In quantum mechanics, particles are found to not obey the
equations of motion that are obeyed by large objects.

Experiments such as
electron diffraction
show that particles have
a wavelike nature. When they are fired through a thin slit, rather
than scattering like hard spheres they interfere like waves.

As a result, Schrodinger developed a wave equation for quantum particles,
which is expressed as

.

Looking at this, the similarity with the classical equation for
conservation of energy, which states that the total energy
of an object is the energy due it motion (kinetic energy)
plus the energy due
to its interaction with its surroundings (potential energy)
is a conserved quantity.

There is a difference in that the momentum of the wave is
specified by how the wave is distributed in space rather
than how the wave is moving in time, so the classical
is replaced by the quantum
.

Schrodinger's equation in 1 dimension

In one dimension, Schrodinger's equation reduces to

For the case of a free wave, this reduces to

for which the solution is a sine wave.

The potential well

Consider a particle which is not free, however, but is confined
between two regions of extremely high potential, so high that
the particle has no chance of penetrating into that region.

The magnitude of the wave function represents the
probability of finding the particle in a given location.
If we require that the chance of finding the particle outside the
well be zero, and we also require that the function for our
wave be continuous, then the boundary conditions for the problem
of a particle trapped in a well with width centered on
are
.

What are the properties of solutions which will satisfy these
boundary conditions?

Exercise

Create a numerical model which integrates Schrodinger's equation
in 1 dimension from to . You may wish to use
CSERD's potential well model.

What are the properties of
the solutions which match the boundary conditions?

What are the limitations on the possible energy levels, if any?