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Quantum Clocks

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Time in quantum mechanics has duly received a lot of attention over the years. Perfect clocks which can turn on/off a particular interaction at a precise time that have been proposed only exist in infinite dimensions and have unphysical Hamiltonians (their spectrum is unbounded from below). It was this observation which led many to conclude that an operator for time cannot exist in quantum mechanics. Here, we prove rigorous results about the accuracy of finite dimensional clocks and show that they can well approximate their infinite dimensional counterparts under the right conditions.