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arXiv:1711.04171 · Quantum Studies Mathematics and Foundations

Comment on “Cornell potential in generalized uncertainty principle formalism: the case of Schrödinger equation”

Djamil Bouaziz

Published Feb 28, 2017• 0 citations
arXivHTMLPDF
DOI
10.1007/s40509-017-0107-0

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1: The Birth of Quantum Theory. 1.1: The radiation of an ideal black body. 1.2: Line spectra and Bohr's model of the atom. 1.3: The quantum conditions of Bohr-Sommerfield. 1.4: Waves of matters. Comments. Problems. 2: Classical Physics - Quantum Physics. 2.1: The difficulties of classical theory. 2.2: The successes of quantum theory. 2.3: Problems with interpretation of quantum theory. 2.4: Quantum theory (1930. Problems. II: Schrodinger's Equation and the Interpretation of its Solutions. 3: Schrodinger's Equation. 3.1: The Schrodinger equation with time. 3.2: The time-dependent Schrodinger equation. Comments. Problems. 4: The Probabilistic Interpretation of the Wavefunction. Comments. Problems. 5: The Motion of a Free Particle in Wave Mechanics. 5.1: The spreading of wave packets. 5.2: The initial problem for Schrodinger's free particle equation. 5.3: The Gaussian wave packet. Comments. Problems. 6: The Momentum of a Particle and the Fourier Transformation of the Wavefunction. Comments. Problems. 7: Description of Experiments in Wave Mechanics and the Heinsberg Uncertainty Principle. 7.1: Heisenberg's uncertainty principle. 7.2: The entropic uncertainty principle. Comments. Problems. 8: A Charged Particle in an Electromagnetic Field. Comments. Problems. 9: The Hydrodynamic Formulation of Wave Mechanics. 9.1: The motion of the probability fluid in an electromagnetic field. Comments. Problems. 10: The Spin of the Elctron and Pauli's Equation. 10.1: The magnetic moment vector. 10.2: Pauli's equation. 10.3: The Stern-Gerlach experiments. Comments. Problems. 11: The Hydrodynamic Picture of Wave Mechanics of a Particle with Spin. Comments. Problems. 12: Ehrenfest's Equations. 12.1: A particle with spin in the electromagnetic field. Problems. 13: Descriptiopn of Particle Motion in Phase Space -- Wigner's Function. Comments. Problems. 14: Mixed States. 14.1: An unpolarized beam as a mixed state. Comments. Problems. III: Solutions of Schrodinger's Equation. 15: Classification of Solutions to Schrodinger's Equation. 15.1: Scattering states. 15.2: The completeness of the set of bound and scattering state wavefunctions. 15.3: Orthogonality of the wavefunctions. 15.4: Development of the system in time. 15.5: Bound and scattered states. Problems. 16: Bound States and Energy. 16.1: The three-dimensional spherically symmetric potential well. Comments. Problems. 17: Scattering States and Collision Cross-sections. 17.1: One-dimensional scattering. 17.2: The method of integral equations. 17.3: Three-dimensional scattering. 17.4: The scattering amplitude. 17.5: The S matrix. 17.6: Collision cross-section. 17.7: The optical theorem. Comments. Problems. 18: Partial Waves and Wave Effects in Scattering. 18.1: Phase shifts. 18.2: The spherical potential hump. 18.3: The Ramsauer-Townsend effect. 18.4: The extinction paradox. Comments. Problems. 19: Resonance States. Problems. 20: The Coulomb Potential. 20.1: Bound states. 20.2: Scattering states. Comments. Problems. 21

Oxford University Press eBooks•Jan 1, 1992•80 citations
Abstract This work investigates analytical solutions of the Schrödinger equation within the framework of the displacement-operator formalism. Three distinct potential models are considered: a hybrid potential combining the Cornell and pseudo-harmonic forms, V 1 ( x ); the pseudo-harmonic potential, V 2 ( x ); and the squared Cornell potential, V 3 ( x ). The proposed approach enables the derivation of expressions for both the energy spectrum, E n , and the corresponding wave functions, ϕ . In each case, the solutions for ϕ are expressed in terms of the confluent Heun function, specifically HeunC C 1 , C 2 , C 3 , C 4 , C 5 , 1 + γ x . Furthermore, we present graphical illustrations of the positive energy spectrum, E n + , accompanied by a detailed discussion of their physical implications. Finally, the thermodynamic properties associated with these potential models are analyzed and interpreted, highlighting their relevance to various quantum systems.
1: The Birth of Quantum Theory. 1.1: The radiation of an ideal black body. 1.2: Line spectra and Bohr's model of the atom. 1.3: The quantum conditions of Bohr-Sommerfield. 1.4: Waves of matters. Comments. Problems. 2: Classical Physics - Quantum Physics. 2.1: The difficulties of classical theory. 2.2: The successes of quantum theory. 2.3: Problems with interpretation of quantum theory. 2.4: Quantum theory (1930. Problems. II: Schrodinger's Equation and the Interpretation of its Solutions. 3: Schrodinger's Equation. 3.1: The Schrodinger equation with time. 3.2: The time-dependent Schrodinger equation. Comments. Problems. 4: The Probabilistic Interpretation of the Wavefunction. Comments. Problems. 5: The Motion of a Free Particle in Wave Mechanics. 5.1: The spreading of wave packets. 5.2: The initial problem for Schrodinger's free particle equation. 5.3: The Gaussian wave packet. Comments. Problems. 6: The Momentum of a Particle and the Fourier Transformation of the Wavefunction. Comments. Problems. 7: Description of Experiments in Wave Mechanics and the Heinsberg Uncertainty Principle. 7.1: Heisenberg's uncertainty principle. 7.2: The entropic uncertainty principle. Comments. Problems. 8: A Charged Particle in an Electromagnetic Field. Comments. Problems. 9: The Hydrodynamic Formulation of Wave Mechanics. 9.1: The motion of the probability fluid in an electromagnetic field. Comments. Problems. 10: The Spin of the Elctron and Pauli's Equation. 10.1: The magnetic moment vector. 10.2: Pauli's equation. 10.3: The Stern-Gerlach experiments. Comments. Problems. 11: The Hydrodynamic Picture of Wave Mechanics of a Particle with Spin. Comments. Problems. 12: Ehrenfest's Equations. 12.1: A particle with spin in the electromagnetic field. Problems. 13: Descriptiopn of Particle Motion in Phase Space -- Wigner's Function. Comments. Problems. 14: Mixed States. 14.1: An unpolarized beam as a mixed state. Comments. Problems. III: Solutions of Schrodinger's Equation. 15: Classification of Solutions to Schrodinger's Equation. 15.1: Scattering states. 15.2: The completeness of the set of bound and scattering state wavefunctions. 15.3: Orthogonality of the wavefunctions. 15.4: Development of the system in time. 15.5: Bound and scattered states. Problems. 16: Bound States and Energy. 16.1: The three-dimensional spherically symmetric potential well. Comments. Problems. 17: Scattering States and Collision Cross-sections. 17.1: One-dimensional scattering. 17.2: The method of integral equations. 17.3: Three-dimensional scattering. 17.4: The scattering amplitude. 17.5: The S matrix. 17.6: Collision cross-section. 17.7: The optical theorem. Comments. Problems. 18: Partial Waves and Wave Effects in Scattering. 18.1: Phase shifts. 18.2: The spherical potential hump. 18.3: The Ramsauer-Townsend effect. 18.4: The extinction paradox. Comments. Problems. 19: Resonance States. Problems. 20: The Coulomb Potential. 20.1: Bound states. 20.2: Scattering states. Comments. Problems. 21