# Visualization of Hydrogen Lesson

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# Lesson Plan - Visualization of Hydrogen Wave Functions

## Audience

The visualization of hydrogen activity is designed for use in an undergraduate level course in physics, chemistry, mathematics, or computational science.

In an undergraduate mathematics or computational science course, this activity could be used to illustrate graphical display of multivariable functions.

In a physics or chemistry course, this activity could be used to introduce the interpretation of the electron cloud and the shapes of different orbitals.

## Solutions

• The l=0 term plotted as a polar plot from 0 to pi will simply look like a half-circle. The l=1 term will vary depending on some choices the students make in visualizing it. The biggest choice will be how to handle the sign. If they simply draw in the opposite direction for negative r, they will get a circle, but half of the circle would actually appear in a quadrant not studied by the possible range of angle in the problem. A better solution would be to either plot the absolute value of the function (for which it would appear as two half circles), to add a constant offset to the function (with an offset of 1, it would appear as half of a cardiod), or to plot the square of the function (which looks similar to the absolute value, but "thinner").

Higher order variants will start to show structure at angles close to 1/2 pi, and changes in sign. The number of different regions of the curve, where the regions are defined by change in sign, will be equal to l+1.

• As the radial energy level increases, the likely position of the electron moves further out from the nucleus, and starts to have multiple peaks, with regions of high and low probability appearing as rings leading out from the center of the atom.
• Looking at the square of the wave function, the student should see that the extra detail towards phi = 1/2 pi shows up as "rings" around the z-axis. The higher angular energy levels will have more rings.
• The energy levels for which m is not equal to zero is characterized by a change in having the electron most likely to be found along the z-axis to having the electron most likely to be found at angles of phi = pi / 2. The l=1, m=1, n=0 state is shaped like a donut.