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Journal of Physics A Mathematical and General

The Kustaanheimo-Stiefel transformation applied to the hydrogen atom: using the constraint equation and resolving a wavefunction discrepancy

Eamonn Cahill

Published May 7, 1990• 7 citations
Publisher

Abstract

The authors show how explicit use of the constraint equation that arises when the Kustaanheimo-Stiefel transformation (1965) is applied to the hydrogen atom in R 3 enables them to determine the wavefunctions for the constrained harmonic oscillator in R 4 with the correct restriction on the quantum numbers in a direct and natural manner. They then highlight and resolve a discrepancy between the R 4 oscillator wavefunctions produced by different authors by considering the measure to be used in R 4 and the operator ordering in the quantised Hamiltonian.

DOI
10.1088/0305-4470/23/9/016

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Physical Review A•Apr 12, 2011•8 citations
The authors show how explicit use of the constraint equation that arises when the Kustaanheimo-Stiefel transformation (1965) is applied to the hydrogen atom in R 3 enables them to determine the wavefunctions for the constrained harmonic oscillator in R 4 with the correct restriction on the quantum numbers in a direct and natural manner. They then highlight and resolve a discrepancy between the R 4 oscillator wavefunctions produced by different authors by considering the measure to be used in R 4 and the operator ordering in the quantised Hamiltonian.
The problem of deriving macroscopic properties from the Hamiltonian of the hydrogen atom is resumed by extending previous results in the literature, which predicted elliptic orbits, into the region of hyperbolic orbits. As a main tool, coherent states of the harmonic oscillator are used which are continued to imaginary frequencies. The Kustaanheimo-Stiefel (KS) map is applied to transform the original configuration space into the product space of four harmonic oscillators with a constraint. The relation derived between real time and oscillator (pseudo) time includes quantum corrections. In the limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$, the time-dependent mean values of position and velocity describe the classical motion on a hyperbola and a circular hodograph, respectively. Moreover, the connection between pseudotime and real time comes out in analogy to Kepler's equation for elliptic orbits. The mean-square-root deviations of position and velocity components behave similarly in time to the corresponding ones of a spreading Gaussian wave packet in free space. To check the approximate treatment of the constraint, its contribution to the mean energy is determined with the result that it is negligible except for energy values close to the parabolic orbit with eccentricity equal to 1. It is inevitable to introduce a suitable scalar product in R4{\mathbb{R}}^{4}R4 which makes both the transformed Hamiltonian and the velocity operators Hermitian. An elementary necessary criterion is given for the energy interval where the constraint can be approximated by averaging.