Stanford APPPHYS 387 - Chapter 2 - Basic Concepts of the Quantum Theory (II): Projection Postulate and Symmetrization Postulate (13 pages)

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Chapter 2 - Basic Concepts of the Quantum Theory (II): Projection Postulate and Symmetrization Postulate



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Chapter 2 - Basic Concepts of the Quantum Theory (II): Projection Postulate and Symmetrization Postulate

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Lecture Notes


Pages:
13
School:
Stanford University
Course:
Appphys 387 - Quantum Optics and Measurements

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Chapter 2 Basic Concepts of the Quantum Theory II Projection Postulate and Symmetrization Postulate 2 1 2 1 1 Quantum measurement Open system vs closed system reservoir dissipation fluctuation unknown force Fex t quantum probe readout quantum system Figure 2 1 A theoretical model for quantum measurement process If a quantum system used for the readout of an unknown force couples to its reservoirs strongly as shown in Fig 2 1 the dissipation process restores the steady state of the quantum system before the next quantum measurement will be attempted In such a case we can say an ensemble of identical quantum systems is prepared from a single physical system and the standard probability interpretation applies On the other hand if a quantum system is well decoupled from reservoirs the state of the quantum system at a time of the next quantum measurement is governed by the results of previous measurements We must develop a new theory for how a quantum system evolves with the simultaneous actions of an unknown external force and quantum measurements 1 2 1 2 Exact measurement An exact quantum measurement with no measurement error is fully characterized by the three questions 1 What is a measurement result It is one of the eigenvalues of a measured observable q defined by q qn i qn qn i 2 1 2 What is the probability of finding a specific result It is given by the q distribution of the state P qn Tr qn ihqn s 2 2 where qn ihqn is a projection X operator If the system consists of a single system p ih mixed state the above formula is reduced s ih pure state or to 2 hq Xn i pure state p qn p hqn i 2 mixed state 2 3 If the system consists of many sub systems H s H 1 H 2 we need to take the trace operations over all sub systems 3 What is the post measurement state It is given by 1 s qn qn ihqn s qn ihqn p qn 2 4 where qn is a measurement result If the system consists of a single system the above formula is reduced to a trivial result s qn qn ihqn This is the von Neumann s recipe



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