A Simple Mathematical Formulation of the Correspondence Principle

Bernal, Jorge; Martín-Ruiz, A.; García-Melgarejo, J. C.

doi:10.4236/jmp.2013.41017

Cited by 21 publications

(38 citation statements)

References 34 publications

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“…In a previous work [13], we have introduced a simple mathematical procedure to connect the classical and quantum probability density functions, were reported analytical results for the quantum harmonic oscillator and the particle in a box. The method is based both on Bohr's correspondence principle and the local averages (coarsegraining) of the quantum distribution.…”

Section: Introductionmentioning

confidence: 99%

The Classical Limit of the Quantum Kepler Problem

Martín-Ruiz¹,

Bernal²,

Frank³

et al. 2013

JMP

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The classical limit of the quantum mechanical Kepler problem is derived by using a simple mathematical procedure recently proposed. The method is based both on Bohr’s correspondence principle and the local averages of the quantum probability distribution. We illustrate in a clear fashion the difference between Planck’s limit and Bohr’s correspondence principle. We discuss the confinement effect in macroscopic systems.

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Section: Introductionmentioning

confidence: 99%

The Classical Limit of the Quantum Kepler Problem

Martín-Ruiz¹,

Bernal²,

Frank³

et al. 2013

JMP

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“…When considering higher orders of approximation in (18) we observe macroscopic quantum behaviour. For example, a residual oscillatory behaviour (in all space) is retained in ( ) , n n x ρ  even for arbitrarily high quantum number n [12]. This is because  keeps a finite value, and thus the Heisenberg's theorem still works.…”

Section: Macroscopic Density Matrix Of the Harmonic Oscillatormentioning

confidence: 99%

“…In most cases, Equation (1) is very difficult to evaluate analytically. We briefly review an alternative procedure to compute the local averages appearing in Equation (1) [12,13]. Supported by the harmonic analysis criteria, we write the classical and quantum distributions as a Fourier expansion, ( ) (1) and (2) is that the Fourier coefficients have a similar behaviour for n large,…”

Section: General Proceduresmentioning

confidence: 99%

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Macroscopic Quantum Behaviour of Periodic Quantum Systems

Martín-Ruiz¹,

Bernal²,

Carbajal-Domínguez³

2014

JMP

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In this paper we introduce a simple procedure for computing the macroscopic quantum behaviour of periodic quantum systems in the high energy regime. The macroscopic quantum coherence is ascribed to a one-particle state, not to a condensate of a many-particle system; and we are referring to a system of high energy but with few degrees of freedom. We show that, in the first order of approximation, the quantum probability distributions converge to its classical counterparts in a clear fashion, and that the interference effects are strongly suppressed. The harmonic oscillator provides a testing ground for these ideas and yields excellent results.

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“…Some particularly useful descriptions of these fundamental issues are provided in Refs. [1][2][3][4][5][6] (to mention only a few).…”

Section: Introductionmentioning

confidence: 99%

Classical path from quantum motion for a particle in a transparent box

Vincenzo

2014

Rev. Bras. Ensino Fís.

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We consider the problem of a free particle inside a one-dimensional box with transparent walls (or equivalently, along a circle with a constant speed) and discuss the classical and quantum descriptions of the problem. After calculating the mean value of the position operator in a time-dependent normalized complex general state and the Fourier series of the function position, we explicitly prove that these two quantities are in accordance by (essentially) imposing the approximation of high principal quantum numbers on the mean value. The presentation is accessible to advanced undergraduate students with a knowledge of the basic ideas of quantum mechanics. Keywords: correspondence principle, classical limit, Ehrenfest theorem.Consideramos o problema de uma partícula livre no interior de uma caixa unidimensional com paredes transparentes (ou equivalentemente, ao longo de um círculo com uma velocidade constante) e discutimos as descrições clássica e quântica do problema. Depois de calcular o valor médio do operador da posição num estado geral complexo normalizado dependente do tempo e a série de Fourier da função de posição, provamos explicitamente que estas duas quantidades estão em correspondência se (essencialmente) impusermos sobre o valor médio a aproximação dos números quânticos principais elevados. A apresentaçãoé acessível a alunos de graduação avançados com conhecimento das idéias básicas da mecânica quântica. Palavras-chave: princípio da correspondência, limite clássico, teorema de Ehrenfest.

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A Simple Mathematical Formulation of the Correspondence Principle

Cited by 21 publications

References 34 publications

The Classical Limit of the Quantum Kepler Problem

The Classical Limit of the Quantum Kepler Problem

Macroscopic Quantum Behaviour of Periodic Quantum Systems

Classical path from quantum motion for a particle in a transparent box

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