Variational Method Applied to Stepped-Infinite-Square-Well Energies
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This Demonstration is computationally intensive and may take some time to complete when 
. An 
basis set (see figure subtitles) approximation of the wavefunction for the 
stepped box yields approximations to the lowest energies of the system. The approximations generally improve with increasing . The energy unit for the axis is the same as that for the 
axis, which has been stretched. The tick marks on the axis are labeled with the squares of the quantum numbers for a flat box of width 
.
Contributed by: M. Hanson (April 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The variational principle guarantees that the expectation value of the Hamiltonian using an approximate wavefunction is an upper bound to the true ground state energy value. Using an approximate wavefunction that is a linear combination of basis functions with adjustable coefficients leads to a secular determinant of dimension 
, which can be solved for approximate energies. The smallest of these is the best approximation to the ground state energy with the other solutions approximating excited states.
The potential for the stepped-infinite-square well can be described with the Mathematica command
V[y_] := Which[0≤y≤π, 0, π
Some adjustment of parameters in the source code could include the eighth energy level in the display.
This Demonstration had its origin in a problem suggested by C. Bardeen.
Permanent Citation
Exact Numerical Solutions for the Stepped-Infinite-Square Well
M. Hanson
Scattering by a Square-Well Potential
M. Hanson
Boundary Conditions for a Semi-Infinite Potential Well
Porscha McRobbie and Eitan Geva
Particle in an Infinite Spherical Well
S. M. Blinder
Time-Evolution of a Wavepacket in a Square Well
Michael Trott
Energy Spectrum for a Finite Potential Well
Adam Strzebonski
Finite Potential Well
Michael R. Braunstein (Central Washington University)
Two Electrons in a Box: Energies
M. Hanson
Particle in an Infinite Vee Potential
S. M. Blinder
Energy Levels of a Morse Oscillator
S. M. Blinder
