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Projection operator method in quantum-mechanical many-particle system

正義 中村

Published Jan 1, 1983• 0 citations
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Projection operator method in quantum-mechanical many-particle system

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Aiming on the preparation of a given physical system in a particular quantum state, we can follow one of several standard approaches. The most straightforward one is to expose the system to a specific coherent evolution that will deterministically bring it into the desired final state. However, in case of microscopic few-particle systems this approach requires very good control of physical parameters at the single particle level. Another method of the state production is based on projective quantum measurements of the system: Quantum Mechanics postulates that the system is randomly projected onto one of the eigenstates of the measurement operator M. By a proper choice of M, the system can end up in the desired state after the measurement. Being less demanding this method is however inherently non-deterministic. To steer the system towards the target state and thus to overcome the randomness of the measurement, we propose to use an active feedback on the system while the measurement is being performed.

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The problem of the quantization of an elementary model with the second class constraints, O(n) rigid rotator model, is investigated by using the Dirac method, the projection operator method (POM) and the nonlinear quantum mechanics (NLQM). It is proved that the quantum effects, which are equivalent to those of NLQM, appear in the equation of motion of POM. Furthermore, it is shown that POM as well as NLQM preserves the isometric symmetry, which the system possesses in the classical level, through the quantization process.

Progress of Theoretical Physics•Jan 1, 1988•2 citations
Aiming on the preparation of a given physical system in a particular quantum state, we can follow one of several standard approaches. The most straightforward one is to expose the system to a specific coherent evolution that will deterministically bring it into the desired final state. However, in case of microscopic few-particle systems this approach requires very good control of physical parameters at the single particle level. Another method of the state production is based on projective quantum measurements of the system: Quantum Mechanics postulates that the system is randomly projected onto one of the eigenstates of the measurement operator M. By a proper choice of M, the system can end up in the desired state after the measurement. Being less demanding this method is however inherently non-deterministic. To steer the system towards the target state and thus to overcome the randomness of the measurement, we propose to use an active feedback on the system while the measurement is being performed.
The problem of the quantization of an elementary model with the second class constraints, O(n) rigid rotator model, is investigated by using the Dirac method, the projection operator method (POM) and the nonlinear quantum mechanics (NLQM). It is proved that the quantum effects, which are equivalent to those of NLQM, appear in the equation of motion of POM. Furthermore, it is shown that POM as well as NLQM preserves the isometric symmetry, which the system possesses in the classical level, through the quantization process.