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Solving the Schrödinger equation for Kosen students : Hydrogen atom

祐二 佐々井

Published Mar 30, 2021• 0 citations
OpenAlex

Abstract

Solving the Schrödinger equation in quantum mechanics is not easy. In this research report, we would like to solve the Schrödinger equation concretely for the energy eigenvalue problem of the hydrogen atom. To help Kosen students understand this problem, we explain general solutions from the calculation of concrete solutions. We also explain the relationship between the principal quantum number n, the azimuthal quantum number l, the magnetic quantum number m, and the spin quantum number s.

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The Schrodinger equation solve the Hydrogen-like Atom, the solution is very complicate mathematically and supply a very little understanding why the atom behave so. In 1913 Bohr proposed a very simple model of the hydrogen atom. I extend this model to see what happens to the atom under a magnetic field. The results obtained using my simple approach are almost the same results obtained from Schrodinger equation with the endless solution, but the physical explanation is entirely different. I leave you the task to check what the correct explanation to the Hydrogen Atom is See also QUANTUM MECHANICS Solved using only Newtonian Mechanics http://vixra.org/abs/1305.0078

•May 1, 2013•0 citations
Solving the Schrödinger equation in quantum mechanics is not easy. In this research report, we would like to solve the Schrödinger equation concretely for the energy eigenvalue problem of the hydrogen atom. To help Kosen students understand this problem, we explain general solutions from the calculation of concrete solutions. We also explain the relationship between the principal quantum number n, the azimuthal quantum number l, the magnetic quantum number m, and the spin quantum number s.
During the solution of the Schrodinger equation for the hydrogen atom by separation of variables, we come across three quantum numbers, namely the azimuthal quantum number l , the magnetic quantum number m l , and the principal quantum number n . The quantum number l arises as a separation constant during the breaking apart of radial and angular segments of the Schrodinger equation and the quantum number m l occurs while separating the azimuthal component from the polar component of the angular part. The quantum number n is necessary to render the series for the non-asymptotic part of the radial wave function convergent. The three quantum numbers are interdependent because after selecting n , the selection of l is restricted to allowed values and similarly after choosing l , limited values are permitted for m l . The fourth quantum number m s representing electron spin is not a direct consequence of solving the Schrodinger equation. The detailed solution procedure of the Schrodinger equation including the normalization of the wave function, restrictions on the values of quantum numbers, and allied analysis are explained.
The Schrödinger equation is the cornerstone of quantum mechanics and is used to determine the energy levels of the hydrogen atom. However, in the teaching of Quantum Chemistry, many students tend to regard the solutions of the Schrödinger equation as purely mathematical results without fully understanding the physical origin of quantum energy levels. In this paper, an approximate model of the hydrogen atom is first developed based on the Heisenberg uncertainty principle and the Coulomb potential. This model is then used to derive the Bohr radius and the ground-state energy by minimizing the total energy function. Subsequently, the time-independent Schrödinger equation is formulated and solved by the method of separation of variables in spherical coordinates to obtain the exact energy spectrum of the hydrogen atom. The results demonstrate that both the approximate model and the exact solution predict the same ground-state energy, E1 = −13.6 eV, while the Schrödinger equation further generalizes this result to the quantized energy spectrum, En = −13.6/n2. These findings indicate that the solutions of the Schrödinger equation are not "miraculous" mathematical outcomes but rather the inevitable consequences of the fundamental principles of quantum mechanics. This approach provides a clearer physical interpretation of quantum energy levels and offers significant pedagogical value for the teaching of Quantum Chemistry.
A hydrogen atom in a constant magnetic field is a problem whose Schrödinger equation has been considered to be analytically unsolvable. However, it has been found for the case of two-dimensional space some specific values of the magnetic field for which the problem has analytical solutions. In this paper, we give a full theory for these analytical solutions. First, we construct an equation to calculate of exact-solution-points, the values of magnetic field which allow exact solutions for the Schrödinger equation. The explicit forms of the wave functions corresponding to these analytical solutions are also given. Second, we analyse the exact-solution-points and discover some important rules. (i) There exist (n − 1)2 different values of the magnetic field so that the Schrödinger equation of a two-dimensional hydrogen atom in such field has exact analytical solutions with the principal quantum number n. (ii) For more details, among these (n − 1)2 exact analytical solutions, there are nr=n−|m|−1 solutions with the same magnetic quantum number m. Both properties are also proved mathematically in this paper. For illustration, some concrete values are shown for the cases of quantum number n up to 8.
This chapter considers Schrodinger's model of hydrogen atom and analyzes in details the quantum wave mechanics approach to the motion of the electron in the hydrogen atom. The stationary Schrodinger equation is not easy to solve as it stands, and the whole problem looks rather complicated when expressed in terms of Cartesian coordinates. An essential difference between Bohr's and Schrodinger's models is that there are three quantum numbers, n , l , m , rather than the one number n describing a state of the electron. Moreover, in Bohr's model the atom is, at each instant, in one and only one energy state, that is., at each instant, the state of the atom is definite, but according to Schrodinger's model, the atom can exist in a superposition of states. In other words, the atom can be simultaneously in more than one energy state at a given instant. This is clearly incompatible with Bohr's model.
In this investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.
The Schrodinger equation solve the Hydrogen-like Atom, the solution is very complicate mathematically and supply a very little understanding why the atom behave so. In 1913 Bohr proposed a very simple model of the hydrogen atom. I extend this model to see what happens to the atom under a magnetic field. The results obtained using my simple approach are almost the same results obtained from Schrodinger equation with the endless solution, but the physical explanation is entirely different. I leave you the task to check what the correct explanation to the Hydrogen Atom is See also QUANTUM MECHANICS Solved using only Newtonian Mechanics http://vixra.org/abs/1305.0078