  Requirements of a conjugate operator are emphasized, especially in its role
in uncertainty relations.It is argued that in many contexts it is necessary to
extend the Hilbert space in order to define a conjugate operator as in gauge
theories. Example of a particle in a box is analysed. This is closely related
to the quantum oscillator through cosine states of Susskind and Glogower.It is
used to justify London's phase wave functions albeit as part of a larger
Hilbert space. A new definition phase uncertainty neccessiated by periodicity
is proposed.It is close to the usual r.m.s. definition.Corresponding number-
phase uncertainty relation is obtained and its implications are discussed.
Hilbert space of an oscillator is identified with the Hilbert space of a planar
rotor with a $Z_2$ gauge invariance.This is used to construct states analogous
to the cosine and sine states and to illustrate unitary equivalence of Hilbert
spaces.
