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Solvable Model for the Quantum Measurement Process

Solvable Model for the Quantum Measurement Process. Armen E. Allahverdyan Roger Balian Theo M. Nieuwenhuizen. Academia Sinica Taipei, June 26, 2004. Setup. The battlefield. The model: system S + apparatus A S=spin-½ A = M + B = magnet + bath.

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Solvable Model for the Quantum Measurement Process

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  1. Solvable Model for theQuantum Measurement Process Armen E. Allahverdyan Roger Balian Theo M. Nieuwenhuizen Academia Sinica Taipei, June 26, 2004

  2. Setup The battlefield The model: system S + apparatus A S=spin-½ A = M + B = magnet + bath Classical measurement Statistical interpretation of QM Selection of collapse basis & collapse Registration of the Q-measurement Post measurement & the Born rule Summary

  3. The battlefield Q-measurement is only contact of QM and experiment Interpretations of QM must be compatible with Q-meas. But no solvable models with enough relevant physics Interpretations Copenhagen: each system has its own wave function of QM multi-universe picture: no collapse but branching (Everett) mind-body problem: observation finishes measurement (Wigner) non-linear extensions of QM needed for collapse: GRW wave function is state of knowledge, state of belief consistent histories Bohmian, Nelsonian QM statistical interpretation of QM

  4. The Hamiltonian Test system: spin ½, no dynamics during measurement: System-Apparatus Apparatus=magnet+bath Magnet: N spins ½, with equal coupling J/4N^3 between all quartets Bath: standard harmonic oscillator bath: each component of each spin couples to its own set of harmonic oscillators

  5. Bath Hamiltonian

  6. Initial density matrix Test system: arbitrary density matrix Magnet: N spins ½, starts as paramagnet (mixed state) Bath: Gibbs state (mixed state) Von Neuman eqn: Initial density matrix -> final density matrix

  7. Intermezzo: Classical measurement of classical Ising spin Classically: only eigenvalues show up: classical statistical physics Measure a spin s_z=+/- 1 with an apparatus of magnet and a bath Dynamics m Free energy F=U-TS: minima are stable states m = tanh h

  8. Statistical interpretation of QM Copenhagen: the wavefunction is the most complete description of the system Statistical interpretation: a density matrix (mixed or pure) describes an ensemble of systems Stern-Gerlach expt: ensemble of particles in upper beam described by |up> Q-measurement describes ensemble of measurements on ensemble of systems

  9. Selection of collapse basis What selects collapse basis: The interaction Hamiltonian Trace out Apparatus (Magnet+Bath) Diagonal terms of r(t) conserved Off-diagonal terms endangered -> disappearence of Schrodinger cats

  10. Fate of Schrodinger cats Consider off-diagonal terms of Initial step in collapse: effect of interaction Hamiltonian only (bath & spin-spin interactions not yet relevant) Cat hides itself after Bath suppresses its returns

  11. Complete solution Mean field Ansatz: Solution: Result: decay of off-diagonal terms confirmed diagonal terms go exactly as in classical setup

  12. Post-measurement state Density matrix: - maximal correlation between S and A - no cat-terms Born rule from classical interpretation

  13. Summary Solution of measurement process in model of apparatus=magnet+bath Apparatus initially in metastable state (mixture) Collapse (vanishing Schrodinger-cats) is physical process, takes finite but short time Collapse basis determined by interaction Hamiltonian Measurement in two steps: Integration of quantum and classical measurements Born rule explained via classical interpretation of pointer readings Observation of outcomes of measurements is irrelevant Solution gives probabilities for outcomes of experiments: system in collapsed state + apparatus in pointer state Statistical interpretation: QM describes ensembles, not single systems Quantum Mechanics is a theory that describes statistics of outcomes of experiments

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