Next: The Preperation of Rydberg
Up: The Aufbau Principal, Kramers
Previous: Kramer's Relation and the
A change in energy levels for a Rydberg atom in a circular
state must obey selection rules so that
or 0, and
. Thus an atom in a Rydberg state under guarded environmental condtions
can only transition as
Thus a Rydberg atom approximates a two-level system.
We demonstrate the theory behind elementary selection rules with a simple example.
A two-state system in the presence of puturbing Hamiltonian can be described as,
![$\displaystyle \Psi(t)=c_a(t)\psi_ae^{-iE_at/\hbar} + c_b(t)\psi_b e^{-iE_bt/\hbar}$](img80.png) |
(3.1) |
Its evolution is described by
from which we have,
![$\displaystyle c_a(t)\langle\psi_a\vert H' \vert\psi_a\rangle e^{-iE_at/\hbar} +...
...\vert H' \vert\psi_b\rangle e^{-iE_bt/\hbar}=i\hbar \dot{c_a(t)}e^{-iE_at/hbar}$](img82.png) |
(3.2) |
Rejoining convention and writing
,
, and assuming (as is warented in
the experiments we discuss) that the diagonal of the perturbing part of the Hamiltonian is zero, we can obtain a
set of equations for the prefactors,
Following [4] we consider a basic hydrogen atom in the
state in an electric field so that
.
As we showed in a previous compendium, the wave functions for hydrogen,
, are
The perturbation matrix is simple in that all but one of these will be even
in
, so the perturbation matrix (
)
will have all zero elements except
![$\displaystyle \langle 100\vert H'\vert 210 \rangle = - \frac{eE}{4\sqrt{2}\pi a...
...2 \cos^2\theta r^2 \sin\theta dr d\theta \ d\phi = -\frac{2^8}{\sqrt{2} 3^5}eEa$](img103.png) |
(3.5) |
|
|
|
|
|
|
|
0 |
0 |
|
0 |
0 |
|
0 |
0 |
0 |
0 |
0 |
|
|
0 |
0 |
0 |
0 |
|
0 |
0 |
0 |
0 |
0 |
|
0 |
0 |
0 |
0 |
0 |
Thus we see that under this perturbation, the
levels are ``selected'
and we have an approximate two-state system under the right conditions (ideal).
In general, we can derive selection rules for m and l transitions. In the case of m, we
consider that
so that
![$\displaystyle 0=\langle n'l'm'\vert[L_z,z]\vert nlm\rangle=(m'-m)\langle n'l'm'\vert z\vert nlm\rangle$](img112.png) |
(3.6) |
That
gives,
![$\displaystyle (m'-m)\langle n'l'm'\vert x\vert nlm\rangle=i\langle n'l'm'\vert y\vert nlm\rangle$](img114.png) |
(3.7) |
Finally,
gives,
![$\displaystyle (m'-m)\langle n'l'm'\vert y\vert nlm\rangle=-i\langle n'l'm'\vert x\vert nlm\rangle$](img116.png) |
(3.8) |
Thus
or
.
For the
case, it can be shown that
From this we can show that
![$\displaystyle 2\hbar^4(l(l+1)+l'(l'+1))\langle n' l' m'\vert r\vert nlm\rangle = \hbar^4(l'(l'_1)-l(l+1))^2)\langle n' l' m'\vert r\vert nlm\rangle$](img121.png) |
(3.9) |
Rewriting
![$\displaystyle l'(l'+1)-l(l+1)=(l'+l+1)(l'-l), \ $](img122.png)
and
We thus conclude that
and
or
Next: The Preperation of Rydberg
Up: The Aufbau Principal, Kramers
Previous: Kramer's Relation and the
tim jones
2007-04-09