Measurement Theory - Quantum Mechanics II | PHY 4605, Study notes of Physics

Download Measurement Theory - Quantum Mechanics II | PHY 4605 and more Study notes Physics in PDF only on Docsity! 1 Measurement Theory For background to this section, reread Griffiths 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 measurement of Q in a physical system represented by ψ = ∑ n anψn yielded qn with probability |an|2 (Q̂ψn ≡ 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 Schrödinger eqn, which is linear diffeq., superposition principle supposed to hold: if |ψ〉 and |φ〉 are allowed states of physical system, so is combination α|ψ〉+ β|φ〉. The state vectors evolve according to S.’s eqn, ih̄ ∂ ∂t |ψ(t)〉 = H|ψ(t)〉, (1) so since both |ψ〉 and |φ〉 are solns so is α|ψ(t)〉+ β|φ(t)〉. 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 figure is probability distribution of particle positions x, e.g. dPA dx = |〈x|A〉|2 (2) for state |A〉, etc. (Recall |x〉 is state with particle definitely at position x.) “Copen- hagen” QM says we don’t add probabilities in |A〉 and |B〉 to get probability in |C〉, but rather prob. amplitudes 1 d e t e c t o r d e t e c t o r d e t e c t o r Figure 1: |A〉 is state with slit 1 closed, |B〉 is state with slit 2 closed, |C〉 is state with both slits opened. |C〉 = 1√ 2 (|A〉+ |B〉), dPC dx = |〈x|C〉|2 (3) So dPC/dx contains not only |〈x|A〉|2 and |〈x|B〉|2 but interference terms: dPC dx = {|〈x|A〉|2 + |〈x|B〉|2 + 〈x|A〉〈B|x〉+ 〈x|B〉〈A|x〉︸ ︷︷ ︸} (4) “interference” Why do these terms give rise to interference pattern? Because the wave function ψA(x) ≡ 〈x|A〉 has oscillatory character like a wave amplitude, roughly eikri/ri (ri measured from slit i, i = 1, 2!). So 1st two terms in (4) are consts., 2nd two vary as ∼ 1 r1 1 r2 (eikr1e−ikr2 + eikr2e−ikr1) ∼ 1 r1 1 r2 cos k(r1 − r2) (5) i.e., classical interference pattern depending only on path difference r1 − r2. ?Point: linear superposition principle crucial to understanding of this expt. Example 2: Stern-Gerlach apparatus: simple device for spatial separation of different-spin particles. For illustration con- sider neutral spin-1/2 particles, e.g. neutrons, place in inhomogeneous magnetic field B(r). Recall energy of spin-1/2 with moment ~µ in magnetic field is U = −~µ ·B (6) Compare energetics with classical case, where any energy between ±µB is allowed. 2 If system is in state |top〉, only one source for sph. wave, so no interference in scattering pattern. If system is in ground state |0〉, there are 2 scattered waves which interfere, causing fringes on photographic plate. Example 4: Schrödinger’s cat 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 Schrödinger: 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 stationary state, therefore system is not, will evolve in time into admixture of state with excited atom and live cat |1, alive〉 and state with grnd. state atom and dead cat |0, dead〉 (Just as in NH3 case, where |top〉 evolves after some time into an admixture of |top〉 and |bot〉). 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 to x̂. If particle with spin along x̂ axis enters, it certainly is detected by camera a. If a particle with spin up (‖ ẑ) enters, according to rules, probability it’s detected by camera a is Pa = |〈χ1|χ〉|2 = | ( 1√ 2 1√ 2 )   1 0   |2 = 1 2 (12) Now when particle leaves the apparatus it is definitely ‖ to x̂, not ẑ, since we know it went through arm 1, as only particles with spins ‖ x̂ 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 |bot〉, there is certainly a scattering. If molecule is in state |top〉 certainly no scattering takes place. If the molecule is in ground state |0〉, scattering is observed with 50% probability. If in given expt. no scattering is observed, molecule is in state |top〉 at end of observation. Thus starting from |0〉, may happen that act of observation forced molecule into |top〉, although no scattering takes place. Not just semantics: |top〉 is a higher energy state than |0〉–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 momentum zero state at rest decays into two spin-1/2 particles, which must be in a spin singlet state, |ψ〉 = (1/2)(| ↑↓〉 − | ↓↑〉) to conserve ang. mom. Therefore as particles fly apart, no matter how far apart they are, each must be considered to be in mixed state of | ↑〉 & | ↓〉! Now suppose one particle detected on Vulcan & found to be | ↑〉 (outcome had prob. 1/2). This collapses wave function instantaneously, such that when the second particle is detected on Klingon home world it is in a state | ↓〉 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 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, à la Star Trek). In 1st universe she measures | ↑〉 , in 2nd, | ↓〉 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 |top〉 and one with |bot〉. The real ground state (confirmed by x-ray expts.) is the mixture |0〉. 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 | ↑〉 all along since its creation, and 2nd one had | ↓〉 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 | ↑〉 , etc. During another decay, it might have a different value leading to | ↓〉 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 along â, b̂ on two planets. Each detector can only measure ±1 in units of h̄/2. Product of measurement A(â) on Vulcan and B(b̂) on Klingon home world is ±1 since A, B = ±1. Note: 1) A does not depend on b̂, etc. because we will hypothesize locality, i.e. just before Vulcan measurement is made, experimenter on Klingon home world may pick favorite orientation b̂ for his detector, such that signal with this information will never make it to Vulcan in time to influence outcome. 2) Only if â = b̂ 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(â, ζ), B = B(b̂, ζ). (Somehow ζ must arrange for antisymmetry of total wave fctn.!) Define average value of product of spins over many measurements to be P (â, b̂) = ∫ dζρ(ζ)A(â, ζ)B(b̂, ζ) (17) where ρ is arbitrary distribution fctn. for hidden variable. Now if detectors aligned, A and B must be perfectly anticorrelated, A(â, ζ) = −B(â, ζ). So can write P (â, b̂) = − ∫ dζρ(ζ)A(â, ζ)A(b̂, ζ) (18) so for any other direction ĉ, P (â, b̂)− P (â, ĉ) = − ∫ dζρ(ζ) [ A(â, ζ)A(b̂, ζ)− A(â, ζ)A(ĉ, ζ) ] (19) = − ∫ dζρ(ζ) [ 1− A(b̂, ζ)A(ĉ, ζ) ] A(â, ζ)A(b̂, ζ) (20) since A(b̂, ζ)2 = 1. Note that since A = ±1 we have |A(â, ζ)A(b̂, ζ)| ≤ 1 and ρ(ζ)[1 − A(â, ζ)A(ĉ, ζ)] ≥ 0, so |P (â, b̂)− P (â, ĉ)| ≤ ∫ dζρ(ζ) [ 1− A(b̂, ζ)A(ĉ, ζ) ] (21) = 1 + P (b̂, ĉ) (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 (â, b̂) = −â · b̂. (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 product is zero. Write down a few trial sets of values for â ‖ z and b̂ ‖ x to convince yourself. And if the detectors are 45◦ apart, P = −1/√2. Apart from sign, similar to classical polarizers! But if â ‖ z and b̂ ‖ x, with ĉ 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 Schrödinger’s Cat and Schrödinger’s Kittens and the Search for Reality, Little, Brown, 1984 and 1995, respectively. Lay account of measurement paradoxes leaning towards hidden variables interpre- tations. 12