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Figure 3.1: Examples of Numerical Simulations: Potential shown with eigenstates
offset by their energy. – a) Eigenstates of coarse harmonic oscillator potential:
V (x) = x 2 , −3.0 ≤ x ≤ 3.0, dx = 1.0. b) Lowest 7 eigenstates of fine harmonic
oscillator potential: V (x) = x 2 , −5.0 ≤ x ≤ 5.0, dx = 0.1 c) Lowest 17 eigenstates
of qubit-like potential: V (δ) = δ 2 − 5 cos δ + 5δ, −8.0 ≤ δ ≤ 3.0, dδ = 0.05. States
localized in the shallow (deep) minimum are shown in green (red), while states
that span both minima are shown in gray.
Plotting the eigenvectors offset by their corresponding energies gives a plot like
Figure 3.1a. For the lower energy states this plot clearly shows the usual oscillating
behavior of the wave functions and for the ground state even the exponential
decay outside the potential. If the x-step-size is decreased from 1 to 0.1, i.e. the
resolution of the approximation is increased by 10×, the first 7 eigenvectors look
like Figure 3.1b. Their corresponding energies are:
1.58, 4.74, 7.89, 11.06, 14.23, 17.45, 20.80
These numbers show the expected equal spacing of the levels fairly well.
The energies of the levels are quite different in the two approximations. Es-
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