Energy eigenvalues and Einstein coefficients for the one-dimensional confined harmonic oscillators

Campoy, G.; Aquino, N.; Granados, V. D.

doi:10.1088/0305-4470/35/23/307

Cited by 41 publications

(75 citation statements)

References 24 publications

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“…In previous works, the method used was successfully applied to the hydrogen atom and harmonic oscillator confined by spherical impenetrable walls 17, 22, 23, as well as to the one‐dimensional harmonic 19 and the double well potential 24. In the present report, this method is applied to the confined 2D hydrogen atom.…”

Section: Discussionmentioning

confidence: 91%

“…The numerical procedure here employed, described in the , is computationally efficient and facilitates implementation. It has previously been applied in the highly accurate calculation of eigenvalues and eigenfunctions for several quantum mechanical problems 17–24. Within this scheme, one can simultaneously obtain energy eigenvalues and eigenfunctions.…”

Section: Computation Of Energy Eigenvalues and Eigenfunctionsmentioning

confidence: 99%

“…The confinement of the system by impenetrable walls partially breaks the energy degeneracy and this in turn introduces changes in the transition energies between states. In addition, new transitions for some of them can also be induced, as in the confined 3D hydrogen atom 32 and in the 1D confined harmonic oscillator 19. Figure 5 shows the energy spectrum for a confinement radius of 2 a.u., where the new transitions are depicted with dashed lines.…”

Section: Filling Order Of One‐electron Statesmentioning

confidence: 99%

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Accurate energy eigenvalues and eigenfunctions for the two‐dimensional confined hydrogen atom

Aquino

Campoy

Flores-Riveros

2005

Int J of Quantum Chemistry

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ABSTRACT:A study of the two-dimensional hydrogen atom confined within a circle of impenetrable walls is presented. The potential inside the box is Coulomb type, whereas outside it is infinite. The energy eigenvalues and some radial wave function properties are computed with high accuracy for different box sizes. We derive the polarizability in the Kirkwood approximation, calculate the Fermi contact term as a function of the confinement radius, and investigate the filling order of the one-electron states. When the electronic configuration of many electrons is constructed by means of the Aufbau principle, the model predicts the inversion 2s-3d levels in the N atom.

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Section: Discussionmentioning

confidence: 91%

Section: Computation Of Energy Eigenvalues and Eigenfunctionsmentioning

confidence: 99%

Section: Filling Order Of One‐electron Statesmentioning

confidence: 99%

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Accurate energy eigenvalues and eigenfunctions for the two‐dimensional confined hydrogen atom

Aquino

Campoy

Flores-Riveros

2005

Int J of Quantum Chemistry

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“…The method that we will present has been used with much success in problems with and without confinement. Some of the problems solved with this method are the one‐dimensional harmonic oscillator confined in symmetric and asymmetric boundaries 33, the confined three‐dimensional isotropic harmonic oscillator 32, the two‐dimensional hydrogen atom confined in circle with impenetrable walls 34, and the three‐dimensional hydrogen atom confined in an impenetrable sphere 11. Other applications to free problems include the hydrogen atom with a harmonic perturbation 35 and the double well potential for the inversion of NH 3 , in which the potential was represented by a polynomial of 20th degree 36.…”

Section: Series Solutionmentioning

confidence: 99%

“…First, we used Maple's algorithms to evaluate the hypergeometric functions and to find their zeros. Second, we used a series method that we have previously applied successfully to several problems 32–36. The energies obtained by both methods differ by less than 1 × 10 − 100 hartrees.…”

Section: Introductionmentioning

confidence: 99%

Highly accurate solutions for the confined hydrogen atom

Aquino¹,

Campoy²,

Montgomery³

2007

Int J of Quantum Chemistry

114

122

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ABSTRACT:The model of the confined hydrogen atom (CHA) was developed by Michels et al. [1] in the mid-1930s to study matter subject to extreme pressure. However, since the eigenvalues cannot be obtained analytically, even the most accurate calculations have yielded little more than 10 figure accuracy. In this work, we show that it is possible to obtain the CHA eigenvalues with extremely high accuracy (up to 100 decimal digits) and we do that using two completely different methods. The first is based on formal solution of the confluent hypergeometric function while the second uses a series method. We also compare radial expectation values obtained by both methods and conclude that the wave functions obtained by these two different approaches are of high quality. In addition, we compute the hyperfine splitting constant, magnetic screening constant, polarizability in the Kirkwood approximation, and pressure as a function of the box radius.

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Information‐Entropic Measures in Confined Isotropic Harmonic Oscillator

Mukherjee

Roy

2018

Advcd Theory and Sims

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Information based uncertainty measures like Rényi entropy (R), Shannon entropy (S) and Onicescu energy (E) (in both position and momentum space) are employed to understand the influence of radial confinement in isotropic harmonic oscillator. The transformation of Hamiltonian in to a dimensionless form gives an idea of the composite effect of oscillation frequency (ω) and confinement radius (r c ). For a given quantum state, accurate results are provided by applying respective exact analytical wave function in r space. The p-space wave functions are produced from Fourier transforms of radial functions. Pilot calculations are done taking order of entropic moments (α, β)as ( 3 5 , 3) in r and p spaces. A detailed, systematic analysis is performed for confined harmonic oscillator (CHO) with respect to state indices n r , l, and r c . It has been found that, CHO acts as a bridge between particle in a spherical box (PISB) and free isotropic harmonic oscillator (IHO). At smaller r c , E r increases and R α r , S r decrease with rise of n r . At moderate r c , there exists an interaction between two competing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n r (delocalization). Most of these results are reported here for the first time, revealing many new interesting features.

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Energy eigenvalues and Einstein coefficients for the one-dimensional confined harmonic oscillators

Cited by 41 publications

References 24 publications

Accurate energy eigenvalues and eigenfunctions for the two‐dimensional confined hydrogen atom

Accurate energy eigenvalues and eigenfunctions for the two‐dimensional confined hydrogen atom

Highly accurate solutions for the confined hydrogen atom

Information‐Entropic Measures in Confined Isotropic Harmonic Oscillator

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