CSERD
  • userhome
  • catalog
  • resources
  • help

Infinite Potential Well Lesson
Shodor > CSERD > Resources > Activities > Infinite Potential Well Lesson
  Lesson  •  Materials  •  Lesson Plan

Lesson - Infinite Potential Well

Schrodinger's Equation

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


\begin{displaymath} \mathbf{H}\Psi \equiv \left[ -\frac{\hbar^2}{2m} \nabla^2 +V \right] \Psi = E \Psi \end{displaymath}

.

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 $1/(2 m) p^2$ is replaced by the quantum $\hbar / (2 m) \nabla^2$.

Schrodinger's equation in 1 dimension

In one dimension, Schrodinger's equation reduces to


\begin{displaymath} - \frac{\hbar^2}{2 m} \frac{d}{d x} \Psi - V(x) \Psi = E \Psi \end{displaymath}

For the case of a free wave, this reduces to


\begin{displaymath} - \frac{\hbar^2}{2 m} \frac{d}{d x} \Psi = E \Psi \end{displaymath}

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 $\Psi^* \Psi$ 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 $2 a$ centered on $x=0$ are $\Psi(-a) = \Psi(a) = 0$.

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 $x=-a$ to $x=a$. 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?

©1994-2024 Shodor