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Solvable Model for the Quantum Measurement Process

This paper presents a solvable model for the quantum measurement process, focusing on the system-apparatus interaction. It discusses classical measurement, statistical interpretation of quantum mechanics, collapse basis selection, and the Born rule. The model includes a detailed setup involving a spin-½ system, magnet, and bath, showcasing the dynamics during measurement. Emphasis is placed on the statistical interpretation of QM, the collapse of quantum states, and the post-measurement state. The solutions provided offer insights into the physical process of measurement and the probabilistic outcomes of experiments.

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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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