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Acta Physica Sinica

COHERENT STATES OF HYDROGEN ATOM

Eng Q

Published Jan 1, 1991• 0 citations
OpenAlex

Abstract

From Kustannheimo-Stiefel transformation, the hygrogen problem in quantum mechanics can be transformed to that of a four dimensional isotropic harmonic oscillator with constrain. The coherent states are defined, for which the expectation values of the position and the momentum are shown to give the classical Kepler orbits. The uncertainty relations for the coherent states are discussed as well.

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Using the harmonic oscillator representation of the hydrogen atom and constructing the appropriate coherent state corresponding to the minimum uncertainty product, the classical limits of the two-dimensional and the three-dimensional hydrogen atom are examined. The method adopted is similar to that used by Bhaumik, Dutta Roy and Ghosh (1986). The authors deduce the classical limit by requiring that the expectation value (r) of the radial variable is large for the two-dimensional hydrogen atom. The resulting trajectory is an ellipse satisfying the conditions of the Kepler orbit in classical mechanics. In order to obtain the classical limit of the three-dimensional hydrogen atom, besides the condition of large (r), they make the uncertainty Delta r=((r 2 )-(r) 2 ) 1/2 a minimum with respect to certain parameters of the coherent state. Imposition of only the first condition leads to an elliptical orbit, which is not a Kepler orbit in general. Imposition of both conditions leads to an orbit identical with that of the two-dimensional hydrogen atom. However, the time evolution of the expectation values of position variables are consistent with the corresponding results of the Kepler problem only for small values of eccentricity, but not in general.

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TURKISH JOURNAL OF PHYSICS•Oct 11, 2000•1 citation
From Kustannheimo-Stiefel transformation, the hygrogen problem in quantum mechanics can be transformed to that of a four dimensional isotropic harmonic oscillator with constrain. The coherent states are defined, for which the expectation values of the position and the momentum are shown to give the classical Kepler orbits. The uncertainty relations for the coherent states are discussed as well.
The Kustaanheino-Stiefel transformation is used to transform the Coulomb problem to a four-dimensional isotropic harmonic oscillator with a constraint. Ordinary coherent states are introduced over this oscillator and are shown to evolve in a fictitious time. When projected back into the physical space, the states follow the classical Kepler orbits in physical time. The resulting quasiclassical picture is obtained without the Bohr correspondence principle of taking only contributions from high principal quantum numbers.
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.
Using the harmonic oscillator representation of the hydrogen atom and constructing the appropriate coherent state corresponding to the minimum uncertainty product, the classical limits of the two-dimensional and the three-dimensional hydrogen atom are examined. The method adopted is similar to that used by Bhaumik, Dutta Roy and Ghosh (1986). The authors deduce the classical limit by requiring that the expectation value (r) of the radial variable is large for the two-dimensional hydrogen atom. The resulting trajectory is an ellipse satisfying the conditions of the Kepler orbit in classical mechanics. In order to obtain the classical limit of the three-dimensional hydrogen atom, besides the condition of large (r), they make the uncertainty Delta r=((r 2 )-(r) 2 ) 1/2 a minimum with respect to certain parameters of the coherent state. Imposition of only the first condition leads to an elliptical orbit, which is not a Kepler orbit in general. Imposition of both conditions leads to an orbit identical with that of the two-dimensional hydrogen atom. However, the time evolution of the expectation values of position variables are consistent with the corresponding results of the Kepler problem only for small values of eccentricity, but not in general.