How does the classical notion of ``phase'' apply to a quantum harmonic oscillator H=1/2(q${\mathrm{^}}^{2}$+p${\mathrm{^}}^{2}$), [q^,p^]=i\ensuremath{\Elzxh}, which cannot have sharp position and momentum? A quantum state \ensuremath{\rho}^ can be assigned a definite classical phase only if it is a large-amplitude localized state. Our only demand, therefore, on a (Hermitian) phase operator \ensuremath{\varphi}^ is that the phase distribution P(\ensuremath{\Phi})= Tr{\ensuremath{\delta}(\ensuremath{\varphi}^-\ensuremath{\Phi})\ensuremath{\rho}^} attribute the correct sharp phase to any such ``classical phase'' state. This requires that the Weyl symbol [\ensuremath{\varphi}^
]w(q,p) of \ensuremath{\varphi}^ tend to \ensuremath{\theta} mod2\ensuremath{\pi} as r\ensuremath{\rightarrow}\ensuremath{\infty}, where \ensuremath{\theta}=${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$(p/q) and r=(
q2+
p2)1/2. There are infinitely many such phase operators. Each is expressible as \ensuremath{\varphi}^=[${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$(p^/q^)${]}_{\mathrm{\ensuremath{\Omega}}}$, where \ensuremath{\Omega} specifies an ordering rule for q^ and p^. The commutator -i[H^,\ensuremath{\varphi}^]=1-2\ensuremath{\pi}[\ensuremath{\delta}(${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$p^/q^)${]}_{\mathrm{\ensuremath{\Omega}}}$ corresponds to the Poisson bracket {H,${\mathrm{\ensuremath{\varphi}}}_{\mathrm{cl}}$${\mathrm{}}}_{\mathrm{}\mathrm{PB}}$=1-2\ensuremath{\pi}\ensuremath{\delta}(\ensuremath{\theta}) for the single-valued classical phase ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{cl}}$=\ensuremath{\theta} mod2\ensuremath{\pi}. Phase states \ensuremath{\Gamma}^(\ensuremath{\Phi}) are defined by the condition that their Weyl symbols [\ensuremath{\Gamma}^(\ensuremath{\Phi})
]w(r,\ensuremath{\theta})\ensuremath{\rightarrow}\ensuremath{\delta}(\ensuremath{\theta}-\ensuremath{\Phi}) as r\ensuremath{\rightarrow}\ensuremath{\infty}. If moreover ${\mathrm{\ensuremath{\int}}}_{0}^{2\mathrm{\ensuremath{\pi}}}$d\ensuremath{\Phi}\ensuremath{\Gamma}^(\ensuremath{\Phi})=1^, then \ensuremath{\Gamma}^(\ensuremath{\Phi}) is a phase probability operator measure (POM). In particular, \ensuremath{\delta}(\ensuremath{\varphi}^-\ensuremath{\Phi}) is a phase POM.Normalizable approximate phase states \ensuremath{\Gamma}${\mathrm{^}}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi}) are defined by [\ensuremath{\Gamma}${\mathrm{^}}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi})
]w(r,\ensuremath{\theta}) =2\ensuremath{\pi}${\mathrm{\ensuremath{\varepsilon}}}^{2}$${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varepsilon}}\mathit{r}}$[\ensuremath{\Gamma}^(\ensuremath{\Phi})
]w(r,\ensuremath{\theta}), \ensuremath{\varepsilon}\ensuremath{\ll}1. Phase states are not, in general, ``pure orthogonal'' in the sense \ensuremath{\Gamma}^(\ensuremath{\Phi})\ensuremath{\Gamma}^(\ensuremath{\Phi}\ensuremath{'})=\ensuremath{\delta}(\ensuremath{\Phi}-\ensuremath{\Phi}\ensuremath{'})\ensuremath{\Gamma}^(\ensuremath{\Phi}), unless they are of the form \ensuremath{\delta}(\ensuremath{\varphi}^-\ensuremath{\Phi}). However, any phase state \ensuremath{\Gamma}${\mathrm{^}}_{1}$(\ensuremath{\Phi}) is trace orthogonal to any phase POM \ensuremath{\Gamma}${\mathrm{^}}_{2}$(\ensuremath{\Phi}), in the sense that Tr{\ensuremath{\Gamma}${\mathrm{^}}_{2}$(\ensuremath{\Phi}\ensuremath{'})\ensuremath{\Gamma}${\mathrm{^}}_{1}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi})}\ensuremath{\rightarrow}\ensuremath{\delta}(\ensuremath{\Phi}-\ensuremath{\Phi}\ensuremath{'}) as \ensuremath{\varepsilon}\ensuremath{\rightarrow}0. This implies that measurement of the POM \ensuremath{\Gamma}${\mathrm{^}}_{2}$(\ensuremath{\Phi}\ensuremath{'}) on the state \ensuremath{\Gamma}${\mathrm{^}}_{1}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi}) yields the outcome \ensuremath{\Phi} with probability 1 as \ensuremath{\varepsilon}\ensuremath{\rightarrow}0. Phase measurements of the first kind are possible in principle; they would allow one to prepare (approximate) phase states and monitor their phase evolution in a quantum (phase) nondemolishing manner. Cases of special interest are the Susskind-Glogower and the Cahill-Glauber ordered phase states and operators. The energy-phase (or number-phase) uncertainty relation is \ensuremath{\Delta}H\ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\ge}0, the lower limit \ensuremath{\Delta}H\ensuremath{\Delta}\ensuremath{\varphi}=0 being realized by pure number states; however, for states whose Wigner functions are localized away from the origin and from the extremities of the (single-valued) phase window (0,2\ensuremath{\pi}), the uncertainty relation is effectively \ensuremath{\Delta}H\ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\ge}1/2. \textcopyright{} 1996 The American Physical Society.
The purpose of this work is to introduce, in a simple, intuitive way, the coherent and squeezed states of the quantum harmonic oscillator (QHO), through a series of exercises, which are solved in detail. Starting from the application of a spatial translation to the ground state of the QHO, we introduce the spatial and momentum translations, focusing on their application to the QHO, which leads us to the displacement operator. Next, we introduce the coherent states and examine their basic aspects. We then proceed to give a simple and purely intuitive introduction to the squeezed states and we conclude by identifying the coherent states as states of minimum energy expectation value compared to the respective squeezed states. The reader is assumed to have a basic knowledge of the postulates and the mathematical formalism of quantum mechanics, including the Dirac notation and the ladder operator method of the QHO.