## `plot::HOrbital` -- visualize the electron orbitals of a hydrogen atom

#### Introduction

`plot::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, ...)`

#### Parameters

`n` | - | 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`

#### Options

`Option1, Option2, ...` | - | options allowed in `plot::Surface3d`

#### Returns

an object of type `plot::Surface3d`.

#### Details

- See the 'Background' below for a physical interpretation of the surface generated by `plot::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.
- The surface generated by `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 1

The 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

- The probability density of an electron is the absolute square of the quantum mechanical wave function Psi(x,y,z). With the usual polar coordinates

             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.

- The orbits corresponding to an energy index `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 ...

#### Changes

- new function

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