plot::HOrbital -- visualize the
electron orbitals of a hydrogen atom
Introductionplot::HOrbital(n, l, m) yields a
visualization of the hydrogen electron orbital with quantum numbers
n, l, m.
Call(s)plot::HOrbital(n, l, m, Option1, Option2, ...)
Parametersn |
- | the principal (energy) quantum number: a positive integer |
l |
- | the angular momentum quantum number: an integer
between 0 and n - 1 |
m |
- | the magnetic quantum number: an integer between
-l and l |
OptionsOption1, Option2, ... |
- | options allowed in plot::Surface3d |
Returnsan object of type plot::Surface3d.
Detailsplot::HOrbital.plot::HOrbital generates a surface of type plot::Surface3d. It uses the
default settings with one exception: a specialized color scheme is used
to depict information on the radial part of the electron orbit (see
``Background''). Using the Color option of plot::Surface3d, this internal color
scheme may be overridden by a user-defined color.plot::HOrbital may be passed
to the function plot::Scene to create a graphical scene.
In the call to plot::Scene, you may specify scene options. Call
plot(...) to display the scene.
Alternatively, you can pass the surface directly to plot together with scene options.
Example
1The following call yields a symbolic surface object:
>> orbit := plot::HOrbital(3, 2, 0)
plot::Surface3d([cos(phi1) sin(theta1)
2 2
(1.5 cos(theta1) - 0.5) , sin(phi1) sin(theta1)
2 2
(1.5 cos(theta1) - 0.5) , cos(theta1)
2 2
(1.5 cos(theta1) - 0.5) ], theta1 = 0..PI, phi1 = 0..2 PI)
We pass this object to plot to render the object:
>> plot(orbit, Ticks = None)
With the Grid-Option of plot::Surface3d, a smoother
surface is generated. The scene option Axis = None is used
in plot to switch off the default box around the graphical
scene:
>> orbit := plot::HOrbital(3, 2, 0, Grid = [30, 30],
Title = "quantum numbers: 3, 2, 0"):
plot(orbit, Axes = None)
The internal coloring is replaced by a new coloring scheme:
>> orbit := plot::HOrbital(3, 2, 0, Grid = [30, 30],
Color = [Height]):
plot(orbit)
>> delete orbit:
Background
x(r, phi, theta) = r * cos(phi) * sin(theta),
y(r, phi, theta) = r * sin(phi) * sin(theta),
z(r, phi, theta) = r * cos(theta),
r = 0..infinity, phi = 0 .. 2*PI, theta = 0 .. PI
the wave function has the form Psi(x,y,z) = R(r) * Y(phi,
theta). The surface is a visualization of the real part of the
function Y(phi, theta): the surface is defined by the
parameterization
x(phi, theta) = Re(Y(phi, theta))^2 * cos(phi) * sin(theta),
y(phi, theta) = Re(Y(phi, theta))^2 * sin(phi) * sin(theta),
z(phi, theta) = Re(Y(phi, theta))^2 * cos(theta) .
The distance of a surface point to the origin is Re(Y(phi,
theta))^2. Hence, the real part of the electron density is
reflected by the shape of the surface: the ``bulges'' indicate high
probabilities.
The radial part R(r) of the wave function Psi(x,y,z) = R(r) * Y(phi, theta) is only used in the coloring scheme of the surface: high values of R(r)^2 yield bright colors, small values yield dark colors. Red colors indicate points where the real part of the wave function Re(Psi) is positive. Blue colors indicate negative values.
n are
referred to as a ``shell''. Traditionally, the following shell symbols
are used:
n | 1 2 3 4 ...
-------------+------------
shell symbol | K L M N ...
The following symbols are associated with the angular momentum:
l | 0 1 2 3 4 5 ...
-------+----------------
symbol | s p d f g h ...