Particle in a Box with a spring

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 kL2/12 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 L, the length of the box. The natural energy unit is , the same as for the simple PIB. In terms of these units we define the force constant to be k = kEnatural/L2 so that k becomes the dimensionless force constant of the dimensionless Schroedinger equation. Numerical integration or expansion in a basis can be used to solve for the eigenenergies and eigenfunctions. For instance, when k=100, the energies and wave functions are depicted in the figure.

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.

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