Visual Quantum Mechanics

< Prev | Next >

Superposition of two Gaussian functions realizing a Dirichlet boundary condition

The Schrödinger equation which governs the motion of wave packets is linear. The sum of two solutions is again a solution of the same equation. Here we see a wave function which is the sum of two Gaussian functions moving in opposite directions. As long as the two Gaussians occupy the same region, we can see a typical interference pattern. This particular superposition has the property of being zero along the diagonal line x+y=0 for all times. We say that the wave function satisfies a "Dirichlet boundary condition" along this line. Boundary conditions are used in quantum mechanics to model inpenetrable potential barriers. The behavior of the wave packet on either side of the line indeed describes the reflection of a particle at an infinitely high potential wall.

"Visual Quantum Mechanics - Book One" also explores the behavior of wave packets under Neumann boundary conditions and periodic boundary conditions.