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Functional Measure in Quantum Mechanics

Explore the probabilistic nature of measurement in quantum mechanics using a functional measure. Discuss the mathematical formulation and historical development of quantum mechanics. Analyze the relationship between wave functions, density matrices, and probability measures. Examine the emergence of classical systems from quantum phenomena.

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Functional Measure in Quantum Mechanics

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  1. On using a functional measure to capture the probabilistic character of measurement in quantum mechanics Erik Deumens UFIT Research Computing & Quantum Theory Project Dept. of Physics and Dept. of Chemistry University of Florida

  2. Some history • 1926 – Birth of quantum mechanics • The correct mathematical formulation! • Never contradicted by any finding since! • Heisenberg, Jordan, Born, Schrödinger, Dirac,… • 1900 – 1926 – today – Discussion about quantum mechanics • Arguments about quantum phenomena… • Invention of contradicting interpretations… • Planck, Einstein, Bohr,…

  3. Scientific method Phenomenology - Description Theory - Explanation 1687 – Newton – Law of force and law of gravity 1913 – Bohr – Atomic theory of elements and chemical bonding 2013 – Allaverdyan, Balian, Nieuwenhuizen – derivation of Born’s rule from Liouville-von Neumann equation • 1619 – Kepler – laws of planetary motion • 1869 – Mendeleev – Periodic table of elements • 1926 – Born – Probability rule

  4. Some philosophy

  5. The mystery of quantum mechanics

  6. Tomomura, Endo, Matsuda, Kawasaki, Exawa Amer. J. Phys, 57, p. 117 (1989)

  7. The math of mechanics

  8. The measurement of systems Classical – direct measurement Quantum – Born probability rule System state function ψ Detector state values r Measurement process is complex Born rule P(q)=|ψ(q)|2 gives probability for q -> r Probability is intrinsic • System state values q • Detector state values r • Measurement process is simple dynamics • Can be approximated by simple transfer of values q -> r • Probability is incidental and comes from inaccuracies and uncertainties

  9. A circular argument

  10. How does this work? • Quantum dynamics evolves wave functions ψ(q) • Interaction with measurement apparatus produces a value q • Value q is recorded from apparatus • clearly q is a property of the apparatus • Einstein-Bohr assumption • q is also a property of the system • How does q line up with the wave function ψ(q) ? • Unending controversy…

  11. Mathematics of mechanics Classical mechanics Quantum mechanics System ↔ ψ(q) in Hilbert space H Dynamics ↔ Schrödinger equation • System ↔ (q,p) in phase space M • Dynamics ↔ Newton equation

  12. Superposition and entanglement

  13. Schmidt decomposition and density matrix

  14. von Neumann density matrix • Gemenge – ensemble of systems • With statistical probability of being in one of several wave functions • Wave functions -> D • Reconstruction not unique: D -> wave functions

  15. Measure theory (mathematics) • Lebesgue measure • translation invariant • universal reference • Probability measures • Finite measure μ(M)=1 • Density matrix • Measure on subspaces of Hilbert space • Measures on ∞-dim spaces • No Lebesgue exists • Finite measures exist

  16. Probability in quantum mechanics Statistical interpretation Functional measure view Same No probability interpretation Deny Born rule Functional probability measure Flow induced by SE Probabilistic nature of QM lies in the measure Same as in classical stat. mech. • Wave functions – Schrödinger equation • Density matrix – Liouville-von-Neumann equation • Probabilistic nature of wave function cannot be separated from that of density matrix

  17. Mathematics of probability and statistics Classical statistics Quantum statistics Can have functional measure, But no density w.r.t. Lebesgue Probability measure μ on H • Probability density ρ(q,p) • Probability measure μ on M

  18. Statistical state in quantum mechanics

  19. Gaussian measures

  20. Observable A, self-adjoint operator

  21. Measurement theory (physics) • Wave functions are the theoretical foundation • Not directly accessible to experiment • System being observed is in a narrow statistical state • Carefully prepared • All system wave functions with weight are very similar, experimentally indistinguishable • Detector is always in a broad statistical state • Impossible to prepare precisely – has Avogadro number of degrees of freedom 1023 • Many detector wave functions with weight are very different, with different macroscopic outcomes for the measurement process

  22. Detector statistics and dynamics • Electron statistical state – sharp – one wave function • Detector statistical state – broad – very many wave functions – many superpositions • Electron interacts with all mini detectors – all in different wave functions – some: fly through – some: excite few molecules, not enough to see – one: excites 500 molecules and sends 500 photons • Photons are captured – fly through optical fiber – excite secondary electrons – current detected – write bit in RAM – display dot on screen

  23. Emergence of classical systems • One mini detector has 500 molecules with 30 atoms – 15,000 degrees of freedom • Variance of the collective variable that tracks excitation of mini detector is square root of N=15,000, which is 122 – two orders less than one molecule • That makes the collective variable effectively dispersionless = classical • We define this to be a q-classical variable • A set of q-classical variables can form a q-classical system • By Ehrenfest theorem, q-classical evolution follows Newton equation

  24. Where do probabilities come from? • From the statistical state of the quantum system! • It gives many random wave functions for every macroscopic system – for all mini detectors in an experimental apparatus • The mini-detector collective variable tracking excitation of 500 photons follows classical dynamics • The mini-detector that gets excited is the random selection of the statistical state evolution • The measurement then proceeds classically – the result is recorded

  25. From deterministic to probabilistic • Wave functions are fundamental – functions • Deterministic dynamics by Schrödinger equation • Probability distribution on Hilbert space of wave functions • Evolution of quantum statistical state • Derive q-classical variables and systems – values • Deterministic dynamics by Newton equation follows from Ehrenfest theorem → Recover classical mechanics • Probability distribution on q-classical phase space • Liouville equation for evolution of q-classical statistical state → Recover classical statistical mechanics → Describe probabilistic quantum measurement process

  26. An analogy: photographs and movies Classical mechanics: values Quantum mechanics: functions ANALOGY 2 hour movie 24 frames per second 172,800 = 24 x 7,200 frames smoothly aligned sequence MATHEMATICS Schrödinger equation ANALOGY • 1 photograph • 1 frame • static MATHEMATICS Newton equation

  27. Questions and discussion Slides and paper at https://people.clas.ufl.edu/deumens

  28. Wave function collapse in space • Position is an operator • Time is a parameter/coordinate • That is inconsistent • Attempts to make time an operator have been made • Make position a coordinate of events • Same as time • Does work well • Even in non-relativistic quantum mechanics

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