Practice problems on the quantum mechanics of atoms and molecules

1. The 2s and 3s electronic wavefunctions of the hydrogen atom have radial nodes, i.e. values of where the radial probability density goes through zero. Locate the values of r at which these nodes occur. Give exact answers. The wavefunctions are

   

   Hint: No integration is required to solve this problem.

2. Calculate the average potential energy of a hydrogen atom in the state. The normalized wavefunction is

   

   Hints: Maple will need to know that . The volume element in spherical polar coordinates is . The variables run over the following ranges: , , .

3. The ground-state wavefunction of a harmonic oscillator is

   

   where . Normalize this wavefunction. Report your normalization factor in terms of .

   Hint: Don't define in your Maple session but do tell Maple that this parameter is positive.

4. Suppose that trigonometric functions had not yet been invented. We would then not be able to solve the particle-in-a-box model exactly. Suppose that we solved it using the following trial wavefunction:

   

   What is the percent error in the ground-state energy using this trial wavefunction?

   Reminder: The particle-in-a-box ground-state energy is .