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### Main Ideas

• Students use Maple to visualize the radial part of probability density of the hydrogen atom
• Students plot Laguerre polynomials

Estimated Time: 20 minutes

Students use Maple to define and plot the radial wave-functions for values of $n$ and $l$ that they choose.

### Prerequisite Knowledge

• Students should have solved or seen a solution of the radial wave-function for the hydrogen atom
• Students should have some acquaintance with the Laguerre polynomials.

### Activity: Introduction

This activity follows easily from the derivation of the radial part of the solution to the hydrogen atom. It is useful to frame students exploration by asking them to investigate how the shape of the radial part of the probability distribution depends on the quantum numbers n and l.

### Activity: Student Conversations

Students are sometimes confused by the labeling of the Laguerre polynomials. In particular, it is helpful to clarify that $L^q_p$ is the Associated Laguerre polynomial with the labels p and q, not $L_p$ to the qth power.

It is useful to help students focus their exploration by encouraging them to vary one parameter at a time and try to draw conclusions based on this.

Students generally are able to conclude that as n is increased with l fixed, the number of “bumps” in the graph increases. They should also recognize that as l is increased, the number of “bumps” decreases.

One common student question that leads to good conversation is, “How can the particle have non-zero energy when it has zero angular momentum?”

### Activity: Wrap-up

Below are some questions that stimulate good wrap-up discussion for this activity.

• How does the shape of the radial probability density depend on n and l.
• What is the effect of adding in the extra factor of $r^2$ in the last expression? Why do we add this factor in?
• What is the difference in physical meaning between the last two plots in the worksheet?
• What happens to the maximum probability as n is increased? Can you explain why you think this happens? Can you explain this by knowing that the all of the eigenstates have to be normalized, i.e., the probability that the electron is somewhere equals one.
• How is the square of the wave function (the probability function) different for the cases when $l=0$?

We make it a point to bring up the following points during the wrap-up if they are not brought up by students.

• The number of bumps in the probability distributions is equal to $n-l$
• The second to last function gives the probability density contribution due to the radial wave function. In order to get the last expression, we multiply by $r^2$ to get the probability that the particle resides in shell of radius r.

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