Using Schrodinger's method: particle in a 1-D Box





  Q:  You put an electron into a 1-D box
      of width L = 1 Angstrom.

      What is the energy of the ground
      state of the electron?  Express
      your answer in eV.

Okay, let's use this wave function to answer a real question.



  Q:  When the particle is in the 
      ground state, n=1, what is its most 
      probable position?

We can answer this in two ways: one graphical, one mathematical. The graphical method involves sketching the wave function, and the square of the wave function, over the space of the box.

The most probable position is where the square of the wave function has the largest value.

The mathematical way of answering this question is to integrate the square of the wave function, times the quantity of interest x, over the entire box.



  Q:  Can you write down this integral?

  Q:  Can you perform this integral?

  Q:  What is the most probable value of position "x" ?

For more information

Wikipedia's entry on "Particle in a box" contains a detailed description of the 1-D, 2-D and 3-D solutions.
Paul Fastad's 1-D Schroedinger applet shows various aspects of the wave function of a particle in a box (and other potentials, too).