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