Periodic Measurement of A Quantum System
Consider a system in a state A, which is the eigenstate of some measurement operator. Say the system under free time evolution will decay with a certain probability into state B. If measurements are made periodically, with some finite interval between each one, at each measurement, the wave function collapses to an eigenstate of the measurement operator. Between the measurements, the system evolves away from this eigenstate into a superposition state of the states A and B. When the superposition state is measured, it will again collapse, either back into state A as in the first measurement, or away into state B. However, its probability of collapsing into state B, after a very short amount of time t, is proportional to t², since probabilities are proportional to squared amplitudes, and amplitudes behave linearly. Thus, in the limit of a large number of short intervals, with a measurement at the end of every interval, the probability of making the transition to B goes to zero.
According to decoherence theory, the collapse of the wave function is not a discrete, instantaneous event. A "measurement" is equivalent to strongly coupling the quantum system to the noisy thermal environment for a brief period of time, and continuous strong coupling is equivalent to frequent "measurement". The time it takes for the wave function to "collapse" is related to the decoherence time of the system when coupled to the environment. The stronger the coupling is, and the shorter the decoherence time, the faster it will collapse. So in the decoherence picture, a perfect implementation of the quantum Zeno effect corresponds to the limit where a quantum system is continuously coupled to the environment, and where that coupling is infinitely strong, and where the "environment" is an infinitely large source of thermal randomness.
Read more about this topic: Quantum Zeno Effect
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