Atomic structure - ground and excited states

In the simplest of atoms, the hydrogen atom, a single electron is
bound by a proton to form a neutral species. The bound electron
occupies a quantum state belonging to one of the hydrogen atomīs
principal shells. The most tightly bound shell has only two quantum
states. One has
spin
up and the other spin down. Otherwise they are identical.
These are the ground states of the atom, traditionally labelled 1s.
The atom has an infinite number of excited states which belong to a series of
excited shells with decreasing binding strengths. This is illustrated below.
Each shell is labelled by an
integer, n, called the principal quantum number. The ground states
belong to the lowest shell with n=1. The binding energies of the shells
are given by I_{n}=
-U_{n}=1/(2n^{2})I_{0},
their geometrical sizes by a_{n}=n^{2}a_{0},
and the number of quantum states in a shell
by 2n^{2}, where I_{0}=27.2 eV and
a_{0}=0.53 ×
10^{-8} cm
are the atomic units of energy and length.

The electron is not well-localized in space. Its position is described by a probability distribution which is often visualized as a cloud, the so-called electron cloud. The 1s electron cloud is shown below. It is spherically symmetric with respect to the proton at the center of the cloud. The first excited shell, with n=2, has eight states. One of them is the 2p-state. The electron cloud of this state is shown next to the 1s-cloud. It is four times larger, and it has the shape of two spherical clouds, one on top of the other.

Rydberg states

Rydberg atoms are highly excited one-electron atoms with a weakly
bound electron occupying a state of a high-lying shell, n>>1. Each
shell has 2n^{2} different electronic states and the size
of each of them is n^{2} times the size of the 1s-cloud.
We show two examples below. One is the spherically symmetic 5s-cloud
which is 25 times larger than the 1s-cloud and has four radial nodes.
The other is a strongly polarized state belonging to the seventh shell.
The cloud is almost entirely above the proton, which is marked by
a black dot. It has six radial nodes. When n is very large this type
of cloud resembles an elliptic orbit with an eccentricity of one, a
so-called linear cloud.

Coherent elliptic Rydberg states

A special class of Rydberg states are the coherent elliptic states whose electronic clouds have the shapes of elliptic orbits. The above linear cloud belongs to this type of state. Another one is shown below. It has eccentricity less than but close to 1.

The eccentricity is a continuous parameter, so there is an infinite number of coherent elliptic Rydberg states for each shell, and the larger n the more the clouds are concentrated near classical Kepler orbits. The animation below shows clouds for n=25. The nucleus is fixed and the probability distribution within the plane is shown for the full range of eccentricities from 0 to 1.

General description of Rydberg states

A hydrogenic Rydberg state is described fully by two conserved
vector quantities, the orbital angular momentum **L**
and the Runge-Lenz vector **A**
(ref.2).
These quantities are
shown below for the case of a coherent elliptic state.

The orbital plane of the electron e is perpendicular to
**L**, and **A** points from the nucleus at
N towards the perihelion at P and thus determines the orientation of
the major axis.

It is adventageous to form two new vector quantities
**j**_{1} and **j**_{2}
from **L** and **A** by the definitions

The components of **j**_{1} and
**j**_{2} satisfy angular momentum commutation
relations and **j**_{1} and
**j**_{2} commute. They are thus independent
pseudo-spins, and they have the same constant size

where n is the principal quantum number.

The pseudospins may be quantized along two arbitrary directions in space,
**z**_{1} and **z**_{2},
separated by the angle 2a as
illustrated below. The projections i_{1} and i_{2}
of the pseudospins take either integer or half-integer values within
the 2j+1=n numbers

The principal quantum number n, the angle
a, and the spin projections
i_{1} and i_{2} determine the quantum state
completely,
|a
i_{1}
i_{2}>_{n}.
The coherent elliptic states mentioned above are particularly
simple in this representation. They have maximum spin projections, so
four possibilities given by
i_{1}=
±i_{2}=
±j
exist, and the excentricity e is simply
e=sin(a). While the states
|a
-j
+j
>_{n}
and
|a
+j
-j
>_{n}
may be degenerate even in the presence of external fields, the states
|a
+j
+j
>_{n}
and
|a
-j
-j
>_{n}
are always non-degenerate when an external fields is present. The
latter states are therefore of particular interest for ApSR.