Transformations of the schrödinger equation with discrete energy spectrum

1962 ◽  
Vol 26 (6) ◽  
pp. 1346-1353 ◽  
Author(s):  
P. Goded
Keyword(s):  
Energy Spectrum ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Discrete Energy

scholarly journals Solution of Non-Autonomous Schrödinger Equation for Quantized de Sitter Klein-Gordon Oscillator Modes Undergoing Attraction-Repulsion Transition

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 943
Author(s):  
Philip Broadbridge ◽  
Kathryn Deutscher
Keyword(s):  
Energy Spectrum ◽  
Scalar Field ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Canonical Quantization ◽  
Elementary Excitation ◽  
Accelerated Expansion ◽  
De Sitter ◽  
Discrete Energy ◽  
Klein Gordon

For a scalar field in an exponentially expanding universe, constituent modes of elementary excitation become unstable consecutively at shorter wavelength. After canonical quantization, a Bogoliubov transformation reduces the minimally coupled scalar field to independent 1D modes of two inequivalent types, leading eventually to a cosmological partitioning of energy. Due to accelerated expansion of the coupled space-time, each underlying mode transits from an attractive oscillator with discrete energy spectrum to a repulsive unit with continuous unbounded energy spectrum. The underlying non-autonomous Schrödinger equation is solved here as the wave function evolves through the attraction-repulsion transition and ceases to oscillate.

Effects of a non-Hermitian potential on the Landau quantization

2019 ◽  
Vol 34 (12) ◽  
pp. 1950072 ◽  
Author(s):  
B. F. Ramos ◽  
I. A. Pedrosa ◽  
K. Bakke
Keyword(s):  
Magnetic Field ◽  
Energy Spectrum ◽  
Schrödinger Equation ◽  
Complex Potential ◽  
Schrodinger Equation ◽  
Uniform Magnetic Field ◽  
Spinless Particle ◽  
Symmetric Complex

In this work, we solve the time-independent Schrödinger equation for a Landau system modulated by a non-Hermitian Hamiltonian. The system consists of a spinless particle in a uniform magnetic field submitted to action of a non-[Formula: see text] symmetric complex potential. Although the Hamiltonian is neither Hermitian nor [Formula: see text]-symmetric, we find that the Landau problem under study exhibits an entirely real energy spectrum.

Superstatistics of two electrons quantum dot

2019 ◽  
Vol 34 (03) ◽  
pp. 1950023 ◽  
Author(s):  
S. Sargolzaeipor ◽  
H. Hassanabadi ◽  
W. S. Chung
Keyword(s):  
Energy Spectrum ◽  
Thermodynamic Properties ◽  
Quantum Dot ◽  
Schrödinger Equation ◽  
Analytical Method ◽  
Schrodinger Equation ◽  
Boltzmann Factor ◽  
Dirac Delta ◽  
Uniform Distributions ◽  
Parabolic Quantum Dot

In this paper, we discussed the Schrödinger equation in the presence of the harmonic two electrons interaction for the parabolic quantum dot and the energy spectrum by an analytical method is obtained, then the effective Boltzmann factor in a deformed formalism for modified Dirac delta and uniform distributions are derived. We make use of the superstatistics for the two distributions in physics and the thermodynamic properties of the system are calculated. Ordinary results are recovered for the vanishing deformed parameter. Furthermore, the effect of all parameters in the problems are calculated and shown graphically.

The Schrödinger equation: total interaction as perturbation

1982 ◽  
Vol 380 (1779) ◽  
pp. 489-494 ◽  
Keyword(s):  
Energy Spectrum ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Eigenvalue Problems ◽  
Free Particle ◽  
Analytic Approximation ◽  
Binding Interaction ◽  
General Applicability ◽  
Total Interaction

The quantization of the energy spectrum of a free particle in the presence of a binding interaction is reconsidered here. These considerations form the basis of a simple analytic approximation of general applicability to eigenvalue problems.

scholarly journals Bound states of the D-dimensional Schrödinger equation for the generalized Woods–Saxon potential

2019 ◽  
Vol 34 (14) ◽  
pp. 1950107 ◽  
Author(s):  
V. H. Badalov ◽  
B. Baris ◽  
K. Uzun
Keyword(s):  
Quantum Mechanics ◽  
Energy Spectrum ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Basis Set ◽  
Saxon Potential ◽  
Radial Wave ◽  
Energy Eigenvalues ◽  
Approximate Analytical Solutions

The formal framework for quantum mechanics is an infinite number of dimensional space. Hereby, in any analytical calculation of the quantum system, the energy eigenvalues and corresponding wave functions can be represented easily in a finite-dimensional basis set. In this work, the approximate analytical solutions of the hyper-radial Schrödinger equation are obtained for the generalized Wood–Saxon potential by implementing the Pekeris approximation to surmount the centrifugal term. The energy eigenvalues and corresponding hyper-radial wave functions are derived for any angular momentum case by means of state-of-the-art Nikiforov–Uvarov and supersymmetric quantum mechanics methods. Hence, the same expressions are obtained for the energy eigenvalues, and the expression of hyper-radial wave functions transforming each other is shown owing to these methods. Furthermore, a finite number energy spectrum depending on the depths of the potential well [Formula: see text] and [Formula: see text], the radial [Formula: see text] and [Formula: see text] orbital quantum numbers and parameters [Formula: see text], [Formula: see text], [Formula: see text] are also identified in detail. Next, the bound state energies and corresponding normalized hyper-radial wave functions for the neutron system of the [Formula: see text]Fe nucleus are calculated in [Formula: see text] and [Formula: see text] as well as the energy spectrum expressions of other higher dimensions are revealed by using the energy spectrum of [Formula: see text] and [Formula: see text].

