October 31, 2007

Quantum mechanical analysis of the hydrogen atom is straightforward; it can be found in introductory quantum mechanics textbooks. However, when we increase the level of complexity by adding one more electron into the system, the problem suddenly becomes non-trivial. Helium atoms, consisting of a positively charged nucleus and two negatively charged electrons, have not allowed an exact solution. So far, only approximation techniques have been developed to predict the energy spectrum of helium atoms.

Doubly-excited helium atoms, the main focus of this
report, do not even allow most of these well-established approximation
techniques. This is because they are not in a bound state; both of the
electrons in a doubly-excited helium atom are at energy levels higher
than their ground state energies, which brings up the total energy in
the system higher than the ionized states, He^{+} +
e^{-}. Therefore, this state is unstable and will eventually
ionize after a certain period of time.

Despite the fact that the doubly-excited state of
helium atoms is only transient, the theoretical description of this
state and its dynamics is an important key to understanding the
scattering properties of helium atoms. Many novel approaches describing
the doubly-excited state of helium atoms have been discussed from the
quantum viewpoint (e.g. see [1,2]), while at the same time,
semi-classical and purely classical approaches have made great progress
in this area of research as well [3]. In this report, I will concentrate
on the analysis of a particular state of helium, commonly known as
the Z^{2+}e^{-}e^{-} configuration, using
classical mechanics.

The helium atom is a three-body coulomb system. Electrons are held in orbit by the attractive coulomb force of a massive nucleus. It resembles the famous three-body problem in astrophysics such as the Sun-Earth-Moon system. However, a system of a helium atom is different from the typical three-body problem in that

- the two electrons exert
*repulsive*forces on each other - the two orbiting electrons have comparable (in fact, equal) masses

Classical analysis of helium atoms is called the Coulomb three-body problem, and as in the gravitational three-body problem, Coulomb three-body problem has no analytic solution. However, the problem can be simplified using a couple of approximation schemes before it is solved numerically.

In the most general case, the dynamics of three bodies can be described by an 18-dimensional phase-space, and the Hamiltonian for the system is

where Z_{i} is the charge of the i-th
particle.

Using the fact that the nucleus is much heavier than
the electron, we can approximate the nucleus mass as infinite and define
the position of the nucleus (which is now the center of the mass) to be
at the origin of the coordinate system. This reduces the problem to six
degrees of freedom (12 dimensions in phase space). Using atomic units (e
= m_{e} = 1) and a proper length scale, the Hamiltonian
becomes

Finally, we focus on the special case of a system
with zero angular momentum (**L**= 0). There have been many studies
on zero angular momentum three-body systems simply because this reduces
the complexity of the problem a great deal. In the study of
highly-excited helium atoms such a choice is not entirely arbitrary, but
rather, it is actually a good approximation. For the Hamiltonian above,
we are free to apply a scaling transformation

which normalizes the problem to an energy-independent form,

But this transformation also rescales the angular
momentum **L** as

Therefore, highly-excited helium atoms can be
approximated by **L**= 0. Now, as proved in [4], three-body systems
with zero angular momentum must have a planetary configuration. That is,
the movements of its constituents are confined in a two dimensional
plane. This cuts down the degree of freedom of our problem to three,
namely three scalar variables r_{1}, r_{2} and
r_{12}.

Now that we have reduced the complexity of the
problem to three degrees of freedom, it is not very difficult to
simulate all types of trajectories of electrons inside a helium atom
using a computer. However, instead of considering all possible
(classical) motions of the electrons inside the doubly-excited helium
atoms, we focus here on a certain group of motions that are more
interesting, namely, what is called the
Z^{2+}e^{-}e^{-} configuration.

