Time operators in two-dimensional Hilbert space

Abstract

We construct infinitely many possible time operators, wherein they form a canonical pair with the Hamiltonian only in a proper subspace of the two-dimensional Hilbert space, and for a specific set of times of total measure zero. When near this set of times, the expectation value of the time operator depends linearly on time. We obtain their uncertainty relation near these times, and show that we can choose a system such that the uncertainty in measuring time is negligible. This provides a way of constructing a quantum clock using two-level systems. As an example, we show that for a Larmor clock, we can measure the spin to access time only for times near the Larmor period. This shows how quantum mechanics limits our ability in measuring time.