Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

European Journal of Physics

Cycle-averaged phase-space states for the harmonic and the Morse oscillators, and the corresponding uncertainty relations

Cleanthes A. Nicolaides, Vassilios Constantoudis

Published Sep 9, 2009• 2 citations
Publisher

Abstract

In Planck's model of the harmonic oscillator (HO) a century ago, both the energy and the phase space were quantized according to epsilon(n) = nh nu, n = 0, 1, 2..., and integral integral dp(x) dx = h. By referring to just these two relations, we show how the adoption of cycle-averaged phase-space states (CAPSSs) leads to the quantum mechanical energy spectrum of the HO, < E(n)> = (n + 1/2)h nu, n = 0, 1, 2,..., where < E(n)> are the average energies, and to < J(n)> = (n + 1/2)h/2 pi, where < J(n)> are the average actions. When anharmonicity to all orders is added in the form of the Morse oscillator (MO), the concept of CAPSS is implemented in terms of action-angle variables and it is shown that the use of < J(n)> of each MO CAPSS also produces the correct discrete spectrum of the MO, again without applying quantum mechanics (QM). In addition, the concept of CAPSS leads to two well-known post-QM relations which are obtained in terms of time averages of the classical trajectories and of < J(n)> : (1) closed integral p dx = 2 pi < J(n)> = (n + 1/2)h, which is the quantum condition of the old quantum theory, albeit with half-integers (i.e. the result of the WKB approximation), and (2) (Delta p Delta x)(n) >= h/4 pi, which is the Heisenberg uncertainty principle. It is shown via numerical computations that, for two MOs, one with intermediate anharmonicity, supporting 22 levels, and another with strong anharmonicity, with 5 levels, the quantities(Delta p Delta x)(n), n = 0, 1,..., 4, which are computed classically for the appropriately chosen trajectories agree very well with the results of computations that apply QM. The introduction of the CAPSS and the concomitant results underline the significance of the concept of the state of the system in physics, both classical and quantum.

DOI
10.1088/0143-0807/30/6/007

Related papers

Linear and nonlinear response functions of the Morse oscillator: Classical divergence and the uncertainty principle

Jianlan Wu, Jianshu Cao

The algebraic structure of the quantum Morse oscillator is explored to formulate the coherent state, the phase-space representations of the annihilation and creation operators, and their classical limits. The formulation allows us to calculate the linear and nonlinear quantum response functions for microcanonical Morse systems and to demonstrate the linear divergence in the corresponding classical response function. On the basis of the uncertainty principle, the classical divergence is removed by phase-space averaging around the microcanonical energy surface. For the Morse oscillator, the classical response function averaged over quantized phase space agrees exactly with the quantum response function for a given eigenstate. Thus, phase-space averaging and quantization provide a useful way to establish the classical-quantum correspondence of anharmonic systems.

The Journal of Chemical Physics•Sep 22, 2001•69 citations

A new look at the quantum mechanics of the harmonic oscillator

H.A. Kastrup

In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr-Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)-model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 -(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = -Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)-model of the HO the three hj correspond to the self-adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)-Hamiltonian . For an m-fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)-model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.

arXiv:quant-ph/0612032•Annalen der Physik•Jun 16, 2007•15 citations

A new look at the quantum mechanics of the harmonic oscillator

H.A. Kastrup

Abstract In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr‐Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)‐model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 ‐(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = ‐Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)‐model of the HO the three hj correspond to the self‐adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)‐Hamiltonian . For an m‐fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)‐model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken‐type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE‐condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.

Annalen der Physik•Jul 6, 2007•10 citations

Phase states and phase operators for the quantum harmonic oscillator

Antoine Royer

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}^${]}_{\mathit{w}}$(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=(${\mathit{q}}^{2}$+${\mathit{p}}^{2}$${)}^{1/2}$. There are infinitely many such phase operators. Each is expressible as \ensuremath{\varphi}^=[${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$(p^/q^)${]}_{\mathrm{\ensuremath{\Omega}}}$, where \ensuremath{\Omega} specifies an ordering rule for q^ and p^. The commutator -i[H^,\ensuremath{\varphi}^]=1-2\ensuremath{\pi}[\ensuremath{\delta}(${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$p^/q^)${]}_{\mathrm{\ensuremath{\Omega}}}$ corresponds to the Poisson bracket {H,${\mathrm{\ensuremath{\varphi}}}_{\mathrm{cl}}$${\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})${]}_{\mathit{w}}$(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})${]}_{\mathit{w}}$(r,\ensuremath{\theta}) =2\ensuremath{\pi}${\mathrm{\ensuremath{\varepsilon}}}^{2}$${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varepsilon}}\mathit{r}}$[\ensuremath{\Gamma}^(\ensuremath{\Phi})${]}_{\mathit{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.