The Hamiltonian above reveals that there are two
discrete symmetries, one associated with the reflection,
(**r**_{1}, **r**_{2}) &rarr
(-**r**_{1}, -**r**_{2}), and the other associated
to the particle exchange transformation, (**r**_{1},
**r**_{2}) &rarr (**r**_{2},**r**_{1}).
This gives rise to an invariant subspaces inside the six dimensional
phase space in which trajectories that start in one of those subspaces
remain there for all times. Examples of such subspaces are

Trajectories that lie on r_{1} +
r_{2} - r_{12} = 0 and r_{1} - r_{2} +
r_{12} = 0 subspace describe helium atoms in which two electrons
are orbiting around the nucleus forming a collinear arrangement.
r_{1} + r_{2} - r_{12}= 0 is the state where two
electrons are on opposite sides of the nucleus (eZe), and r_{1}
- r_{2} + r_{12}= 0 is where two electrons are on the
same side of the nucleus (Zee). The third example, r_{1} -
r_{2}= 0, describes trajectories where the two electrons stay
equidistance from the nucleus at all times (Wannier ridge [5]).

Of the three simplest examples of symmetric motions
given above, only the "Zee" case, r_{1} - r_{2} +
r_{12}= 0, is a stable trajectory for the electrons. Simulations
show that other symmetric motions are chaotic in nature and soon lead to
autoionization [6].

The Zee state has many interesting features, other than the fact that two electrons are orbiting the neutron with the same frequency (collinear arrangement). First of all, the two electrons are always distinguishable as the inner electron and the outer electron. These two cannot swap their positions because their degrees of freedom are confined to one dimension and they remain separated due to their repulsive forces on each other.

Depending on the initial conditions, the inner electron exhibits varying degrese of elliptical motions, but the distance between the outer electron and the nucleus converges to a stable value over time. Since the outer electron stays nearly fixed when viewed in a rotational frame, it is usually referred to as a "frozen planet orbit."

Another critical feature the Zee state has is its
robustness against perturbations. Not only does it maintain stability
when the initial conditions are varied within the symmetry constraints,
r_{1} - r_{2} + r_{12}= 0, it is also stable
when it is perturbed away from the above subspace, or even when the
initial assumption **L** = 0 is slightly relaxed. This is a strong
indication that this Zee state occurs in Nature.

The notion of the frozen planet orbit has given birth to a new approximation technique within quantum mechanics that explains the energy levels of two-electron atoms. This technique utilizes the fact that the outer electron is almost fixed in space, thus only weakly coupled with other electron and the nucleus. The state of this frozen electron is approximated as an harmonic-oscillator and therefore it can be quantized via normal modes. Parameters used in this approximation are borrowed from the classical properties of the frozen planet orbit. It has been shown that this approximation is superior to other known approximations in a semi-classical limit (high energy) [7].

The idea of frozen planet orbit can be also found in spectroscopy on other helium-like atoms, which contain two electrons in their outermost shells such as Calcium (atomic number 20) and Barium (56). For example, in a laser spectroscopic study on Barium by Eichmann et al.[8], Barium atoms were highly excited using six-laser beams into planetary states and after the autoionization their energies were measured. Eichmann notes that the resulting spectrum could be understood only when we assume that there is a "frozen electron."

© 2007 Yeong Dae Kwon. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] C.E. Wulfman, "R_{4} Invariants for
Doubly Excited States of Helium," Phys. Lett. A **26**, 9
(1968).

[2] J. Macek, "Properties of Autoionizing States of
He," J. Phys. B **1**, 2 (1968).

[3] G. Tanner, "The Theory of Two-Electron Atoms:
Between Ground State and Complete Fragmentation," Rev. Mod. Phys.
**72** (2000).

[4] L.A. Pars, *A Treaties on Analytical
Dynamics* (Heinemann, 1965).

[5] G.H. Wannier, "The Threshold Law for Single
Ionization of Atoms or Ions by Electrons," Phys. Rev. **90**, 5
(1953).

[6] D. Wintgen, K. Richter and G. Tanner, "Classical
Mechanics of Two-Electron Atoms," *Phys. Rev. A* **48**, 6
(1993).

[7] D. Wintgen, E.A. Soloven, K. Richter and J.S.
Briggs, "Novel Dynamical Symmetries of Asymmetrically Doubly Excited
Two-Electron Atoms," J. Phys. B **25** (1992).

[8] V. Lange, U. Eichmann and W. Sandner, "Positional
Correlation in Laser-Excited Three-Body Coulomb Systems," Phys. Rev.
Lett. **64**, 3 (1990).