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Register a new userOn the energy of electric field in hydrogen atom
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Authors
Yuri Kornyushin
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It is shown that hydrogen atom is a unique object in physics having negative energy of electric field, which is present in the atom. This refers also to some hydrogen-type atoms: hydrogen anti-atom, atom composed of proton and antiproton, and positronium.
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Closed-orbit theory provides a general approach to the semiclassical description of photo-absorption spectra of arbitrary atoms in external fields, the simplest of which is the hydrogen atom in an electric field. Yet, despite its apparent simplicity, a semiclassical quantization of this system by means of closed-orbit theory has not been achieved so far. It is the aim of this paper to close that gap. We first present a detailed analytic study of the closed classical orbits and their bifurcations. We then derive a simple form of the uniform semiclassical approximation for the bifurcations that is suitable for an inclusion into a closed-orbit summation. By means of a generalized version of the semiclassical quantization by harmonic inversion, we succeed in calculating high-quality semiclassical spectra for the hydrogen atom in an electric field.
Semiclassical periodic-orbit theory and closed-orbit theory represent a quantum spectrum as a superposition of contributions from individual classical orbits. Close to a bifurcation, these contributions diverge and have to be replaced with a uniform approximation. Its construction requires a normal form that provides a local description of the bifurcation scenario. Usually, the normal form is constructed in flat space. We present an example taken from the hydrogen atom in an electric field where the normal form must be chosen to be defined on a sphere instead of a Euclidean plane. In the example, the necessity to base the normal form on a topologically non-trivial configuration space reveals a subtle interplay between local and global aspects of the phase space structure. We show that a uniform approximation for a bifurcation scenario with non-trivial topology can be constructed using the established uniformization techniques. Semiclassical photo-absorption spectra of the hydrogen atom in an electric field are significantly improved when based on the extended uniform approximations.
The S-matrix theory formulation of closed-orbit theory recently proposed by Granger and Greene is extended to atoms in crossed electric and magnetic fields. We then present a semiclassical quantization of the hydrogen atom in crossed fields, which succeeds in resolving individual lines in the spectrum, but is restricted to the strongest lines of each n-manifold. By means of a detailed semiclassical analysis of the quantum spectrum, we demonstrate that it is the abundance of bifurcations of closed orbits that precludes the resolution of finer details. They necessitate the inclusion of uniform semiclassical approximations into the quantization process. Uniform approximations for the generic types of closed-orbit bifurcation are derived, and a general method for including them in a high-resolution semiclassical quantization is devised.
In this work we investigate the $q$-deformation of the $so(4)$ dynamical symmetry of the hydrogen atom using the theory of the quantum group $su_q(2)$. We derive the energy spectrum in a physically consistent manner and find a degeneracy breaking as well as a smaller Hilbert space. We point out that using the deformed Casimir as was done before leads to inconsistencies in the physical interpretation of the theory.
We obtain the following analytical formula which describes the dependence of the electric potential of a point-like charge on the distance away from it in the direction of an external magnetic field B: Phi(z) = e/|z| [ 1- exp(-sqrt{6m_e^2}|z|) + exp(-sqrt{(2/pi) e^3 B + 6m_e^2} |z|) ]. The deviation from Coulombs law becomes essential for B > 3pi B_{cr}/alpha = 3 pi m_e^2/e^3 approx 6 10^{16} G. In such superstrong fields, electrons are ultra-relativistic except those which occupy the lowest Landau level (LLL) and which have the energy epsilon_0^2 = m_e^2 + p_z^2. The energy spectrum on which LLL splits in the presence of the atomic nucleus is found analytically. For B > 3 pi B_{cr}/alpha, it substantially differs from the one obtained without accounting for the modification of the atomic potential.
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