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arXiv:quant-ph/0211143

On the number-phase problem

Spiridon Dumitru

Published Nov 22, 2002• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0211143

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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.
The minimization problem of finding the number-phase minimum uncertainty states (MUS) is considered and its solutions are found either numerically or, under some special conditions, analytically. The phase uncertainty measure is based on the Bandilla-Paul dispersion. The problem is treated (i) in a finite-dimensional Hilbert space and (ii) for a countably infinite-dimensional Hilbert space (i.e. the standard quantum harmonic oscillator), with the constraint of a given mean photon number. The MUS relations between the photon number uncertainty and phase uncertainty are presented. Connections to some other minimization problems are discussed.
Discussions on uncertainty relations (UR) and quantum measurements (QMS) persisted until nowadays in publications about quantum mechanics (QM). They originate mainly from the conventional interpretation of UR (CIUR). In the most of the QM literarure, it is underestimated the fact that, over the years, a lot of deficiencies regarding CIUR were signaled. As a rule the alluded deficiencies were remarked disparately and discussed as punctual and non-essential questions. Here we approach an investigation of the mentioned deficiencies collected in a conclusive ensemble. Subsequently we expose a reconsideration of the major problems referring to UR and QMS. We reveal that all the basic presumption of CIUR are troubled by insurmountable deficiencies which require the indubitable failure of CIUR and its necessary abandonment. Therefore the UR must be deprived of their statute of crucial pieces for physics. So, the aboriginal versions of UR appear as being in postures of either (i) thought-experimental fictions or (ii) simple QM formulae and, any other versions of them, have no connection with the QMS. Then the QMS must be viewed as an additional subject comparatively with the usual questions of QM. For a theoretical description of QMS we propose an informationtransmission model, in which the quantum observables are considered as random variables. Our approach directs to natural solutions and simplifications for many problems regarding UR and QMS.
The quantum-mechanical description of phase and angle variables is reviewed, with emphasis on the proper mathematical description of these coordinates. The relations among the operators and state vectors under consideration are clarified in the context of the Heisenberg uncertainty relations. The familiar case of the azimuthal angle variable $\ensuremath{\phi}$ and its "conjugate" angular momentum Lz{L}_{z}Lz​ is discussed. Various pitfalls associated with the periodicity problem are avoided by employing periodic variables ($sin\ensuremath{\phi}$ and $cos\ensuremath{\phi}$ to describe the phase variable. Well-defined uncertainty relations are derived and discussed. A detailed analysis of the three-dimensional harmonic oscillator excited in coherent states is given. A detailed analysis of the simple harmonic oscillator is given. The usual assumption that a (Hermitian) phase operator $\ensuremath{\varphi}$ (conjugate to the number operator NNN) exists is shown to be erroneous. However, cosine and sine operators CCC and SSS exist and are the appr\'opriate phase variables. A Poisson bracket argument using action-angle (rather JJJ, $cos\ensuremath{\varphi}$, $sin\ensuremath{\varphi}$) variables is used to deduce CCC and SSS. The spectra and eigenfunctions of these operators are investigated, along with the important "phase-difference" periodic variables. The properties of the oscillator variables in the various types of states are analyzed with special attention to the uncertainty relations and the transition to the classical limit. The utility of coherent states as a basis for the description of the evolution of the density matrix is emphasized. In this basis it is easy to identify the classical Liouville equation in action-angle variables along with quantum-mechanical "corrections." Mention is made of possible physical applications to superfluid systems.
This survey tries to investigate the truths and deficiencies of prevalent philosophy about Uncertainty Relations (UR) and Quantum Measurements (QMS). The respective philosophy, known as being eclipsed by unfinished controversies, is revealed to be grounded on six basic precepts. But one finds that all the respective precepts are discredited by insurmountable deficiencies. So, in regard to UR, the alluded philosophy discloses oneself to be an unjustified mythology. Then UR appear either as short-lived historical conventions or as simple and limited mathematical formulas, without any essential significance for physics. Such a finding reinforces the Dirac's prediction that UR "`in their present form will not survive in the physics of future"'. The noted facets of UR motivate reconsiderations of associated debates on QMS. Mainly one reveals that, properly, UR have not any essential connection with genuine descriptions of QMS. For such descriptions, it is necessary that, mathematically, the quantum observables to be considered as random variables. The measuring scenarios with a single sampling, such are wave function collapse or Schrodinger's cat thought experiment, are revealed as being useless exercises. We propose to describe QMS as transmission processes for stochastic data. Note that the above-announced revaluation of UR and QMS philosophy does not disturb in any way the practical framework of the usual quantum mechanics.