new rules must be found to describe macroscopic physics. • Application of human consciousness which constitutes measurement. Con- sciousness must be considered external to qm, accounted for in description of measurement process. 3. Many-worlds approach (Everett) Here idea is bizarre and sci-fi like. When physicist measures spin in mixed state, instead of being placed in mixed state herself, the universe forks into two copies of itself (you may call them “parallel” if you wish, `ala Star Trek). In 1st universe she measures | ↑i , in 2nd, | ↓i with probability 1. Two questions I don’t understand: 1) what happens to the amplitude factors α and β weighting the two pure states in the microscopic wave function? If |α| ¿ |β| is one universe less likely? 2) How does this work at the microscopic level? In the NH3 case we don’t want the universe to split into one copy with |topi and one with |boti. The real ground state (confirmed by x-ray expts.) is the mixture |0i. How does nature decide when to split and when not?

4. Hidden variables approach (Einstein, Bohm) Some other variables ζ are as- sumed to characterize system completely, in addition to wave fctn. ψ. No idea how to measure ζ =⇒ “hidden” variable. For example, EPR “paradox” now resolved by saying, 1st particle on Vulcan had spin | ↑i all along since its creation, and 2nd one had | ↓i all along. During one such decay, ζ might have one value (as determined by the hidden variables of the initial state & presumably some conservation laws), determining the spin of the particle on Vulcan to be | ↑i , etc. During another decay, it might have a different value leading to | ↓i on Vulcan. Local means ζ was set at the site of the decay, and the information is carried with travelling particles. Information then obviously travels at sublight speeds, no problem with causality. Bell’s Theorem

10 ? Bell: If local hidden variables theory exists, must satisfy Bell’s inequality (see discussion below, based on Griffiths p. 377-8). But QM predictions violate inequality =⇒ QM is not just incomplete, but wrong. Reverse implication: if QM is right (i.e., confirmed by all expts), no local hidden variable theory is allowed. Surprising further implication: qm inherently nonlocal (but not acausal, because no information can be transferred due to collapse of wave fctn.!) pf.: EPR expt., let alignment of 2 detectors be general (measure spin component alonga ˆ, ˆb on two planets. Each detector can only measure ±1 in units ofh/ ¯ 2. Product of measurement A(ˆa) on Vulcan and B(ˆb) on Klingon home world is ±1 since A, B = ±1. Note: 1) A does not depend on ˆb, etc. because we will hypothesize locality, i.e. just before Vulcan measurement is made, experimenter on Klingon home world may pick favorite orientation ˆb for his detector, such that signal with this information will never make it to Vulcan in time to influence outcome. 2) Only ifa ˆ = ˆb do we have A = −B and p ≡ A · B = −1 with 100% certainty.

Now suppose that given decay is characterized by value of hidden variable ζ. The value of the measurements on the two planets will now be assumed to depend not only on the detector orientation, but also on ζ, A = A(ˆa, ζ), B = B(ˆb, ζ). (Somehow ζ must arrange for antisymmetry of total wave fctn.!) Define average value of product of spins over many measurements to be Z P (ˆa, ˆb) = dζρ(ζ)A(ˆa, ζ)B(ˆb, ζ) (17) where ρ is arbitrary distribution fctn. for hidden variable. Now if detectors aligned, A and B must be perfectly anticorrelated, A(ˆa, ζ) = −B(ˆa, ζ). So can write Z P (ˆa, ˆb) = − dζρ(ζ)A(ˆa, ζ)A(ˆb, ζ) (18) so for any other directionc ˆ,

Z h i P (ˆa,ˆb) − P (ˆa, cˆ) = − dζρ(ζ) A(ˆa, ζ)A(ˆb, ζ) − A(ˆa, ζ)A(ˆc, ζ) (19) Z h i = − dζρ(ζ) 1 − A(ˆb, ζ)A(ˆc, ζ) A(ˆa, ζ)A(ˆb, ζ) (20) since A(ˆb, ζ)2 = 1. Note that since A = ±1 we have |A(ˆa, ζ)A(ˆb, ζ)| ≤ 1 and ρ(ζ)[1 − A(ˆa, ζ)A(ˆc, ζ)] ≥ 0, so

Z h i |P (ˆa, ˆb) − P (ˆa, cˆ)| ≤ dζρ(ζ) 1 − A(ˆb, ζ)A(ˆc, ζ) (21) = 1 + P (ˆb, cˆ) (22)

11 This is Bell’s inequality applicable to local hidden variable theories. Now show quantum mechanics gives examples incompatible with (21-22). First note qm =⇒ P (ˆa, ˆb) = −aˆ · ˆb. (Prove this on prob. set 1!) Example. If detector b is oriented perpendicular to detector a, although each measurement yields ±1, the average or expectation value of the ˆ product is zero. Write down a few trial sets√ of values fora ˆ k z and b k x to convince yourself. ◦ And if the detectors are 45 apart, P = −1/ 2. Apart from sign, similar√ to classical√ polarizers! But ifa ˆ k z and ˆb k x, withc ˆ at 45◦ between them, (21-22) says 1/ 2 ≤ 1 − 1/ 2, which isn’t true. Endnotes “Professor Wigner, are there any laws of nature which we cannot know?” —Anthony Zee, currently Institute of Theoretical Physics, Santa Barbara “I do not know of any.” —Eugene Wigner, formerly Princeton University, dec. 1985(?)

“... it is entirely possible that future generations will look back from the vantage point of a more sophisticated theory, and wonder how we could have been so gullible.” — Griffiths

References on measurement theory: 1. Bohm, David, Quantum Theory, Dover, NY, 1989. General discussion of measurement theory by adherent of hidden variables viewpoint.

2. Wigner, Eugene, Symmetries and Reflections, Indiana U. Press, Bloomington 1967. Essays on quantum physics including discussions of role of observer by one of founders of qm. 3. N.D. Mermin, Physics Today p. 38 (April 1985). Mermin writes often for Physics Today on the foundations of quantum theory, and he’s always worth reading. His summary of developments regarding hidden variables in Rev. Mod. Physics 65 (1993), p. 803 is probably the most up-to-date high-level review.

4. M. Jammer, The Philosophy of Quantum Mechanics, Wiley, NY 1974. 5. J.S. Bell, Rev. Mod. Phys. 38, 447 (1966). 6. J. Gribbin, In Search of Schr¨odinger’sCat and Schr¨odinger’sKittens and the Search for Reality, Little, Brown, 1984 and 1995, respectively. Lay account of measurement paradoxes leaning towards hidden variables interpre- tations.

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