A number-phase uncertainty relation connecting the excitation number of an oscillator to the phase angle is formulated in terms of S and C operators. The new expression for the so-called coherent states is evaluated.
The known approaches of number-phase problem (for a quantum oscillator) are mutually contradictory. All of them are subsequent in respect with the Robertson-Schrödinger uncertainty relation (RSUR). In oposition here it is proposed a new approacch aimed to be aboriginal as regard RSRUR. From the new perspective the Dirac's operators for vibrational number and phase appear as correct mathematical tools while the alluded problem receives a natural solution. PACS codes: 03.65.-w, 03.65.Ca, 03.65.Fd, 03.65.Ta Keywords: quantum oscillator, number and phase, uncertainty relations.
Recently a new uncertainty relation was found as an alternative to a number-phase uncertainty relation for a harmonic oscillator. In this paper we determine numerically, via the discrete-variable-representation method known from quantum chemistry, the exact states that saturate this new uncertainty relation. We analyze the physical properties of the states and compare them to the intelligent states of the Pegg-Barnett uncertainty relation. We find that for a given set of expectation values of the physical parameters, which are the particle number and the two quadratures, the two kinds of intelligent states are equivalent. The intelligent states are the eigenstates corresponding to the lowest eigenvalue of a Hermitian operator, which, when interpreted as a Hamiltonian of a physical sytem, describes a nonlinear driven harmonic oscillator, for example, a Duffing oscillator for a certain parameter range. Hence, our results can be interpreted as the determination of the ground state of such physical systems in an explicit analytic form as well. As the Pegg-Barnett intelligent states we use are expressed in terms of a coherent-state superposition facilitating experimental synthesis, we relate the states determined here to experimentally feasible ones.