Variational Method Applied to Stepped-Infinite-Square-Well Energies

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 .

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<y≤2π, vo, True, ∞].

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.