Variational Principle for Quantum Particle in a Box

This Demonstration shows the variational principle applied to the quantum particle-in-a-box problem. The Hamiltonian describing the particle is , and the eigenfunctions and eigenvalues are given by and , respectively. If is a trial wavefunction that depends on the variational parameter , then minimizing the energy functional with respect to leads to an estimate for the energy. In this example, the values of that minimize are and , . The left panel shows the energy estimate and the three lowest eigenenergies, where the red are located at the , and the right graphic shows the normalized trial wavefunction for the ground and second excited states, which are the lowest even functions with respect to the central point.