On the nondegeneracy theorem for a particle in a box

Vincenzo, Salvatore De

doi:10.1590/s0103-97332008000300009

Cited by 4 publications

(3 citation statements)

References 24 publications

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“…This shows that we should not take the non-degeneracy theorem for granted particularly for singular interactions. This was first realized for the so-called one-dimensional Hydrogen atom [8], where the non-degeneracy theorem breaks down and has been studied for other one-dimensional singular potentials since then [9,10,11,12,13,14]. In contrast to the degeneracies that appear in bound states, we give elementary proof that the ground state is non-degenerate and the ground state wave function can be always chosen as real-valued and strictly positive.…”

Section: Introductionmentioning

confidence: 94%

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A singular one-dimensional bound state problem and its degeneracies

et al. 2017

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We give a brief exposition of the formulation of the bound state problem for the one-dimensional system of N attractive Dirac delta potentials, as an N ×N matrix eigenvalue problem (ΦA = ωA). The main aim of this paper is to illustrate that the non-degeneracy theorem in one dimension breaks down for the equidistantly distributed Dirac delta potential, where the matrix Φ becomes a special form of the circulant matrix. We then give elementary proof that the ground state is always non-degenerate and the associated wave function may be chosen to be positive by using the Perron-Frobenius theorem. We also prove that removing a single center from the system of N delta centers shifts all the bound state energy levels upward as a simple consequence of the Cauchy interlacing theorem.

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Section: Introductionmentioning

confidence: 94%

“…Now we can find the wave function associated with this bound state in the coordinate space by taking its Fourier transform. We perform the integration exactly as we did in (13), and obtain [6] ψ…”

Section: Bound States For N Dirac Delta Potentialsmentioning

confidence: 99%

A singular one-dimensional bound state problem and its degeneracies

et al. 2017

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“…[9] and brief comments in Refs. [5,15,16], the problem of a classical particle inside a penetrable box is rarely discussed. Clearly, in each of these problems, the particle 1 E-mail: salvatore.devincenzo@ucv.ve.…”

Section: Introductionmentioning

confidence: 99%

Classical-quantum versus exact quantum results for a particle in a box

Vincenzo

2012

Rev. Bras. Ensino Fís.

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Classical-quantum versus exact quantum results for a particle in a box (Resultados clássico-quânticos versus resultados quânticos exatos para uma partícula em uma caixa)The problems of a free classical particle inside a one-dimensional box: (i) with impenetrable walls and (ii) with penetrable walls, were considered. For each problem, the classical amplitude and mechanical frequency of the τ -th harmonic of the motion of the particle were identified from the Fourier series of the position function. After using the Bohr-Sommerfeld-Wilson quantization rule, the respective quantized amplitudes and frequencies (i.e., as a function of the quantum label n) were obtained. Finally, the classical-quantum results were compared to those obtained from modern quantum mechanics, and a clear correspondence was observed in the limit of n ≫ τ .Foram considerados os problemas de uma partícula livre clássica dentro de uma caixa unidimensional: (i) com paredes impenetráveis e (ii) com paredes penetráveis. Para cada problema, foram identificados a partir da série de Fourier da função de posição, a amplitude clássica ea freqüência mecânica clássica do τ -ésimo harmônico do movimento da partícula. Depois de usar a regra de quantização de Bohr-Sommerfeld-Wilson, foram obtidos a respectivas amplitudes e freqüências quantizadas (istoé, como uma função do rótulo quantum n). Finalmente, os resultados clássico-quânticos foram comparados com aqueles obtidos a partir da moderna mecânica quântica, e uma clara correspondência foi observada no limite de n ≫ τ . Palavras-chave: mecânica clássica, partícula em uma caixa, harmônicos de Fourier, harmônicos de Heisenberg.

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Composite Majorana fermion wave functions in nanowires

2012

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We consider Majorana fermions (MFs) in quasi-one-dimensional nanowire systems containing normal and superconducting sections where the topological phase based on Rashba spin orbit interaction can be tuned by magnetic fields. We derive explicit analytic solutions of the MF wavefunction in the weak and strong spin orbit interaction regimes. We find that the wavefunction for one single MF is a composite object formed by superpositions of different MF wavefunctions which have nearly disjoint supports in momentum space. These contributions are coming from the extrema of the spectrum, one centered around zero momentum and the other around the two Fermi points. As a result, the various MF wavefunctions have different localization lengths in real space and interference among them leads to pronounced oscillations of the MF probability density. For a transparent normal-superconducting junction we find that in the topological phase the MF leaks out from the superconducting into the normal section of the wire and is delocalized over the entire normal section, in agreement with recent numerical results by Chevallier et al..

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On the nondegeneracy theorem for a particle in a box

Cited by 4 publications

References 24 publications

A singular one-dimensional bound state problem and its degeneracies

A singular one-dimensional bound state problem and its degeneracies

Classical-quantum versus exact quantum results for a particle in a box

Composite Majorana fermion wave functions in nanowires

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