Solutions of Many-Body Atomic Systems Using Green's Function and Variational Methods


Student thesis: Doctoral Thesis

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Award date24 Apr 2017


The Schrödinger wave equation is the fundamental wave equation in quantum mechanics. It is important in the study of atomic and molecular physics as it gives information about the state of a microscopic system, such as an atom or a molecule. However, solving the Schrödinger wave equation is not an easy task. The exact analytical solution of the wave equation is possible only for a few well-known cases, such as quantum harmonic oscillator and hydrogen atom. In the single electron case of hydrogen atom, the exact analytical solution is obtained after a lengthy calculation. In the case of a system with more than two particles, the exact analytical solution is almost impossible. Therefore, approximate methods are used to study the many-body wave equation. Among the available methods, the perturbation theory, the Hartree-Fock solution, and the variational method are the most well-known approximate methods usually used to study the many-body Schrödinger wave equation. Besides analytical solutions, there have been many different attempts to solve the equation numerically. Finite difference and finite element methods are used to solve the equation computationally. Owing to its importance, the search for an efficient method for solving the Schrödinger wave equation is still a hot topic for researchers.

In this connection, we studied the wave equation computationally by using two different methods: The Green’s function technique and the variational method. In our first attempt using the Green’s function approach, we solved the non-relativistic radial wave equation for the hydrogen atom. The Green’s function is renowned for the solution of differential equations as in some cases it can make the solution of the equations much simpler.

In the application of Green’s function technique to the radial hydrogen wave equation, we solved the equation using the Lippmann-Schwinger type equation, which requires the Green’s function for the solution. The Green’s function for the radial wave equation is worked out using the Laplace transform technique. A singularity appeared in the final form of the Green’s function, to avoid the singular solution we did substitution which leads to the determination of the discrete energy eigenvalue of the hydrogen wave equation. Further, the wave function was proposed with the help of boundary conditions r=0, and r=∞, where r is the radial distance. With the help of the Green’s function and the proposed wave function, the Lippmann-Schwinger type equation were integrated using a computational programing. Further, the iterative integration technique was used to refine the result to the standard known solution. The normalized resultant wave after the initial three iterations is plotted against the standard radial wave for comparison and it is deduced that our resultant wave function converges to the standard solution as the number of iterations increases.

In our second attempt, the Schrödinger wave equation was solved using the linear principle of variation. The basic aim of this work is to obtain the three-dimensional wave from the linear combination of x,y,z−one-dimensional wave functions. Basically, the presence of the potential term prevents the splitting of the three-dimensional hydrogen atom wave equation into three one-dimensional equations using the separation of variables technique in the Cartesian coordinates, therefore the exact three-dimensional wave function of the hydrogen atom cannot be obtained from any proposed components in the (x,y,z) coordinate system. However, the variational method provides an alternative technique to obtain the approximate three-dimensional wave function from the linear combination of one-dimensional waves. Therefore, we considered the one-dimensional wave functions along the axes x,y,z as basis and constructed a proposed wave function from the linear combination of these basis with variable coefficients. The variational method of minimization was used to determine these unknown coefficients along with the energy eigenvalues of the proposed wave function. Further, the iteration approach was used to reach the convergence of the result to the true solution, i.e. the resultant wave is inserted into the Schrödinger wave equation and the residual term is obtained as Hψ−Eψ. A new wave function was proposed by adding the residual term to the wave function ψ with variable coefficients. For the next iteration, the process was repeated. The iterations were continued until the residual vector becomes significantly reduced to a very low value. The work is extended to helium atom too and the ground state energy of the helium atom was calculated. We also revised the work done by Nicholas et al. and found out the energy eigenvalue of the helium atom computationally. In this case, the finite difference method was used to solve the non-relativistic radial wave equation for the helium atom. We successfully determined the ground state energy of the helium atom with this method.