Quantum Mechanics 1

Schrodinger's equation in one dimension

 Scaling

The square potential well is characterised by two parameters - its width 2a and depth Vo. On the face of it therefore, a systematic numerical investigation to find the eigenvalues and eigenfunctions for all possible square wells would involve a two-dimensional search in which the depth and width would be varied independently.

Fortunately the above assumption is incorrect, and it is sufficient to vary only one of the well parameters to get all relevant information, from which the eigenvalues and eigenfunctions for an arbitrary combination of the well parameters may be deduced. (It is important that this point be appreciated, since a two-dimensional search would obviously involve much more work and numerical tedium.) 

To see this, begin by finding the binding energies for the three lowest lying states for the two cases (i) depth = 300eV,  width = 0.2 nm (i.e. the default option) and (ii) depth = 192 eV, width = 0.25 nm. (You will find that you can vary the well parameters  via the "parameters" menu.) The numbers you end up with should look quite different, but when you calculate the ratio of binding energy to well depth the two sets of ratios should match. (Not quite exactly, in practice; the amount of mismatch will give you some idea of the computational error introduced by the computer.)

Why should this result be? To answer this,  you should return to the analytic treatment of the finite potential well covered in the lectures, and show (actually, most of the work has already been done for you)  that the ratio of binding energy to well depth depends on the single parameter b which involves both the width and depth. Check that the value of b is the same for the two combinations of parameters you worked with above. 

Notice that the mass of the particle is also a parameter in the formula for b.  In the above manipulation  we assumed no change in the choice of particle (an electron) so only the well width and depth changed. But we could easily deal with cases where different masses had to be considered in addition to different well parameters.

Put this into practice by considering a proton inside a nucleus whose potential is modelled as a one dimensional  square well of depth 40 MeV and width 2.0 fm (1 fm = 10-15 m.). Calculate the well width in nm for the equivalent problem of an electron in a well of depth 300 eV. Finally, use the computer to find the electronic ground state binding energy, and so  deduce the binding energy for the proton in its ground state. Comment on why there are no bound excited states for this case.