Other Applications of Neural Networks to Quantum Mechanical Problems

2012 ◽  
Author(s):  
Lionel Raff ◽  
Ranga Komanduri ◽  
Martin Hagan ◽  
Satish Bukkapatnam
Keyword(s):  
Neural Networks ◽  
Wave Function ◽  
Linear Combination ◽  
Vibrational Spectra ◽  
Schrodinger Equation ◽  
Usual Method ◽  
Basis Functions ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Vibrational Wave Function

The usual method for solving the vibrational Schrödinger equation to obtain molecular vibrational spectra and the associated wave functions generally involves the expansion of the vibrational wave function, ψk(y), in terms of a linear combination of a set of basis functions.

Exact solutions to solitonic profile mass Schrödinger problem with a modified Pöschl–Teller potential

2016 ◽  
Vol 31 (04) ◽  
pp. 1650017 ◽  
Author(s):  
Shishan Dong ◽  
Qin Fang ◽  
B. J. Falaye ◽  
Guo-Hua Sun ◽  
C. Yáñez-Márquez ◽  
...  
Keyword(s):  
Wave Function ◽  
Exact Solutions ◽  
Ground State ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Wave Functions ◽  
Numerical Calculations ◽  
Potential Parameter ◽  
Heun Functions ◽  
Single Well

We present exact solutions of solitonic profile mass Schrödinger equation with a modified Pöschl–Teller potential. We find that the solutions can be expressed analytically in terms of confluent Heun functions. However, the energy levels are not analytically obtainable except via numerical calculations. The properties of the wave functions, which depend on the values of potential parameter [Formula: see text] are illustrated graphically. We find that the potential changes from single well to a double well when parameter [Formula: see text] changes from minus to positive. Initially, the crest of wave function for the ground state diminishes gradually with increasing [Formula: see text] and then becomes negative. We notice that the parities of the wave functions for [Formula: see text] also change.

scholarly journals SCHRÖDINGER EQUATION FOR HEAVY MESONS EXPANDED IN 1/mQ

1996 ◽  
Vol 11 (03) ◽  
pp. 257-266 ◽  
Author(s):  
TAKAYUKI MATSUKI
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Higher Order ◽  
Wave Functions ◽  
Heavy Mesons ◽  
Higher Order Corrections

Operating just once the naive Foldy-Wouthuysen-Tani transformation on the Schrödinger equation for [Formula: see text] bound states described by a Hamiltonian, we systematically develop a perturbation theory in 1/mQ which enables one to solve the Schrödinger equation to obtain masses and wave functions of the bound states in any order of 1/mQ. There also appear negative components of the wave function in our formulation which contribute also to higher order corrections to masses.

Stationary properties of approximate wave functions

1969 ◽  
Vol 47 (21) ◽  
pp. 2355-2361 ◽  
Author(s):  
A. R. Ruffa
Keyword(s):  
Wave Function ◽  
Total Energy ◽  
Helium Atom ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Mechanical Wave ◽  
Expectation Value ◽  
Charge Densities ◽  
Hartree Fock ◽  
Approximate Wave

The accuracy of quantum mechanical wave functions is examined in terms of certain stationary properties. The most elementary of these, namely that displayed by the class of wave functions which yields a stationary value for the total energy of the system, is demonstrated to necessarily require few other stationary properties, and none of these appear to be particularly useful. However, the class of wave functions which yields both stationary energies and charge densities has very important stationary properties. A theorem is proven which states that any wave function in this class yields a stationary expectation value for any operator which can be expressed as a sum of one-particle operators. Since the Hartree–Fock wave function is known to possess these same stationary properties, this theorem demonstrates that the Hartree–Fock wave function is one of the infinitely many wave functions of the class. Methods for generating other wave functions in this class by modifying the Hartree–Fock wave function without changing its stationary properties are applied to the calculation of wave functions for the helium atom.

scholarly journals On the Quantum Mechanical Wave Function as a Link Between Cognition and the Physical World: A Role for Psychology

2020 ◽  
Author(s):  
Douglas Michael Snyder
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Physical World ◽  
Wave Functions ◽  
Human Cognition ◽  
Quantum Mechanical ◽  
Mechanical Wave ◽  
Empirical Level ◽  
And Behavior ◽  
Empirical Demonstration

A straightforward explanation of fundamental tenets of quantum mechanics concerning the wave function results in the thesis that the quantum mechanical wave function is a link between human cognition and the physical world. The reticence on the part of physicists to adopt this thesis is discussed. A comparison is made to the behaviorists’ consideration of mind, and the historical roots of how the problem concerning the quantum mechanical wave function arose are discussed. The basis for an empirical demonstration that the wave function is a link between human cognition and the physical world is provided through developing an experiment using methodology from psychology and physics. Based on research in psychology and physics that relied on this methodology, it is likely that Einstein, Podolsky, and Rosen’s theoretical result that mutually exclusive wave functions can simultaneously apply to the same concrete physical circumstances can be implemented on an empirical level. Original article in The Journal of Mind and Behavior is on JSTOR at https://www.jstor.org/stable/pdf/43853678.pdf?seq=1 . Preprint on CERN preprint server at https://cds.cern.ch/record/569426 .

scholarly journals On Whether People Have the Capacity to Make Observations of Mutually Exclusive Physical Phenomena

2021 ◽  
Author(s):  
Douglas Michael Snyder
Keyword(s):  
Wave Function ◽  
Sigmund Freud ◽  
Physical System ◽  
William James ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Human Observer ◽  
Physical Interaction ◽  
Mechanical Wave ◽  
Physical Phenomena