The “accidental” degeneracy of the hydrogen atom is no accident

2015 ◽  
Vol 93 (3) ◽  
pp. 312-317
Author(s):  
P.C. Deshmukh ◽  
Aarthi Ganesan ◽  
N. Shanthi ◽  
Blake Jones ◽  
James Nicholson ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Energy Spectrum ◽  
Hydrogen Atom ◽  
Schrödinger Equation ◽  
Atomic Physics ◽  
Schrodinger Equation ◽  
Spin States ◽  
Interesting Aspect ◽  
Accidental Degeneracy ◽  
Pedagogical Analysis

The Schrödinger equation does not account for the 2n2degeneracy of the hydrogen atom, which it dismisses as an “accidental” degeneracy. The factor of “2” in the 2n2degeneracy is well-accounted-for in the relativistic formulation by the two spin states of the electron. The n2degeneracy is nevertheless not quite an “accident”; it is due to the SO(4), rather than SO(3), symmetry of the hydrogen atom. This result is well known, but is inadequately commented upon in most courses in quantum mechanics and atomic physics, leaving the student wondering about the origins of the n2degeneracy of the hydrogen atom. A pedagogical analysis of this interesting aspect, which highlights the fundamental principles of quantum mechanics, is presented in this article. While doing so, not only is the n2degeneracy of the hydrogen atom explained, but its energy spectrum and eigenfunctions are obtained without even using the Schrödinger equation, employing only the fundamental principles of quantum mechanics rather than the Schrödinger equation.

scholarly journals The application bispherical coordinate in Schrödinger equation for Mobius square plus modified Yukawa potential using Nikiforov Uvarov Functional Analysis (NUFA) method

2020 ◽  
Vol 4 (2) ◽  
pp. 48
Author(s):  
Briant Sabathino Harya Wibawa ◽  
A Suparmi ◽  
C Cari
Keyword(s):  
Functional Analysis ◽  
Wave Function ◽  
Energy Spectrum ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Yukawa Potential ◽  
Radial Part ◽  
Angular Part ◽  
Variable Separation Method ◽  
Bispherical Coordinates

<p class="Abstract">The application bispherical coordinates in Schrödinger equation for the Mobius square plus modified Yukawa potential have been obtained. The Schrödinger equation in bispherical coordinates for the separable Mobius square plus modified Yukawa potential consisting of the radial part and the angular part for the Mobius square plus modified Yukawa potential is solved using the variable separation method to reduce it to the radial part and angular part Schrödinger equation. The aim of this study was to solve the Schrödinger's equation of radial in bispherical coordinates for the Mobius square plus modified Yukawa potential using the Nikiforov Uvarov Functional Analysis (NUFA) method. Nikiforov Uvarov Functional Analysis (NUFA) method used to obtained energy spectrum equation and wave function for the Mobius square plus modified Yukawa potential. The result of energy spectrum equation for Mobius square plus modified Yukawa potential can be shown in Equation (50). The result of un-normalized wave function equation for Mobius square plus modified Yukawa potential can be shown in Table 1.</p>

Effective mass of the discrete values as a hidden feature of the one-dimensional harmonic oscillator model: Exact solution of the Schrödinger equation with a mass varying by position

2021 ◽  
pp. 2150206
Author(s):  
E. I. Jafarov ◽  
S. M. Nagiyev
Keyword(s):  
Energy Spectrum ◽  
Positive Integer ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Effective Mass ◽  
Schrodinger Equation ◽  
Solvable Model ◽  
Wave Functions ◽  
Stationary States ◽  
Quantum Harmonic Oscillator

In this paper, exactly solvable model of the quantum harmonic oscillator is proposed. Wave functions of the stationary states and energy spectrum of the model are obtained through the solution of the corresponding Schrödinger equation with the assumption that the mass of the quantum oscillator system varies with position. We have shown that the solution of the Schrödinger equation in terms of the wave functions of the stationary states is expressed by the pseudo Jacobi polynomials and the mass varying with position depends from the positive integer [Formula: see text]. As a consequence of the positive integer [Formula: see text], energy spectrum is not only non-equidistant, but also there are only a finite number of energy levels. Under the limit, when [Formula: see text], the dependence of effective mass from the position disappears and the system recovers known non-relativistic quantum harmonic oscillator in the canonical approach where wave functions are expressed by the Hermite polynomials.

Study of Schrödinger equation in terms of Heun functions in the commutative vs. non-commutative spaces

2019 ◽  
Vol 34 (23) ◽  
pp. 1950183 ◽  
Author(s):  
S. Sargolzaeipor ◽  
H. Hassanabadi ◽  
W. S. Chung
Keyword(s):  
Magnetic Field ◽  
Energy Spectrum ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Heun Functions ◽  
Commutative Space

In this paper, we solved the Schrödinger equation in the commutative and non-commutative (NC) spaces under the presence of magnetic field. In other words, we obtained the energy spectrum and wave functions in terms of Heun functions. When we considered the case [Formula: see text], we observed that the NC space converts to the commutative space.

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