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Journal of Physics Conference Series

Improved variational method that solves the energy eigenvalue problem of the hydrogen atom

L. M. León Hilario, E. Aldo Arroyo

Published Dec 1, 2018• 0 citations
Publisher

Abstract

In most quantum mechanics textbooks for graduate studies, the hydrogen atom is studied in an approximate way by means of the variational method. The type of trial functions commonly used are the Gaussian and the Lorenzian. In this paper we consider a natural generalization of the Mei method to improve the trial wave functions [1] applied to hydrogen atom. We propose a sequence of functions as trial wave functions to calculate the eigen-energies of the hydrogen atom. These trial wavefunctions are given in terms of three variational parameters, one of them is fixed by means of the normalization condition and the other two are adjustable parameters. One of these parameters can be chosen as being an integer number, then this parameters will define the sequence of functions. We will show that when the integer parameter approaches to infinity, the ground state, first excited state and second excited state converge to the exact results.

DOI
10.1088/1742-6596/1143/1/012031

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In most quantum mechanics textbooks for graduate studies, the hydrogen atom is studied in an approximate way by means of the variational method. The type of trial functions commonly used are the Gaussian and the Lorenzian. In this paper we consider a natural generalization of the Mei method to improve the trial wave functions [1] applied to hydrogen atom. We propose a sequence of functions as trial wave functions to calculate the eigen-energies of the hydrogen atom. These trial wavefunctions are given in terms of three variational parameters, one of them is fixed by means of the normalization condition and the other two are adjustable parameters. One of these parameters can be chosen as being an integer number, then this parameters will define the sequence of functions. We will show that when the integer parameter approaches to infinity, the ground state, first excited state and second excited state converge to the exact results.
A simple and accurate variational method is formulated and applied to the ground and excited states of the hydrogen atom as well as the hydrogen molecule ion in a homogeneous magnetic field of arbitrary strength. For the ground state, the electronic wave function o is assumed to be a function of a single, appropriately chosen variable R which contains a variational parameter. The functional form of o, and the corresponding energy, are determined by solving a Schr6dinger- like equation, in which the variational parameter appears explicitly in the kinetic and potential energy terms. The variational parameter is optimized by minimizing the lowest eigenvalue of the equation. Excited states may also be obtained by a slight modification of the method, and explicit calculations are carried out for the m1 = - 1 first excited state of hydrogen and the ( )-state of H2 +. Numerical comparisons are made with some recent papers in the literature. Subject headings: atomic processes - magnetic fields - molecular processes - quantum mechanics
In applying the variational method, we demonstrate that, with careful study of the general features of the trial wave functions, we can achieve systematic improvements in calculating the eigenvalues of quantum mechanical systems. We apply five different sets of trial wave functions to calculate the ground state and first excited state energies of weakly bound potentials, such as V(x) = ∣x∣, √∣x∣ and ln (∣x∣). We demonstrate that accurate results can be obtained through a thorough understanding of the asymptotic behaviours. With proper adjustments, we show how to improve trial wave functions. Finally, by calculating to the fifth excited states, we compare the exact and variational results with those obtained from the semi-classical method, and discuss the merits of the different approaches.
When applying the variational method, it is demonstrated that the systematic improvements in calculating the eigen‐energies of quantum mechanical systems can be achieved by careful study of the general features of the trial wave functions. Two sets of trial wave functions were used for the harmonic oscillator potential. In the first example, it is shown, even with a crude initial guess, that accurate results will be obtained by adjusting the wave function without introducing any additional variational parameters. Even though the second example requires more algebraic manipulation, it is shown that it converges to the exact result much faster. It is suggested that these calculations be incorporated in computer algebra software packages for the teaching of quantum mechanics.
Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.
The direct variational method is used to study two simple confined systems, namely, the hydrogen atom and the harmonic oscillator within impenetrable spherical boxes. The trial wave functions have been assumed as the product of the ‘‘free’’ solutions of the corresponding Schrödinger equation and a simple function that satisfies the respective boundary conditions. The energy levels obtained in this way are extremely close to the exact ones, thus proving the utility of the proposed method. This might constitute a complementary exercise for undergraduate students since usual quantum mechanics textbooks do not explore the use of the variational method to study confined systems.