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The quantum measurement process in an exactly solvable model
Statistical Mechanics · Submitted 13 Aug 2004
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The von Neumann collapse and the Born rule are derived rather than postulated.
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The abstract, sentence by sentence
An exactly solvable model for a quantum measurement is discussed which is governed by hamiltonian quantum dynamics. The $z$-component $\hat s_z$ of a spin-1/2 is measured with an apparatus, which itself consists of magnet coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state according to the sign of $s_z$ of the tested spin. The quantum measurement goes in two steps. On a timescale $1/\sqrt{N}$ the off-diagonal elements of the spin's density matrix vanish due to a unitary evolution of the tested spin and the $N$ apparatus spins; on a larger but still short timescale this is made definite by the bath. Then the system is in a `classical' state, having a diagonal density matrix. The registration of that state is a quantum process which can already be understood from classical statistical mechanics. The von Neumann collapse and the Born rule are derived rather than postulated.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | An exactly solvable model for a quantum measurement is discussed which is governed by hamiltonian quantum dynamics. | contribution 0.91 | 0.72 | 0.06 | asserted 0.99 |
| s1 | The $z$-component $\hat s_z$ of a spin-1/2 is measured with an apparatus, which itself consists of magnet coupled to a bath. | method 0.94 | 0.53 | 0.03 | asserted 1.00 |
| s2 | The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. | method 0.97 | 0.39 | 0.03 | asserted 1.00 |
| s3 | Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state according to the sign of $s_z$ of the tested spin. | result 0.60 | 0.66 | 0.13 | asserted 1.00 |
| s4 | The quantum measurement goes in two steps. | result 0.61 | 0.60 | 0.05 | asserted 1.00 |
| s5 | On a timescale $1/\sqrt{N}$ the off-diagonal elements of the spin's density matrix vanish due to a unitary evolution of the tested spin and the $N$ apparatus spins; on a larger but still short timescale this is made definite by the bath. | result 0.84 | 0.80 | 0.19 | asserted 1.00 |
| s6 | Then the system is in a `classical' state, having a diagonal density matrix. | result 0.87 | 0.76 | 0.11 | asserted 1.00 |
| s7 | The registration of that state is a quantum process which can already be understood from classical statistical mechanics. | interpretation 0.92 | 0.71 | 0.14 | asserted 0.92 |
| s8 | The von Neumann collapse and the Born rule are derived rather than postulated. | result 0.82 | 0.89 | 0.74 | asserted 1.00 |
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Original abstract
An exactly solvable model for a quantum measurement is discussed which is governed by hamiltonian quantum dynamics. The $z$-component $\hat s_z$ of a spin-1/2 is measured with an apparatus, which itself consists of magnet coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state according to the sign of $s_z$ of the tested spin. The quantum measurement goes in two steps. On a timescale $1/\sqrt{N}$ the off-diagonal elements of the spin's density matrix vanish due to a unitary evolution of the tested spin and the $N$ apparatus spins; on a larger but still short timescale this is made definite by the bath. Then the system is in a `classical' state, having a diagonal density matrix. The registration of that state is a quantum process which can already be understood from classical statistical mechanics. The von Neumann collapse and the Born rule are derived rather than postulated.
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