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

Nicolaides, CA; Constantoudis, V

dc.contributor.author	Nicolaides, CA	en
dc.contributor.author	Constantoudis, V	en
dc.date.accessioned	2014-03-01T01:30:05Z
dc.date.available	2014-03-01T01:30:05Z
dc.date.issued	2009	en
dc.identifier.issn	0143-0807	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/19468
dc.subject.classification	Education, Scientific Disciplines	en
dc.subject.classification	Physics, Multidisciplinary	en
dc.subject.other	5-level	en
dc.subject.other	Action-angle variables	en
dc.subject.other	Anharmonicities	en
dc.subject.other	Average energy	en
dc.subject.other	Classical trajectories	en
dc.subject.other	Discrete spectrum	en
dc.subject.other	Harmonic oscillators	en
dc.subject.other	Heisenberg uncertainty principle	en
dc.subject.other	Morse oscillator	en
dc.subject.other	Numerical computations	en
dc.subject.other	Phase spaces	en
dc.subject.other	Quantum mechanical energy	en
dc.subject.other	Quantum mechanics	en
dc.subject.other	Time averages	en
dc.subject.other	Uncertainty relation	en
dc.subject.other	WKB approximations	en
dc.subject.other	Approximation theory	en
dc.subject.other	Phase space methods	en
dc.subject.other	Schrodinger equation	en
dc.subject.other	Spectroscopy	en
dc.subject.other	Oscillators (electronic)	en
dc.title	Cycle-averaged phase-space states for the harmonic and the Morse oscillators, and the corresponding uncertainty relations	en
heal.type	journalArticle	en
heal.identifier.primary	10.1088/0143-0807/30/6/007	en
heal.identifier.secondary	http://dx.doi.org/10.1088/0143-0807/30/6/007	en
heal.language	English	en
heal.publicationDate	2009	en
heal.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.	en
heal.publisher	IOP PUBLISHING LTD	en
heal.journalName	European Journal of Physics	en
dc.identifier.doi	10.1088/0143-0807/30/6/007	en
dc.identifier.isi	ISI:000271266400011	en
dc.identifier.volume	30	en
dc.identifier.issue	6	en
dc.identifier.spage	1277	en
dc.identifier.epage	1294	en