Note: Right click on any applet to make a copy of the image.  Choosing the Show Multiple option and Resetting allows up to 4 plots to be viewed simultaneously.  The mouse coordinates may be observed by left-clicking within the graph.

Radial Wave Functions

The set of solutions to the radial equation can be found using standard methods of differential equations.  Notice that the radial solutions are determined by two quantum numbers n and l.  The solutions are the unnormalized associated Laguerre polynomials.  


  1. Make a multiple plot of the possible radial wave functions for n = 3.  How is the behavior for l = 0 different than for l = 1 or 2?  Try this for a few other values for the principal quantum number and see if your conclusion holds.

  2. For n = 3, how many times does the radial wave function cross zero (change signs) for each possible value of l?  Try this for a few other values for the principal quantum number and see if you conclusion holds.

  3. For a given principal quantum number, there is a maximum value for l.  The graph of the radial wave function in this case should have only one maximum value.  Obtain a general formula relating the radius for this max value for all n

  4. Using some math plotting software (e.g., MathCad, Mathematica, Excel) plot r2R(r)2 for n = 1 and l = 0.  The wave function R(r) may be found in your text.  What is the most probable radius for the electron.  Do the same for n = 2 and l = 1 and n = 3 and l = 2.  Do these values agree with the corresponding radii predicted by the Bohr model?  (Remember that r is in units of a0.)

  5. Find the expectation value <r> for the three states mentioned above in Exercise 4.  How do your answers compare to those of Exercise 4.  Explain your conclusions.

  6. Calculate the most probable radii and the expectation value of r for the 2s state and compare it to the values for the 2p state calculated in Exercise 4.  Explain your conclusions.