Physical Review A•Jan 1, 1996•38 citations

Linkage between thermodynamic quantities and the uncertainty relation in harmonic oscillator model

Shoichi Nagata

The Heisenberg uncertainty relation, ΔxΔpx ⩾ ℏ/2 (ℏ ≡ h/2π), is well known. The purpose of this paper is to study a linkage between macroscopic thermodynamic quantities and the uncertainty relation. To simplify matters, we consider one-dimensional harmonic oscillator model. The calculations are carried out by two steps. First, the thermal average of the uncertainty relation is obtained for canonical system in thermal equilibrium at temperature T, which is denoted by (Δx)T(Δpx)T. Second, the thermodynamic quantities are expressed analytically as a function of (Δx)T(Δpx)T within the harmonic oscillator model. Finally, a connection between the energy fluctuations and the uncertainty relation is clarified for the Einstein’s heat capacity model. The analysis is made on the basis of fluctuation theory. The uncertainty relation is valid not only in quantum state but also at finite temperature.

Results in Physics•Jan 1, 2016•7 citations

An analysis of states in the phase space: the anharmonic oscillator

Sebastiano Tosto

The paper introduces a simple quantum model to calculate in a general way allowed frequencies and energy levels of the anharmonic oscillator. The theoretical basis of the approach has been introduced in two early papers aimed to infer the properties of quantum systems exploiting the uncertainty principle only. For clarity the anharmonic oscillator is described having in mind the lattice oscillations of atoms/ions, yet quantum formalism of the model and approach have general character and can be extended to any oscillating system. The results show that the harmonic energy levels split into a complex system of energy levels dependent upon the number of anharmonic terms that characterize the oscillator.

