Solution of Schrödinger Equation Not Implementing Conventional Separation of Variables: Using the Trial and Error Brute Force Permutation Method

2019 ◽  
pp. 96-107
Author(s):  
J. M. Ramírez-Velásquez ◽  
Ismael Villegas
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Separation Of Variables ◽  
Brute Force ◽  
Trial And Error

scholarly journals Description of a free quantum-mechanical particle in the Lobachevsky space based on the integral equation

2019 ◽  
Vol 55 (3) ◽  
pp. 319-324
Author(s):  
Yu. A. Kurochkin
Keyword(s):  
Integral Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Scattering Amplitude ◽  
Separation Of Variables ◽  
Free Particle ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Quantum Mechanical ◽  
Lobachevsky Space

The quantum mechanical problem of the motion of a free particle in the three-dimensional Lobachevsky space is interpreted as space scattering. The quantum case is considered on the basis of the integral equation derived from the Schrödinger equation. The work continues the problem considered in [1] studied within the framework of classical mechanics and on the basis of solving the Schrödinger equation in quasi-Cartesian coordinates. The proposed article also uses a quasi-Cartesian coordinate system; however after the separation of variables, the integral equation is derived for the motion along the axis of symmetry horosphere axis coinciding with the z axis. The relationship between the scattering amplitude and the analytical functions is established. The iteration method and finite differences for solution of the integral equation are proposed.

scholarly journals Separability of the Planar 1/ρ2 Potential in Multiple Coordinate Systems

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1312
Author(s):  
Richard DeCosta ◽  
Brett Altschul
Keyword(s):  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
Separation Of Variables ◽  
Bound State ◽  
Three Dimensions ◽  
Attractive Potential ◽  
Coordinate Systems ◽  
Repulsive Potential ◽  
Isotropic Harmonic Oscillator

With a number of special Hamiltonians, solutions of the Schrödinger equation may be found by separation of variables in more than one coordinate system. The class of potentials involved includes a number of important examples, including the isotropic harmonic oscillator and the Coulomb potential. Multiply separable Hamiltonians exhibit a number of interesting features, including “accidental” degeneracies in their bound state spectra and often classical bound state orbits that always close. We examine another potential, for which the Schrödinger equation is separable in both cylindrical and parabolic coordinates: A z-independent V∝1/ρ2=1/(x2+y2) in three dimensions. All the persistent, bound classical orbits in this potential close, because all other orbits with negative energies fall to the center at ρ=0. When separated in parabolic coordinates, the Schrödinger equation splits into three individual equations, two of which are equivalent to the radial equation in a Coulomb potential—one equation with an attractive potential, the other with an equally strong repulsive potential.

Accidental Degeneracy of the Hydrogen Atom and its Non-accidental Solution in Parabolic Coordinates

2021 ◽  
Author(s):  
P.C. Deshmukh ◽  
Aarthi Ganesan ◽  
Sourav Banerjee ◽  
Ankur Mandal
Keyword(s):  
Quantum Mechanics ◽  
Hydrogen Atom ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Physics ◽  
Separation Of Variables ◽  
Classical Mechanics ◽  
Dynamical Symmetry ◽  
Coordinate Systems ◽  
Casimir Operators

The degeneracy associated with dynamical symmetry of a potential can be identified in quantum mechanics, by solving the Schrödinger equation analytically, using the method of separation of variables in at least two different coordinate systems, and in classical mechanics by solving the Hamilton-Jacobi equation. In the present pedagogical article, the notion of separability and superintegrability of a potential, with profound implications is discussed. In an earlier tutorial paper, we had addressed the n<sup>2</sup>-fold degeneracy of the hydrogen atom using the Casimir operators corresponding to the SO(4) symmetry of the 1/r potential. The present paper is a sequel to it, in which we solve the Schrödinger equation for the hydrogen atom using separation of variables in the parabolic coordinate systems. In doing so, we take the opportunity to revisit some excellent classical works on symmetry and degeneracy in classical and quantum physics, if only to draw attention to these insightful studies which unfortunately miss even a mention in most undergraduate and even graduate level courses in quantum mechanics and atomic physics.

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