hw3a2 - PHYS 455 Answers to Homework III 1. Quantum Zeno...

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Unformatted text preview: PHYS 455 Answers to Homework III 1. Quantum Zeno Effect is the quantum analog of the Zeno paradox. One of many alternative versions of the paradox is summarized by the statement “A watched pot never boils”. It is possible to claim that the quantum analog of the paradox is more convincing than the classical ones. In this problem, we will see on a 2-level system what the strange feature of the paradox is. Consider a two-level system that we may call as a “pot”. We will use the 2 × 1 matrix representation. Let us identify the two states in the standard basis as the “cold” state and the “boiled” state. φ cold = • 1 ‚ , φ boiled = • 1 ‚ . Let the time development operator for the pot be given as U ( t 1 + Δ t,t 1 ) = • cos ω Δ t- sin ω Δ t sin ω Δ t cos ω Δ t ‚ . This might me considered as describing a “heating” process as it transforms the “cold” state φ cold into the “boiled” state φ boiled in a certain time. But, of course, this is not a perfect heating as the same process will convert the boiled state into the cold state eventually. But, in a small 2-level system, where there is not much “phase space”, these types of behavior is expected. Suppose that at the initial time t = 0, the pot is prepared in the cold state. ψ ( t = 0) = φ cold = • 1 ‚ ....
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hw3a2 - PHYS 455 Answers to Homework III 1. Quantum Zeno...

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