Wavefunction Illustrations:
y(r,J,j)
= R(r) .Y(J,j)
There are a lot of possibilities for illustrating total wavefunction
y(r,J,j)
for hydrogen-like atoms which consists of radial part
Rn,l(r) and angle part Yl,m(J,j):
yn,l,m
= Rn,l(r) Yl,m(J,j).
Since there are 3 dimensions it's quite hard to illustrate wavefunctions in
three-dimensional space.
-
The small illustrations below shows the wavefunction y
as the point density. The wavefunction of s-Electrons
(l = 0) doesn't depend on angles and therefore it has spherical symmetry.
But it's clearly seen that there are small spherical shells in inner regions of 2s-
and 3s-Orbitals.
Radial and Angle parts:
yn,l,m = Rn,l(r)
Yl,m(J,j) |
For higher y values one will have
higher point density. |
 |
-
It's also one more possibility: we plot the wavefunction "surface" and surface
area shows wavefunction value for particular r,J,j
values: y(r,J,j)
= c. It's talked about iso-surface. If we combine such surfaces for different r,J,j
values we will have the following clip which can be seen here for 3dz-electron: 3dz-Isofläche: 0.82Mb
-
We can cut the wavefunction into different sections, for instance, for y-value (y = yo)
we will have function y(x, y=yo, z) which
can be then plotted in the (x,z)-plane. If we combine such sections for
different yo-values then we will have a clip. One of them is shown for 3dz-electron here: 3dz-Schnitte:
0.60Mb
-
The new way to illustrate the wavefunction is called "Volume
rendering". Moreover each point (r,J,j)
is colored by its own color for the whole wavefunction y(r,J,j).
This illustration shows diffuse character of wavefunction, that makes impossible
to determine direct position of electron. Here one can find wavefunction
volume rendering for 3dz-electrons: 3dz-Rendering:
0.70Mb
Here one can also find some clips that show the above-mentioned electron
orbitals 2s (1,0Mb), 2px(1,2Mb),
3px(1,2Mb),
3dxz(1,6Mb).
You can find more information in Internet: Orbitron.
Normalized wavefunctions of hydrogen-like atoms. For nuclear charge Z>1
one must substitute ao with ao/Z
everywhere
| n |
l |
m |
ynlm |
Rnl(r) |
Ylm(J,j) |
| 1 |
0 |
0 |
1 s |
 |
 |
| 2 |
0 |
0 |
2 s |
 |
|
| 2 |
1 |
0 |
2 pz |
 |
 |
| 2 |
1 |
1 |
2 px |
|
 |
| 2 |
1 |
-1 |
2 py |
|
 |
| 3 |
0 |
0 |
3 s |
 |
|
| 3 |
1 |
0 |
3 pz |
 |
|
| 3 |
1 |
1 |
3 px |
|
|
| 3 |
1 |
-1 |
3 py |
|
|
| 3 |
2 |
0 |
3 dz2 |
 |
 |
| 3 |
2 |
1 |
3 dxz |
|
 |
| 3 |
2 |
-1 |
3 dyz |
|
 |
| 3 |
2 |
2 |
3 dx2-y2 |
|
 |
| 3 |
2 |
-2 |
3 dxy |
|
 |
| To the classical illustration Radial + Angle parts: |
 |
| The probability clouds for the first states of hydrogen atom. Right
figures correspond to Bohr radii. |
