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Physics
Quantum Physics
General Questions Related to Quantum Measurement
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[QUOTE="Denis, post: 5758935, member: 620146"] What is named "measurement" in many QT introductions, and described by self-adjoint operators, or projector-valued measures (PVM), is in fact only a very special type of measurement, which can be used also for state preparation - the state after the measurement is defined by the measurement result. There is another notion of measurement, which does not care at all about what happens after this, thus, cannot be used for state preparation, only gets results. The object which is measured may be even destroyed. Such measurements are described by a positive-operator valued measure (POVM). Mathematically, it appears that there is nonetheless not much new, because a POVM appears to be a PVM on some larger space. The most interesting example is imprecise common measurement of position and momentum. It appears to reduce to the scheme that you use some additional test particle, which with some approximation is at rest near the point ##q_t=0##, and measure the operators ##p+p_t, q-q_t##, which commute, as approximations for p and q. This has rather beautiful mathematics if you use the ground state of a harmonic oscillator for the test particle, because this gives holomorph representations of the commutation relations. But this does not give any information about the resulting state. So, if you want such information, you have no other choice than to model the imprecise measurement by some interaction and to solve the Schroedinger equation for the whole process. The result will, then, depend on the initial wave function too (different from the standard measurement, where the initial wave function defines only the probabilities of the possible outcomes, but not the resulting state for a given value of the outcome). And will probably depend on a lot of details about the interaction and the measurement device. [/QUOTE]
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General Questions Related to Quantum Measurement
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