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Journal of Mathematical Physics

Exact analytical solutions of a two-dimensional hydrogen atom in a constant magnetic field

Dai-Nam Le, Ngoc-Tram D. Hoang, Van-Hoang Le

Published Apr 1, 2017• 19 citations

Abstract

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.

DOI: 10.1063/1.4979618

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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.

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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.

Abstract The problem of the hydrogen atom in a magnetic field can be treated satisfactorily by approximate methods which are quite accurate for very low or very high magnetic field strengths. Before this paper the only useful treatments for intermediate field strengths were variational methods giving upper bounds to the energies. As described in the paper the diffusion QMC method was applied in the same way as in earlier solutions for the hydrogen atom, but with additional magnetic terms added to the Hamiltonian expression. These include complex expressions and the required wavefunction includes a corresponding factor. With the magnetic field aligned along one axis, integration over the angle ϕ in (r, θ, ϕ) coordinates reduces the additional terms to real functions of r and z as well as the magnetic quantum number m. In the diffusion/multiplication processes of a random walk procedure these may be simply included with the potential energy term. In this study the energy at each step was determined from the average multiplication factor for the walkers. Importance sampling was not used. Results were obtained for quantum numbers m = 0, 1 at each of three magnetic field strengths. A comparison of results showed good agreement with the lowest values obtained in several of the earlier analytic variational calculations.

In this work the Schrödinger equation of the hydrogen-like atom is analytically solved. Three sets of analytical solution are obtained if the factor r−l is not neglected. The first solution is the same as the traditional radial wave function; another one diverges; the last one is far different from the traditional solution. On the consideration of the finite size of the nucleus, the third wave function does not diverge while r approaches to zero. Its radial wave function has below characteristics: (1) the angular-momentum quantum number l must be greater than the principal quantum number n; (2) l must not be 0 or 1; (3) the electron-cloud distribution differs from the traditional one; (4) the electron is closer to the nucleus by comparison with that in traditional results. On the other hand, the validity of solutions needs to be verified experimentally. 1.

The two-dimensional hydrogen atom in an external magnetic field is considered in the context of phase space. Using the solution of the Schrödinger equation in phase space, the Wigner function related to the Zeeman effect is calculated. For this purpose, the Bohlin mapping is used to transform the Coulomb potential into a harmonic oscillator problem. Then, it is possible to solve the Schrödinger equation easier by using the perturbation theory. The negativity parameter for this system is realised.