Consider a particle in a box of length l in the wavefunction, (x) = ( 30 / l 3)x

Question

Consider a particle in a box of length l in the wavefunction, (x) = ( 30 / l 3)x(1 - x / l), Determine the expectation values (x) and (x 2) Determine the expectation value (E) for a particle in a box described by this wavefunction Consider a particle trapped in the following three dimensional rectangular box: Provide the ground state wavefunction for this system Determine the energies (in units h 2 / ma 2) and the quantum numbers tor the live lowest energy levels? ma' c. What are the degeneracies of each of the six lowest energy levels?

Explanation / Answer

This section provides a very rough review of the quantum chemistry needed to continue in this program. There are a number of good on-line reviews, my favorite being the materials developed by David Sherrill at the Center for Computational Quantum Chemistry. An additional, but slow (heavy use of GIF files), is entitled "Quantum Mechanics" at the Wilson Institute, University of California, San Diego. There are basically two different models of atomic structure, the Bohr model and the quantum model. In both models, we are often interested in evaluating the energy of a particular system. Bohr Model: Electrons are "particles" that revolve around the nucleus in orbits. These orbits are at fixed distances from the nucleus. Electrons can move between orbits, using or releasing energy in the process. The energy needed to jump to a different orbital is given in the equation: where h = Planck's constant n = quantum number (orbit number) Using the Bohr model, we are able to determine exactly where each electron is on its path around the nucleus. Below is an example of what a a Bohr orbital might look like. Quantum Model: The quantum model says that electrons are not particles, but have wavelike characteristics. It uses the Schro

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