  It is shown that a canonical time observable may be defined for any quantum
system having a discrete set of energy eigenvalues, thus significantly
generalising the known case of time observables for periodic quantum systems
(such as the harmonic oscillator). The general case requires the introduction
of almost-periodic probability operator measures (POMs), which allow the
expectation value of any almost-periodic function to be calculated. An entropic
uncertainty relation for energy and time is obtained which generalises the
known uncertainty relation for periodic quantum systems. While non-periodic
quantum systems with discrete energy spectra, such as hydrogen atoms, typically
make poor clocks that yield no more than 1 bit of time information, the
anisotropic oscillator provides an interesting exception. More generally, a
canonically conjugate observable may be defined for any Hermitian operator
having a discrete spectrum.
