  We demonstrate that the time operator that measures the time of arrival of a
quantum particle into chosen state can be defined as a self-adjoint
quantum-mechanical operator using periodic boundary conditions on applied to
wavefuncions in energy representation. The time becomes quantized into discreet
eigenvalues and the eigenstates of the time operator, the stroboscopic
wavepackets introduced recently [Phys. Rev. Lett. 101, 046402 (2008).] form
orthogonal system of states. The formalism provides simple physical
interpretation of the time-measurement process and direct construction of
normalized, positive definite probability distribution for the quantized values
of the arrival time. The average value of the time is equal to the phase time
but in general depends on the choise of zero time eigenstate, whereas the
uncertainity of the average is related to the traversal time and is independent
of this choise. The general fromalism is applied to a particle tunneling
through resonant tunneling barrier in 1D.
