Lesson - Visualization of Hydrogen Wave Functions

Methods

The wave function of Hydrogen is a complex function of three spatial variables and one time variable. The problem in attempting to visualize such a function is that paper and screen is limited to two dimensions on which to draw.

In addition, the variables used in the solutions for Hydrogen are often not x, y, and z, but r, theta, and phi.

There are a variety of methods you can use to go about visualizing the results of the Hydrogen wave function.

Contour/Density Plots

Contour and Density plots are two methods of doing the same thing, plotting one variable as a function of two independent variables on a 2 dimensional surface. Typically this would be done by drawing a function f(x,y) on an x-y grid as either contours (lines representing points for which f(x,y) is equal), or by coloring in different colors or brightnesses where the function is higher in value. The first method is referred to as a contour plot, and the second is referred to as a density plot. These are commonly used in geology to represent the topography of the Earth's surface.

Surface Plots

A surface plot is similar to a contour plot, except that a projection of a surface whose height at each x,y point is given by the function f(x,y)a.

3-D Contour Plots

It is also possible to draw the projection of a surface in three dimensions for which every point (x,y,z) on the surface has the same value of f(x,y,z).

Polar Plots

When plotting a function of an angular variable theta, it is often convenient to draw the function on an x-y plane as a polar plot, where the distance from the origin of the curve at a given angle theta is the same as the function f(theta) that you want to visualize.

Slicing 3-D Functions

It can sometimes be useful to see what a contour or density plot of the function along a single plane through your 3 dimensional function looks like. You can think of this as a "cross-section" of your function. Common slices to look at for a 3 dimensional function are the X, Y, and Z planes.

3-D Density Plots

For functions of three spatial variables, it is also possible to treat the function as if it represented a "cloud" with a density specified by the function. The function can then be visualized as if it were a cloud, by tracing rays of light through a cloud, given off by dense spots in the cloud, to an observer some distance away.

Exercises

For these exercises it may be useful to be able to generate functions which are commonly used in the solution of Schrodinger's equation for Hydrogen. There is a tool to format equations for the solution of Schrodinger's equation for Hydrogen available for your use.

The polar component of the Hydrogen wavefunction is given by the associated Legendre functions. Make polar plots of the m=0 terms for the first 4 angular energy levels of Hydrogen. What trends do you notice? You may want to use CSERD's polar plot tool for this.

Make plots of the first three energy levels of the radial wave function for Hydrogen. What happens to the likely distances of the atom from the nucleus as the energy level increases? You may want to use Interactivate's Graphit for this (note: Graphit expects functions of a variable "x"), or a graphing tool of your own choice.

Using CSERD's 3-D Density plot tool (or another tool of your choosing), look at the m=0 and n=0 terms of the square of the wave function for hydrogen (the probability) distribution for the first 4 levels of l. Describe the appearance of each.

For the l=1, m=1, n=0 state, make images of the wave function. Describe the shape of the most likely place to find the electron.