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Abstract: It is shown that a canonical time observable may be defined for any quantumsystem having a discrete set of energy eigenvalues, thus significantlygeneralising the known case of time observables for periodic quantum systemssuch as the harmonic oscillator. The general case requires the introductionof almost-periodic probability operator measures POMs, which allow theexpectation value of any almost-periodic function to be calculated. An entropicuncertainty relation for energy and time is obtained which generalises theknown uncertainty relation for periodic quantum systems. While non-periodicquantum systems with discrete energy spectra, such as hydrogen atoms, typicallymake poor clocks that yield no more than 1 bit of time information, theanisotropic oscillator provides an interesting exception. More generally, acanonically conjugate observable may be defined for any Hermitian operatorhaving a discrete spectrum.



Author: Michael J. W. Hall

Source: https://arxiv.org/







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