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American Journal of Physics

Coherent states composed of Stark eigenfunctions of the hydrogen atom

Jonathan Baugh, D. A. Edmonds, P. T. Nellesen, Charles E. Burkhardt, Jacob J. Leventhal

Published Nov 1, 1997• 3 citations

Abstract

It is well known that for the nonrelativistic hydrogen atom it is possible to separate the Schrödinger equation in parabolic as well as spherical coordinates. The eigenfunctions obtained in these coordinate systems are each a suitable basis set in the absence of an electric field, but only the parabolic states, the Stark eigenfunctions, retain their character in the presence of a weak field. The properties of coherent superpositions of these Stark states are investigated and the motion of the resulting wave packet described. It is shown that a properly constituted superposition will mimic classical motion. It is also shown that the constant energy separation between adjacent Stark states of a given principal quantum number leads to periodic motion. In general, they split into distinct “clumps” of probability, but revive to form a single packet after each period. It is, however, possible to construct a packet that maintains its shape for all time.

DOI: 10.1119/1.18728

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It is well known that for the nonrelativistic hydrogen atom it is possible to separate the Schrödinger equation in parabolic as well as spherical coordinates. The eigenfunctions obtained in these coordinate systems are each a suitable basis set in the absence of an electric field, but only the parabolic states, the Stark eigenfunctions, retain their character in the presence of a weak field. The properties of coherent superpositions of these Stark states are investigated and the motion of the resulting wave packet described. It is shown that a properly constituted superposition will mimic classical motion. It is also shown that the constant energy separation between adjacent Stark states of a given principal quantum number leads to periodic motion. In general, they split into distinct “clumps” of probability, but revive to form a single packet after each period. It is, however, possible to construct a packet that maintains its shape for all time.

We explore the classical limit of the hydrogen atom by constructing a minimum-uncertainty wave packet that travels along a Kepler orbit. The dynamics of the wave packet display both classical and quantum-mechanical properties. Over a limited period of time, during which the dynamics of the wave packet may be considered linear, the motion of the wave packet can be described by classical equations of motion. After this time, the quantum dynamics, whose most prominent features are decays and revivals of the wave packet, becomes dominant. We discuss decays and revivals of the wave packet in detail.

The Stark effect of the fine structure of hydrogenic atoms is discussed in the light of the quantum mechanics. Explicit formulas are given for the energy levels in either a weak or a strong electric field, including in each case both the relativity corrections and the Stark effect. (The conventional treatment neglects either one or the other.) The results are, except in form, similar to those recently given by Schlapp. In the new mechanics, in contradistinction to the old Bohr theory, there is a linear Stark effect even in weak fields, because of the identity of energy for the

\mathrm{ns}

and

n{p}_{1}

levels in the absence of external fields. This degeneracy is shown to account for the results of the recent experiments of Ornstein, Zernike, and Snoek, who found that the

2s

level in

H

is not metastable.The coefficients are found for the development of the "parabolic" eigenfunctions in terms of the "polar" ones, and a set of numerical values of these coefficients is tabulated.Methods are given for the calculation of the relative intensities in the Stark effect in terms of the already available theoretical fine structure intensities in the absence of fields.

The revival structure of Stark wave packets is considered. These wave packets have energies depending on two quantum numbers and are characterized by two sets of classical periods and revival times. The additional time scales result in revival structures different from those of free Rydberg wave packets. We show that Stark wave packets can exhibit fractional revivals. We also show that these wave packets exhibit particular features unique to the Stark effect. For instance, the wave functions can be separated into distinct sums over even and odd values of the principal quantum number. These even and odd superpositions interfere in different ways, resulting in unexpected periodicities in the interferograms of Stark wave packets.

We present the quantal dynamics of electronic wave packets in the hydrogen atom in the presence of circularly polarized microwave and magnetic fields in regions of classical stability. Whereas wave packets may disperse without the stabilizing influence of a magnetic field, in its presence stable motion can be maintained for wave packets that are localized either at global equilibria that are either a maximum or minimum in a zero-velocity surface. Because these extrema are locally harmonic their vacuum states are truly coherent states in the original sense of Schr\"odinger.

Using a unique method, we create coherent states in an atom by periodically (\ensuremath{\sim}100--200 MHz) applying a low-amplitude, \ensuremath{\sim}5-V/cm, ns electric-field pulse that drives the atoms to a realm where many stationary states lie within a very narrow energy range. Using a cw laser we excite a coherent superposition of these coherent states and probe the attributes of such a state in the frequency domain (rather than in the time domain). We show that such coherent superpositions exhibit localized electronic probability densities and, therefore, qualify as wave packets.