Two-well oscillator

Banerjee, Kalyan; Bhatnagar, Sakshi

doi:10.1103/physrevd.18.4767

Cited by 83 publications

(53 citation statements)

References 13 publications

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“…This value strictly depends on the number of the basis functions and will increase for a larger set of them. Since the potential shifted by 1/(4λ) is positive definite, some authors have reported the highly accurate positive definite eigenvalues E n + 1/(4λ) [14].…”

Section: Resultsmentioning

confidence: 99%

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Accurate energy spectrum for double-well potential: periodic basis

2010

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We present a variational study of employing the trigonometric basis functions satisfying periodic boundary condition for the accurate calculation of eigenvalues and eigenfunctions of quartic double-well oscillators. Contrary to usual Dirichlet boundary condition, imposing periodic boundary condition on the basis functions results in the existence of an inflection point with vanishing curvature in the graph of the energy versus the domain of the variable. We show that this boundary condition results in a higher accuracy in comparison to Dirichlet boundary condition. This is due to the fact that the periodic basis functions are not necessarily forced to vanish at the boundaries and can properly fit themselves to the exact solutions.

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Section: Resultsmentioning

confidence: 99%

“…We can also use the scaling properties of the double-well potential to find the properties of a general Hamiltonian H(k, λ) = −p 2 − kx 2 + λx 4 from a more simpler one which we have studied here H(1, β) = −p 2 − x 2 + βx 4 . Using the transformation of variable from x to k 1/4 x, we find the following scaling properties [14,26] …”

Section: Resultsmentioning

confidence: 99%

Accurate energy spectrum for double-well potential: periodic basis

2010

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“…The most common nonperturbative methods such as the variational method [6], semi-classical approximation method, e.g. the WKBJ method [9], the Hill determinant method [11] etc, suffer from the limitation that these are seldom systematically improvable and are often tailored for the specific problem under investigation.. Hence the need arises to develop a nonperturbative method that is, in principle, applicable to general quantum systems, as well as, systematically improvable.…”

Section: Introductionmentioning

confidence: 99%

“…In this direction, several schemes have been forwarded which go beyond the ordinary perturbation theory. These include: variational [6] methods, variation-perturbation method [7], Gaussian approximation scheme [8], the WKBJ method [9], the Hartree approximation scheme [10], the Hill-determinant method [11] and its variants [12], the method of modified perturbation theory [13], the Boguliobov-transform methods [14] and many more [15]. The most common nonperturbative methods such as the variational method [6], semi-classical approximation method, e.g.…”

Section: Introductionmentioning

confidence: 99%

A New General Approximation Scheme in Quantum Theory: Application to the Anharmonic and the Double Well Oscillators

Mahapatra

Santi²,

Pradhan³

2005

Int. J. Mod. Phys. A

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A self-consistent, non-perturbative approximation scheme is proposed which i s potentially applicable to arbitrary interacting quantum systems. For the case of selfinteraction, the scheme consists in approximating the original interaction H I (φ) by a suitable 'potential' V(φ ) which satisfies the following two basic requirements, ( i ) exact solvability (ES): the 'effective' Hamiltonian, H 0 generated by V(φ ) is exactly solvable i.e., the spectrum of states | n > and the eigen-values E n are known and (ii) equality of quantum averages (EQA):The leading order (LO) results for | n > and E n are thus readily obtained and are found to be accurate to within a few percent of the 'exact' results. These LO-results are systematically improvable by the construction of an improved perturbation theory (IPT) with the choice of H 0 as the unperturbed Hamiltonian and the modified interaction,, as the perturbation where λ is the coupling strength. The condition of convergence of the IPT for arbitrary λ is satisfied due to the EQA requirement which ensures that < n | λ H ′ (φ ) | n > = 0 for arbitrary 'λ' and 'n'. This is in contrast to the divergence (which occurs even for infinitesimal λ!) in the naïve perturbation theory where the original interaction λH I (φ ) is chosen as the perturbation. We apply the method to the different cases of the anharmonic-and double well potentials, e.g. quartic-, sextic-and octic-anharmonic oscillators and quartic-, sexticdouble well oscillators. Uniformly accurate results for the energy levels over the full allowed range of 'λ' and 'n' are obtained. The results compare well with the exact results predicted by super symmetry for the case of the sextic anharmonic-and the double well partner potentials. Further improvement in the accuracy of the results by the use of IPT, is demonstrated. We also discuss the vacuum structure and stability of the resulting theory in the above approximation scheme.

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“…in accordance with (8). It is interesting to notice that the first N + 1 equations in (25) may be written as an inhomogeneous linear system …”

Section: Matrix Representation Of the Schrödinger Operator In Hpmmentioning

confidence: 94%

The scaled Hermite–Weber basis in the spectral and pseudospectral pictures

Taşeli

Alıcı

2005

J Math Chem

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Computational efficiencies of the discrete (pseudospectral, collocation) and continuous (spectral, Rayleigh-Ritz, Galerkin) variable representations of the scaled HermiteWeber basis in finding the energy eigenvalues of Schrödinger operators with several potential functions have been compared. It is well known that the so-called differentiation matrices are neither skew-symmetric nor symmetric in a pseudospectral formulation of a differential equation, unlike their Rayleigh-Ritz counterparts. In spite of this fact, it is shown here that the spectra of matrix Hamiltonians generated by Hermite collocation method may be determined by way of diagonalizing symmetric matrices. Furthermore, the symmetric matrix elements do not require the evaluation of Hermite polynomials at the grid points. Surprisingly, the present numerical results suggest that the convergence rates of collocation and Rayleigh-Ritz methods are entirely the same. KEY WORDS:Schrödinger operator, quantum mechanical oscillators, singular SturmLiouville problems on the real line, spectral and pseudospectral methods, HermiteWeber functions, Hermite collocation points

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Two-well oscillator

Cited by 83 publications

References 13 publications

Accurate energy spectrum for double-well potential: periodic basis

Accurate energy spectrum for double-well potential: periodic basis

A New General Approximation Scheme in Quantum Theory: Application to the Anharmonic and the Double Well Oscillators

The scaled Hermite–Weber basis in the spectral and pseudospectral pictures

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