[Paper Review] Analysis of the Stern-Gerlach Measurement

Anu Venugopalan, Deepak Kumar+1 more|ArXiv.org|Jan 27, 1995

Stochastic processes and statistical mechanics3 references3 citations

TL;DR

This paper proposes a dynamical model for wave function collapse in quantum measurement by modeling the interaction of a spin-1/2 particle with a macroscopic quantum apparatus coupled to a dissipative environment. Decoherence from environmental coupling diagonalizes the apparatus density matrix, enabling a momentum measurement to correlate with spin states in a Stern-Gerlach setup, thus reproducing quantum probabilities through classical-like outcomes while preserving quantum principles.

ABSTRACT

A dynamical model for the collapse of the wave function in a quantum measurement process is proposed by considering the interaction of a quantum system (spin-1/2) with a macroscopic quantum apparatus interacting with an environment in a dissipative manner. The dissipative interaction leads to decoherence in the superposition states of the apparatus, making its behaviour classical in the sense that the density matrix becomes diagonal with time. Since the apparatus is also interacting with the system, the probabilities of the diagonal density matrix are determined by the state vector of the system. We consider a Stern-Gerlach type model, where a spin- 1/2 particle is in an inhomogeneous magnetic field, the whole set up being in contact with a large environment. Here we find that the density matrix of the combined system and apparatus becomes diagonal and the momentum of the particle becomes correlated with a spin operator, selected by the choice of the system-apparatus interaction. This allows for a measurement of spin via a momentum measurement on the particle with associated probabilities in accordance with quantum principles.

Motivation & Objective

• To develop a dynamical model explaining wave function collapse in quantum measurements.
• To address the measurement problem by incorporating environmental decoherence in a macroscopic apparatus.
• To demonstrate how classical-like outcomes emerge from quantum dynamics in a Stern-Gerlach setup.
• To show that measurement probabilities align with quantum mechanics through apparatus-state diagonalization.
• To unify the role of system-apparatus interaction and environmental dissipation in achieving definite measurement records.

Proposed method

• Model the spin-1/2 particle interacting with an inhomogeneous magnetic field as a quantum measurement setup.
• Introduce a macroscopic quantum apparatus coupled to the particle and to a large environment with dissipative interactions.
• Use a master equation approach to describe the time evolution of the combined system-apparatus density matrix under environmental influence.
• Show that environmental dissipation causes decoherence, diagonalizing the apparatus density matrix over time.
• Demonstrate that the diagonal elements of the density matrix correspond to probabilities dictated by the system's state vector.
• Establish a momentum-spin correlation via the system-apparatus interaction Hamiltonian, enabling spin measurement through momentum detection.

Experimental results

Research questions

• RQ1 How can wave function collapse be dynamically explained in a quantum measurement process?
• RQ2 What role does environmental dissipation play in inducing classical behavior in a macroscopic measuring apparatus?
• RQ3 How does the interaction between a quantum system and a macroscopic apparatus lead to definite measurement outcomes?
• RQ4 Can the probabilities of measurement outcomes be derived from the dynamics of the system-apparatus-environment tripartite system?
• RQ5 How is the correlation between particle momentum and spin state established in a Stern-Gerlach experiment under decoherence?

Key findings

• The density matrix of the combined system and apparatus becomes diagonal over time due to environmental dissipation, indicating decoherence.
• The diagonal elements of the apparatus density matrix correspond to probabilities determined by the initial state vector of the spin-1/2 system.
• A correlation between the particle's momentum and a specific spin operator emerges, selected by the form of the system-apparatus interaction Hamiltonian.
• The measurement outcome—determined by momentum detection—yields probabilities consistent with standard quantum mechanics.
• The model demonstrates that classical behavior in measurement outcomes arises dynamically from quantum interactions with the environment.
• The framework provides a dynamical explanation for the emergence of definite outcomes in quantum measurements without postulating collapse.