It has been shown by Einstein, Podolsky, and Rosen that in quantum mechanics one of two different wave functions predicting specific values for quantities represented by non-commuting Hermitian operators can characterize the same physical system, without a physical interaction responsible for which wave function is realized in a measurement. This result means that one can make predictions regarding mutually exclusive features of a physical system. It is important to ask whether people can make observations of mutually exclusive phenomena. Our everyday experience informs us that a human observer is capable of observing one set of physical circumstances at a time. Evidence from psychology, though, indicates that people may have the capacity to make observations of mutually exclusive physical phenomena, even though this capacity in not generally recognized. Working independently, Sigmund Freud and William James provided some of this evidence. How the nature of the quantum mechanical wave function is associated with the problem posed by Einstein, Podolsky, and Rosen is addressed at the end of the paper.

scholarly journals The Third Five-Parametric Hypergeometric Quantum-Mechanical Potential

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
T. A. Ishkhanyan ◽  
A. M. Ishkhanyan
Keyword(s):  
General Solution ◽  
Linear Combination ◽  
Schrodinger Equation ◽  
Short Range ◽  
Hypergeometric Functions ◽  
Variable Parameter ◽  
Fundamental Solutions ◽  
Quantum Mechanical ◽  
Square Root ◽  
The Third

We introduce the third five-parametric ordinary hypergeometric energy-independent quantum-mechanical potential, after the Eckart and Pöschl-Teller potentials, which is proportional to an arbitrary variable parameter and has a shape that is independent of that parameter. Depending on an involved parameter, the potential presents either a short-range singular well (which behaves as inverse square root at the origin and vanishes exponentially at infinity) or a smooth asymmetric step-barrier (with variable height and steepness). The general solution of the Schrödinger equation for this potential, which is a member of a general Heun family of potentials, is written through fundamental solutions each of which presents an irreducible linear combination of two Gauss ordinary hypergeometric functions.

A new way of finding locations of zeros of wave functions

2004 ◽  
Vol 82 (7) ◽  
pp. 549-560 ◽  
Author(s):  
A Nanayakkara
Keyword(s):  
Wave Function ◽  
Complex Systems ◽  
Schrodinger Equation ◽  
Equation Of Motion ◽  
Wave Functions ◽  
Analytic Method ◽  
04.20 Jb ◽  
Complex Zeros ◽  
03.65 Ge ◽  
03.65 Sq

A new analytic method is presented for evaluating zeros of wave functions. In this method, locating the zeros of wave functions of the Schrodinger equation is converted to finding the roots of a polynomials. The coefficient of this polynomial can be evaluated analytically for a class of potentials. The speciality of this method is that the zeros are located without solving an equation of motion for the wave function. The method is valid for both real and complex systems and can be applied for locating both real and complex zeros. Examples are given to illustrate the method. PACS Nos.: 02.30.Mv, 03.65.Ge, 03.65.Sq, 03.65.–w, 04.20.Jb, 04.20.Ha, 05.45.Mt

scholarly journals Solutions of the D-Dimensional Schrödinger Equation with the Hyperbolic Pöschl-Teller Potential plus Modified Ring-Shaped Term

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Ibsal A. Assi ◽  
Akpan N. Ikot ◽  
E. O. Chukwuocha
Keyword(s):  
Wave Function ◽  
Energy Level ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Radial Equation ◽  
Energy Eigenvalues ◽  
Separation Of Variable ◽  
Nu Method ◽  
Hyperspherical Coordinate

We solve the D-dimensional Schrödinger equation with hyperbolic Pöschl-Teller potential plus a generalized ring-shaped potential. After the separation of variable in the hyperspherical coordinate, we used Nikiforov-Uvarov (NU) method to solve the resulting radial equation and obtain explicitly the energy level and the corresponding wave function in closed form. The solutions to the energy eigenvalues and the corresponding wave functions are obtained using the NU method as well.

scholarly journals SOLVING THE SCHRÖDINGER EQUATION FOR BOUND STATES WITH MATHEMATICA 3.0

1999 ◽  
Vol 10 (04) ◽  
pp. 607-619 ◽  
Author(s):  
WOLFGANG LUCHA ◽  
FRANZ F. SCHÖBERL
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Interaction Potential ◽  
Schrodinger Equation ◽  
Bound States ◽  
Energy Eigenvalue ◽  
Wave Functions ◽  
Spherically Symmetric ◽  
Method Of Solution ◽  
Numerical Integration Procedure

Using Mathematica 3.0, the Schrödinger equation for bound states is solved. The method of solution is based on a numerical integration procedure together with convexity arguments and the nodal theorem for wave functions. The interaction potential has to be spherically symmetric. The solving procedure is simply defined as some Mathematica function. The output is the energy eigenvalue and the reduced wave function, which is provided as an interpolated function (and can thus be used for the calculation of, e.g., moments by using any Mathematica built-in function) as well as plotted automatically. The corresponding program schroedinger.nb can be obtained from [email protected].

scholarly journals Isgur–Wise function within a QCD quark model with Airy's function as the wave function of heavy–light mesons

2013 ◽  
Vol 91 (1) ◽  
pp. 34-42 ◽  
Author(s):  
Sabyasachi Roy ◽  
N.S. Bordoloi ◽  
D.K. Choudhury
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Quark Model ◽  
Infinite Series ◽  
Schrodinger Equation ◽  
Quantum Mechanical ◽  
Perturbation Techniques ◽  
Mechanical Perturbation ◽  
Light Mesons

We report a somewhat improved wave function for mesons by taking the linear confinement term in standard QCD potential as a parent and the coulombic term as a perturbation while applying quantum mechanical perturbation techniques in solving the Schrödinger equation with such a potential. We find that Airy's infinite series appears in the wave function of the mesons. We report our calculations on the Isgur–Wise function and its derivatives for heavy–light mesons within this framework.

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