Variational Principle for Quantum Particle in a Box

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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.
Contributed by: Porscha McRobbie and Eitan Geva (March 2011)
Open content licensed under CC BY-NC-SA
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"Variational Principle for Quantum Particle in a Box"
http://demonstrations.wolfram.com/VariationalPrincipleForQuantumParticleInABox/
Wolfram Demonstrations Project
Published: March 7 2011