arXiv:1105.3615•May 18, 2011•2 citations
In Planck's model of the harmonic oscillator (HO) a century ago, both the energy and the phase space were quantized according to epsilon(n) = nh nu, n = 0, 1, 2..., and integral integral dp(x) dx = h. By referring to just these two relations, we show how the adoption of cycle-averaged phase-space states (CAPSSs) leads to the quantum mechanical energy spectrum of the HO, < E(n)> = (n + 1/2)h nu, n = 0, 1, 2,..., where < E(n)> are the average energies, and to < J(n)> = (n + 1/2)h/2 pi, where < J(n)> are the average actions. When anharmonicity to all orders is added in the form of the Morse oscillator (MO), the concept of CAPSS is implemented in terms of action-angle variables and it is shown that the use of < J(n)> of each MO CAPSS also produces the correct discrete spectrum of the MO, again without applying quantum mechanics (QM). In addition, the concept of CAPSS leads to two well-known post-QM relations which are obtained in terms of time averages of the classical trajectories and of < J(n)> : (1) closed integral p dx = 2 pi < J(n)> = (n + 1/2)h, which is the quantum condition of the old quantum theory, albeit with half-integers (i.e. the result of the WKB approximation), and (2) (Delta p Delta x)(n) >= h/4 pi, which is the Heisenberg uncertainty principle. It is shown via numerical computations that, for two MOs, one with intermediate anharmonicity, supporting 22 levels, and another with strong anharmonicity, with 5 levels, the quantities(Delta p Delta x)(n), n = 0, 1,..., 4, which are computed classically for the appropriately chosen trajectories agree very well with the results of computations that apply QM. The introduction of the CAPSS and the concomitant results underline the significance of the concept of the state of the system in physics, both classical and quantum.
The algebraic structure of the quantum Morse oscillator is explored to formulate the coherent state, the phase-space representations of the annihilation and creation operators, and their classical limits. The formulation allows us to calculate the linear and nonlinear quantum response functions for microcanonical Morse systems and to demonstrate the linear divergence in the corresponding classical response function. On the basis of the uncertainty principle, the classical divergence is removed by phase-space averaging around the microcanonical energy surface. For the Morse oscillator, the classical response function averaged over quantized phase space agrees exactly with the quantum response function for a given eigenstate. Thus, phase-space averaging and quantization provide a useful way to establish the classical-quantum correspondence of anharmonic systems.
In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr-Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)-model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 -(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = -Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)-model of the HO the three hj correspond to the self-adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)-Hamiltonian . For an m-fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)-model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.
Abstract In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr‐Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)‐model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 ‐(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = ‐Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)‐model of the HO the three hj correspond to the self‐adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)‐Hamiltonian . For an m‐fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)‐model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken‐type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE‐condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.
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{]}_{\mathit{w}}]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{\mathit{q}}^{2}q2+p2{\mathit{p}}^{2}p2)1/2{)}^{1/2})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}}$${\mathrm{}}}_{\mathrm{}\mathrm{PB}}$=1-2\ensuremath{\pi}\ensuremath{\delta}(\ensuremath{\theta}) for the single-valued classical phase ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{cl}}$=\ensuremath{\theta} mod2\ensuremath{\pi}. Phase states \ensuremath{\Gamma}^(\ensuremath{\Phi}) are defined by the condition that their Weyl symbols [\ensuremath{\Gamma}^(\ensuremath{\Phi})]w{]}_{\mathit{w}}]w​(r,\ensuremath{\theta})\ensuremath{\rightarrow}\ensuremath{\delta}(\ensuremath{\theta}-\ensuremath{\Phi}) as r\ensuremath{\rightarrow}\ensuremath{\infty}. If moreover ${\mathrm{\ensuremath{\int}}}_{0}^{2\mathrm{\ensuremath{\pi}}}$d\ensuremath{\Phi}\ensuremath{\Gamma}^(\ensuremath{\Phi})=1^, then \ensuremath{\Gamma}^(\ensuremath{\Phi}) is a phase probability operator measure (POM). In particular, \ensuremath{\delta}(\ensuremath{\varphi}^-\ensuremath{\Phi}) is a phase POM.Normalizable approximate phase states \ensuremath{\Gamma}${\mathrm{^}}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi}) are defined by [\ensuremath{\Gamma}${\mathrm{^}}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi})]w{]}_{\mathit{w}}]w​(r,\ensuremath{\theta}) =2\ensuremath{\pi}${\mathrm{\ensuremath{\varepsilon}}}^{2}$${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varepsilon}}\mathit{r}}$[\ensuremath{\Gamma}^(\ensuremath{\Phi})]w{]}_{\mathit{w}}]w​(r,\ensuremath{\theta}), \ensuremath{\varepsilon}\ensuremath{\ll}1. Phase states are not, in general, ``pure orthogonal'' in the sense \ensuremath{\Gamma}^(\ensuremath{\Phi})\ensuremath{\Gamma}^(\ensuremath{\Phi}\ensuremath{'})=\ensuremath{\delta}(\ensuremath{\Phi}-\ensuremath{\Phi}\ensuremath{'})\ensuremath{\Gamma}^(\ensuremath{\Phi}), unless they are of the form \ensuremath{\delta}(\ensuremath{\varphi}^-\ensuremath{\Phi}). However, any phase state \ensuremath{\Gamma}${\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 Heisenberg uncertainty relation, ΔxΔpx ⩾ ℏ/2 (ℏ ≡ h/2π), is well known. The purpose of this paper is to study a linkage between macroscopic thermodynamic quantities and the uncertainty relation. To simplify matters, we consider one-dimensional harmonic oscillator model. The calculations are carried out by two steps. First, the thermal average of the uncertainty relation is obtained for canonical system in thermal equilibrium at temperature T, which is denoted by (Δx)T(Δpx)T. Second, the thermodynamic quantities are expressed analytically as a function of (Δx)T(Δpx)T within the harmonic oscillator model. Finally, a connection between the energy fluctuations and the uncertainty relation is clarified for the Einstein’s heat capacity model. The analysis is made on the basis of fluctuation theory. The uncertainty relation is valid not only in quantum state but also at finite temperature.
The paper introduces a simple quantum model to calculate in a general way allowed frequencies and energy levels of the anharmonic oscillator. The theoretical basis of the approach has been introduced in two early papers aimed to infer the properties of quantum systems exploiting the uncertainty principle only. For clarity the anharmonic oscillator is described having in mind the lattice oscillations of atoms/ions, yet quantum formalism of the model and approach have general character and can be extended to any oscillating system. The results show that the harmonic energy levels split into a complex system of energy levels dependent upon the number of anharmonic terms that characterize the oscillator.