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International Journal of Theoretical and Mathematical Physics

Riemannian Quantum Theory of a Particle in a Finite-Potential Well

I. I. Ewa, Lucas Williams Lumbi, S. X. K. Howusu

Published Jan 1, 2018• 0 citations
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Abstract

In this article, the golden Riemannian Laplacian operator was constructed using the golden metric tensor in spherical polar coordinate and was applied to the Schrodinger wave equation in order to obtain the golden Riemannian Schrodinger equation for a particle in a finite-potential well. The results are that the golden Riemannian Laplacian operator and golden Riemannian Schrodinger equation were augmented with additional correction terms; which are not found in the existing equations and can be applied to a finite-potential well problem, so as to obtain the expression for the allowed energy values.

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In this article, the golden Riemannian Laplacian operator was constructed using the golden metric tensor in spherical polar coordinate and was applied to the Schrodinger wave equation in order to obtain the golden Riemannian Schrodinger equation for a particle in a finite-potential well. The results are that the golden Riemannian Laplacian operator and golden Riemannian Schrodinger equation were augmented with additional correction terms; which are not found in the existing equations and can be applied to a finite-potential well problem, so as to obtain the expression for the allowed energy values.
In our previous work titled “Riemannian Quantum Theory of a Particle in a Finite-Potential Well", we constructed the Riemannian Laplacian operator and used it to obtain the Riemannian Schrodinger equation for a particle in a finite-potential well. In this work, we solved the golden Riemannian Schrodinger equation analytically to obtain the particle energy. The solution resulted in two expressions for the energy of a particle in a finite-potential well. One of the expressions is for the odd energy levels while the other is for the even energy levels.
We show that the first order form of the Schrodinger equation proposed in [1] can be obtained from the Dirac equation in the non-relativistic limit. We also show that the Pauli Hamiltonian is obtained from this equation by requiring local gauge invariance. In addition, we study the problem of a spin up particle incident on a finite potential barrier and show that the known quantum mechanical results are obtained. Finally, we consider the symmetric potential well and show that the quantum mechanical expression for the quantized energy levels of a particle is obtained with periodic boundary conditions. Based on these conclusions, we propose that the equation introduced in [1] is the non-relativistic limit of the Dirac equation and more appropriately describes spin 1/2 particles in the non-relativistic limit.
We consider the quantum-mechanical problem of a particle bound to a configuration of spherical potentials, each of finite range. If the Schr\"odinger equation can be solved for each potential by itself, then it is shown how to solve it for the configuration, provided the potentials do not overlap. The energy levels are the zeros of a determinant of formally infinite order, but in practice this is always well approximated by a finite determinant and often by one of small order. As illustrative examples we consider some states of a particle bound to three square wells with the configuration of an equilateral triangle, and to two truncated Coulomb potentials. The possibility of extending an approximate version of the method to overlapping potentials is pointed out.
The Schrödinger operators on the Newtonian spacetime are defined in a way which make them independent of the class of inertial observers. In this picture the Schrödinger operators act not on functions on the spacetime but on sections of a certain one-dimensional complex vector bundle—the Schrödinger line bundle . This line bundle has trivializations indexed by inertial observers and is associated with an U (1)-principal bundle with an analogous list of trivializations—the Schrödinger principal bundle . If an inertial frame is fixed, the Schrödinger bundle can be identified with the trivial bundle over spacetime, but as there is no canonical trivialization (inertial frame), these sections interpreted as 'wavefunctions' cannot be viewed as actual functions on the spacetime. In this approach, the change of an observer results not only in the change of actual coordinates in the spacetime but also in a change of the phase of wavefunctions. For the Schrödinger principal bundle, a natural differential calculus for 'wave forms' is developed that leads to a natural generalization of the concept of the Laplace–Beltrami operator associated with a pseudo-Riemannian metric. The free Schrödinger operator turns out to be the Laplace–Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schrödinger principal bundle. The presented framework does not involve any ad hoc or axiomatically introduced geometrical structures. It is based on the traditional understanding of the Schrödinger operator in a given reference frame—which is supported by producing right physics predictions—and it is proven to be strictly related to the frame-independent formulation of analytical Newtonian mechanics and Hamilton–Jacobi equations that makes a bridge between the classical and quantum theory.
We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.
The quantum model of finite potential well is a most subject in the book of quantum mechanics. The solutions maybe derive from one dimension Schrödinger’s equation for describe a bound state energy of confined particle in the finite potential well. Sometime, the problem of confined particle in the finite potential well does not have an exact solution yet, but there are in fact exact solution.In this work a numerical technique is adopted to find the exact solution for the quantum model of finite potential well. To achieve this numerical solution, I use Fortran code computer program by the Newton-Raphson method for calculating the Eigen state energy for a particle (electron) confined in the finite potential well. Five values of potential width and depth have studied, the results showed that large potential width and large potential depth led me to more bound states, while smaller potential width and depth led me to fewer bound states.A comparison between infinite potential and finite potential has been presented. The results bound state energy and penetration. The results of comparison showed that the wave function of particle confined in the infinite potential cannot penetrate in the well, while the wave function of particle in the finite potential was penetrate in the well; the energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well. For confined particle in 2 or 3-dimension, we can reduce from result of 1-dimension.