Degeneracy in one dimension: Role of singular potentials

Bhattacharyya, Krishnendu; Pathak, R.K.

doi:10.1002/(sici)1097-461x(1996)59:3<219::aid-qua5>3.0.co;2-0

Cited by 15 publications

(21 citation statements)

References 38 publications

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“…We have not been able to obtain a better understanding of this curious feature of our solutions. However, the singularity in the potentials may provide the key to the understanding of degeneracy in one dimensions [9].…”

Section: This Turns Out To Bementioning

confidence: 99%

Exact bound states in volcano potentials

Koley¹,

Kar²

2007

Physics Letters A

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Quantum mechanics in a one-parameter family of volcano potentials is investigated. After a discussion on their construction and classical mechanics, we obtain exact, normalisable bound states for specific values of the energy. The nature of the wave functions and probability densities, as well as some curious features of the solutions are highlighted. * Electronic address: ratna@cts.iitkgp.ernet.in † Electronic address : sayan@cts.iitkgp.ernet.in

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Section: This Turns Out To Bementioning

confidence: 99%

Exact bound states in volcano potentials

Koley¹,

Kar²

2007

Physics Letters A

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“…In spite of this fact, the proof of this theorem in some books makes no mention of this specific point [6]. We recently found another reference in which the appearance of degeneracy in the presence of certain singular potentials is explored in onedimensional quantum systems, although the authors of this reference also showed cases of nondegeneracy in the presence of singular potentials [7]. We have also seen a study that found that bound states, degenerate in energy, may exist even if the potential is unbounded from below at infinity [8].…”

Section: Introductionmentioning

confidence: 99%

On the nondegeneracy theorem for a particle in a box

Vincenzo¹

2008

Braz. J. Phys.

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We present some essential results for the Hamiltonian of a particle in a box. We discuss the invariance of this operator under time-reversalT , the possibility of choosing real eigenfunctions for it and the degeneracy of its energy eigenvalues. Once these results have been presented, we introduce the usual nondegeneracy theorem and discuss some issues surrounding it. We find that the nondegeneracy theorem is true if the boundary conditions areT -invariant but "confining" (i.e., the particle is in a real impenetrable box). If the boundary conditions are notT -invariant (belonging to a family of so-called "not confining" boundary conditions), the respective eigenfunctions are strictly complex and there is no degeneracy. Consistently, we verify the validity of the theorem also in this case. Finally, if the boundary conditions are alsoT -invariant, but "not confining", then we can have degeneracy in the energy levels only if the respective eigenfunctions can be specifically written as complex. We find that the nondegeneracy theorem fails in these cases. If the respective eigenfunctions can be written as only real, then we do not have degeneracy and the nondegeneracy theorem is true.

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“…The singular or spiked oscillator is defined by the family of quantum Hamiltonians where α and λ take positive values only and the domain of x is [0,∞). This problem has received enough attention since the middle 70s 1–42 because of its applications in physical chemistry and nuclear physics 1, 3, 38–42 and mainly for its interesting mathematical behavior. As Aguilera‐Navarro and Guardiola pointed out 9, none of the two terms on the potential dominate for the extreme values of λ.…”

Section: Introductionmentioning

confidence: 99%

“…The number of works on this topic has been steadily increasing 1–42. A variety of methods have been used to solve this problem, and few of them are as follows: perturbation theory, the linear variational method, Rayleigh‐Ritz variational method, and numerical approaches.…”

Section: Introductionmentioning

confidence: 99%

Accurate solutions for the spiked oscillators

Campoy¹,

Aquino²

2008

Int J of Quantum Chemistry

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ABSTRACT:The one-dimensional harmonic oscillator potential plus a term of the form /x ␣ is known as the spiked oscillator potential; it constitutes a very interesting system because of its difficulties to accept perturbative and variational solutions for certain regimes of the ␣ parameter. By the use of a numerical method, we obtain accurate energy eigenvalues and eigenfunctions for a wide range of values and a few ␣ values. The accuracy of the present results is by much higher than previously published results.

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Degeneracy in one dimension: Role of singular potentials

Cited by 15 publications

References 38 publications

Exact bound states in volcano potentials

Exact bound states in volcano potentials

On the nondegeneracy theorem for a particle in a box

Accurate solutions for the spiked oscillators

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