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Radial Solution of Hydrogen Lesson
Shodor > CSERD > Resources > Activities > Radial Solution of Hydrogen Lesson
  Lesson  •  Materials  •  Lesson Plan

Lesson Plan - Radial Solution, Hydrogen Wavefunction

Suggestions

Students when using the provided model should remember that the numerical solution is typically solved from the outside (high values of r) to the inside (low values of r). For some numerical integration packages, such as Stella, the user does not have control over the direction used in the integration. Stella, which was used for this prebuilt model, only allows for an independent variable, and only allows that variable to be time. This can be circumvented using a substitution of variables (in this case r = 50 - t + 0.0001, integrated from t = 0 to 50 to bring r from far away to very close to the center) but a graph of y will appear "backwards" to some students.

Audience

The radial solution of hydrogen activity is designed for use in an undergraduate level course in physics, chemistry, or mathematics.

In an upper level undergraduate mathematics course, this activity could be used to illustrate eigenvalue problems in applied mathematics.

In a lower level undergraduate physics course, this activity could be used to introduce Schrodinger's equation and particle wave duality.

In an upper level undergraduate course in modern physics, quantum mechanics, or chemistry, this activity could be used to introduce numerical solutions to Schrodinger's equation. In this case, it might be appropriate to require students to build their own model rather than using the one supplied.

Solutions

Using the provided model, students should find that only integer values of $l$ and $\nu$ are acceptable. Furthermore, for a given value of $l$, $\nu$ should not be less that $l$.

For solutions that do not match the given boundary conditions, if $l>0$ the typical result is that the solution "blows up" close to the center of the atom. For $l=0$, the solution does not "blow up", but does fail to meet the criterion that as you get closer to the center of the atom the chance of finding the electron should fall to zero. This is seen most easily for $l=0$ by viewing the bar graph animation, and leaving the time on the animation set to t=50 (the end of the simulation, or the value close to the center of the atom).

If $\nu \propto \frac{1}{\sqrt{E}}$, then $E \propto \frac{1}{\nu^2}$.

Since $\nu$ is always an integer, this makes sense. If the energy is expressed as a negative value (the electron is bound to the hydrogen atom, a positive value would imply that the electron was free), then a lower value of the energy level would imply a more negative, and thus lower, energy. The difference in two energy levels n and m, with a constant of proportionality for the solution to the last problem of R, would give the observed energy levels of the Hydrogen atom.

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