This is a "decorated" particle in a box problem. In addition to the
infinitely hard walls, the particle is also held to the center of the box by a Hooke's law
spring. Let the total potential be given by . The constant *kL ^{2}/12* is
included to make the average potential across the box equal to zero; this makes
comparisons with the simple PIB easier. The natural unit of length for this problem is

**Figure.**
The 4 lowest energy states of the particle in a box with a spring. The potential is
infinite outside the range *0<x<1* and it is parabolic inside the range as
depicted by the blue curve. The vertical scale is energy (in multiples of the natural
energy unit defined above). The horizontal scale is the particle coordinate, *x*.
Energy eigenvalues are shown as horizontal lines at -1.874,
+2.973, 8.703, and 15.927. Wave functions are drawn referenced to their respective
eigenvalues.

Wave function shapes resemble the simple PIB solutions with the
resemblence getting stronger as excitation increases. Note that there is a non-vanishing
probability to find the particle beyond the classical turning points for states with *E<V(x)*
in parts of the box (tunnelling). It appears from the graph
that energy levels of this decorated PIB are lower than corresponding energies of the
simple PIB system: -1.874 vs 1.000, 2.973 vs 4.000, 8.703 vs 9.000, and 15.927 vs 16.000.
But as the energy increases, the influence of the potential on the simple PIB eigenvalue
decreases.

Created or up-dated 08/03/99
by R.D. Poshusta