Particle in a Box

Question: Can a simple function like y(x) = A*Sin[Pi*x/a] describe an electron in a molecule?

Answer; yes.

The Basic Model System

1) Consider a box with no lid with infinitely high walls. This box represents the "molecular frame" for which the electron can exist.

2) When a particle/electron is placed inside this box, it has a particular "energy." Considering that "energy" is a measurable quantity in systems described by classical mechanics, Q.M. postulate 2 tells us that there exists a corresponding operator that can be used to predict the energy or as we say this observable.

3) As with all "operator algebra" problems, there are 3 parts:

- the operator

- the (eigen)function

- the eigenvalue = observable.

The operator in this case, like many, is the Hamiltonian operator, the eigenfunction was given above, and we need to solve for the eigenvalue or energy.

Initial Tasks

1) Convince yourself that the function provided above is an eigenfunction of the Hamiltonian operator.

The Chemical System: