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Physical Review A

Rutherford atom in quantum theory

Maciej Kaliński, Joseph H. Eberly, J. A. West, Carlos R. Stroud

Published Mar 10, 2003• 26 citations
Publisher

Abstract

We predict the existence of a self-sustained one-electron wave packet moving on a circular orbit in the helium atom. The wave packet is localized in space, but does not spread in time. This is a realization within quantum theory of a classical object that has been called a ``Rutherford atom,'' a localized planetary electron on an unquantized circular orbit under the influence of a massive charged core.

DOI
10.1103/physreva.67.032503

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The quantum theory of atoms is based on a Hamiltonian which is just that of the classical Kepler theory of planetary orbits. In spite of this common foundation, the predictions of the quantum theory ordinarily do not much resemble those of the classical theory. Even in the limit of large quantum numbers the energy eigenstates of quantum theory do not look at all like the classical elliptical orbits. Recently, Parker and Stroud1 showed that it is possible to excite a Rydberg atomic wave packet localized in the radial coordinate. Also, Yeazell and Stroud2 showed that a wave packet localized in the angular coordinates can be formed. Both wave packets exhibit an evolution describable, at least in part, by classical theory. However, neither wave packet truly corresponds to a wave packet traveling in a classical elliptical orbit. We discuss the manner in which such a wave packet might be formed using a short-pulsed laser and arf dressing field. The evolution of the resulting wave packet is compared with that of its classical counterpart.

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We predict the existence of a self-sustained one-electron wave packet moving on a circular orbit in the helium atom. The wave packet is localized in space, but does not spread in time. This is a realization within quantum theory of a classical object that has been called a ``Rutherford atom,'' a localized planetary electron on an unquantized circular orbit under the influence of a massive charged core.
Rydberg atoms are atoms with a highly excited electron. In his Perspective, Villeneuve explains that by creating a coherent superposition of Rydberg states, the electron can be made to behave almost like a classical Rutherford atom. He highlights the report by Maeda et al., who show that they can force the electron wave packet to remain localized for thousands of highly elliptical orbits around the nucleus, much like a miniature planetary system. Such experiments help to bridge the gap between the quantum mechanical world and our everyday classical world.
We demonstrate a protocol to create localized wave packets in very-high-n Rydberg states which travel in nearly circular orbits around the nucleus. Although these wave packets slowly dephase and eventually lose their localization, their motion can be monitored over several orbital periods. These wave packets represent the closest analog yet achieved to the original Bohr model of the hydrogen atom, i.e., an electron in a circular classical orbit around the nucleus. The possible extension of the approach to create "planetary atoms" in highly correlated stable multiply excited states is discussed.
In Bohr's original planetary model of the atom the electron moves along orbits of special geometric simplicity. While wave mechanics precludes the idea that a physical path could be ascribed to the electron, a classical or planetary atom can still be envisaged in which the electronic wavepacket neither spreads nor disperses as its center moves along the Kepler orbit, and this orbit is conned to a single plane in space. We show theoretically how an electronic wavepacket may be localized in this fashion in a similar way to ion confinement in a Penning trap. Because external fields are needed to keep the packet confined, a more fitting analogy than a planetary orbit is the motion of a charged dust grain in one of the rings of a giant planet such as Saturn.
A model which I wish to show you was constructed to illustrate the motion of the electrons around the nucleus of the Rutherford atom, and the changing dimensions and varying plane of their orbit. More violent disturbance produces a series of secondary waves in each orbit, which divide the orbit in the manner suggested by the wave and quantum theories. The model is a very simple one, and my only excuse for bringing it to your notice is the pleasure which it has given to myself, and the fact that it has enabled me to realise as I could not previously do, Rutherford's theory of atomic structure. Rutherford's atom, which at present holds the field, consists essentially of a nucleus around which rotate a number of electrons. This number is the atomic number of the atom, so that with hydrogen there is one circulating electron, with helium there are two, and the number of electrons steadily rises with the atomic number until we reach uranium, which has ninety-two circulating electrons. Each of these electrons rotates independently, and its orbit is subject to very great variation. Indeed the motion of the electron is so complex that it is exceedingly difficult to form any concrete idea of what a variable orbit really means, or what perturbations in the orbit signify. It is with the idea of assisting the imagination that I have constructed this model, consisting of two prisms so placed that a horizontal beam of light is reflected downwards by the first prism on to the second, by which it is again reflected horizontally.
In 1897, Thomson demonstrated that cathode rays (observed to be emitted from the cathodes of vacuum tubes) were in fact charged particles which he called electrons. Thomson proposed that the atom consisted of a positively charged sphere in which were embedded negatively charged electrons. Rutherford subsequently found in 1911 that the electrons orbited at some distance from a central positively charged nucleus. Rutherford proposed that electrostatic attraction between the nucleus and the electron was balanced by the centrifugal force arising from the orbital motion. However, if this were the case, then the electrons (being accelerated inwards towards the center of rotation) would continuously radiate all their energy as electromagnetic waves and very quickly fall into the nucleus. In 1913, Bohr postulated two important additions to Rutherford’s theory of atomic structure: 1. Electrons can orbit the nucleus in what are called stationary states in which no emission of radiation occurs and in which the angular momentum L is constrained to have values: 2. Electrons can make transitions from one state to another accompanied by the emission or absorption of a single photon of energy E = hf thus leading to absorption and emission spectra.
The quantum theory of atoms is based on a Hamiltonian which is just that of the classical Kepler theory of planetary orbits. In spite of this common foundation, the predictions of the quantum theory ordinarily do not much resemble those of the classical theory. Even in the limit of large quantum numbers the energy eigenstates of quantum theory do not look at all like the classical elliptical orbits. Recently, Parker and Stroud1 showed that it is possible to excite a Rydberg atomic wave packet localized in the radial coordinate. Also, Yeazell and Stroud2 showed that a wave packet localized in the angular coordinates can be formed. Both wave packets exhibit an evolution describable, at least in part, by classical theory. However, neither wave packet truly corresponds to a wave packet traveling in a classical elliptical orbit. We discuss the manner in which such a wave packet might be formed using a short-pulsed laser and arf dressing field. The evolution of the resulting wave packet is compared with that of its classical counterpart.