Energies for Particle in a Gaussian Potential Well

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The Schrödinger equation for a particle in a one-dimensional Gaussian potential well , given by
, has never been solved analytically. This Demonstration derives an approximation for the first few bound-state energies,
, using the linear variational method. The wavefunction is approximated by a linear combination
. It is convenient to take the basis functions
as the corresponding orthonormalized eigenfunction of the linear harmonic oscillator:
, where
is the
Hermite polynomial and
is a scaling constant to be determined variationally. After evaluating the matrix elements
over the selected set of
basis functions, Mathematica can calculate the
eigenvalues in a single step, from which we select only those with negative values. For convenience, we set
, so that all distances are expressed in bohrs (Bohr radii) and energy quantities in hartrees.
Contributed by: S. M. Blinder (August 2012)
Open content licensed under CC BY-NC-SA
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Reference:
D.A. McQuarrie, Quantum Chemistry, Sausalito, CA: University Science Books, 1983, pp. 266-275. Or numerous other texts on quantum mechanics or quantum chemistry.
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