Hydrogen-like Atoms

Peter M. Bell

In the strange world of low‐temperature physics, few more unusual phenomena have been discovered than monatomic hydrogen gas. According to a recent report (New Scientist, Jan. 1981, p. 204), hydrogen atoms do not tend to pair up to form molecules under special low‐temperature‐cryogenic experimental conditions. The reasons for the existence of atomic hydrogen gas at ultra‐low temperature, and for its implications, lie in an understanding of its almost pure quantum mechanical properties. Practical uses of atomic hydrogen will await more knowledge of its properties, but right now, if available in sufficient abundance, atomic hydrogen could be used in maser clocks because it emits highly defined frequencies of microwave radiation at room temperature, and atomic hydrogen gas could be used as an energy conservative fuel, as it recombines to the diatomic molecule. For every recombination between two hydrogen atoms, 4.5 eV of energy are released, which calculates to be over 1 million watts of power per second for 10 grams of atomic hydrogen recombined. The large impulse relative to its light mass could be significant in applications of atomic hydrogen as a rocket fuel. Practical applications aside, the main interest in atomic hydrogen at this moment is in the insight into quantum mechanics that study of its properties may provide. Atomic hydrogen atoms ‘abound throughout the universe,’ even though not on Earth, and thus quantum effects in space, and the hydrogen‐rich planets as well, may be elucidated by results of these studies.

Shu Hotta

Toshihiro Iwai

Quantization of the conformal Kepler problem is defined and studied in order that the quantized system, which will be referred to as a conformal hydrogen atom, may associate the harmonic oscillator with the hydrogen atom. The conformal hydrogen atom shares with the harmonic oscillator the eigenspaces of negative energies. The four-dimensional conformal hydrogen atom reduces to the three-dimensional ordinary hydrogen atom. The symmetry group SO(4) of the hydrogen atom is brought out from the symmetry subgroup of the harmonic oscillator. The conformal hydrogen atom gives an example of those quantum systems of which the configuration spaces are curved Riemannian spaces with nonconstant scalar curvatures and of which the Hamiltonian operators depend on the scalar curvatures.

Shu Hotta

Yoshifumi Ito

In this paper, we study the derivation and the solution of Schrodinger equation of the hydrogen atoms using the theory of Natural Statistical Physics. Using these results, we study the phenomena of the spectra of hydrogen atoms and the phenomena of the stability of hydrogen atoms. Here we consider the system of hydrogen atoms for which we need not to consider the in uence of the outer electro-magnetic eld. This is the case where there is no in uence of outer electro-magnetic eld or where we can neglect the in uence of the outer electro-magnetic eld. In this paper, we succeeded in deriving the Schrodinger equation in the natural and reasonable way by the method of variational calculus. Thereby we can obtain the complete understanding of the phenomena of the spectra of hydrogen atoms and the phenomena of the stability of hydrogen atoms. We remark that the model of the system of hydrogen atoms considered in this paper does not concern with the scattering state of hydrogen atoms. In this paper, Fourier's method plays the fundamental role. For the results of this paper, we refer to Ito[16], [19], [20], [22], [24], [25], [26]. 2010 Mathematics Subject Classi cation. Primary 82D99, 82B99, 81Q99.