Solvable Model for theQuantum Measurement Process
Armen E. Allahverdyan
Roger Balian
Theo M. Nieuwenhuizen
Academia SinicaTaipei, June 26, 2004
Setup
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
Summary
Post measurement & the Born rule
The battlefield
The battlefieldQ-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
The Hamiltonian ASAS HHHH
0SHTest system: spin ½, no dynamics during measurement:
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
System-Apparatus
4^)4/( mNJHM
Apparatus=magnet+bath
Bath Hamiltonian
Initial density matrix
Von Neuman eqn: Initial density matrix -> final density matrix
Test system: arbitrary density matrix
Magnet: N spins ½, starts as paramagnet (mixed state)
Bath: Gibbs state (mixed state)
Classically: only eigenvalues show up: classical statistical physics
Measure a spin s_z=+/- 1 with an apparatus of magnet and a bath
1,1 z
Intermezzo: Classical measurement of classical Ising spin
Dynamics
Free energy F=U-TS: minima are stable states m = tanh h
m
Statistical interpretation of QM
Q-measurement describes ensemble of measurements on ensemble of systems
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>
Copenhagen: the wavefunction is the most complete description of the system
Selection of collapse basisWhat 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
Fate of Schrodinger catsConsider off-diagonal terms of
Cat hides itself after
Bath suppresses its returns
Initial step in collapse: effect of interaction Hamiltonian only (bath & spin-spin interactions not yet relevant)
Complete solution
Mean field Ansatz:
Solution:
Result: decay of off-diagonal terms confirmed diagonal terms go exactly as in classical setup
Post-measurement stateDensity matrix: - maximal correlation between S and A - no cat-terms
Born rule from classical interpretation
SummarySolution of measurement process in model of apparatus=magnet+bathApparatus 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 measurementsBorn rule explained via classical interpretation of pointer readings
Observation of outcomes of measurements is irrelevant
Quantum Mechanics is a theory that describes statistics of outcomes of experiments
Statistical interpretation: QM describes ensembles, not single systems
Solution gives probabilities for outcomes of experiments: system in collapsed state + apparatus in pointer state
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