delta function potential
is a very useful one to make simple models of molecules and solids.
First we solve the problem with one attractive delta function
Since the bound state has negative energy, the solutions that are normalizable
continuous and its first derivative have a discontinuity
computed from the Schr
dinger equation at
gives us exactly one bound state with
Next we use two delta functions to model a molecule, . Solving this problem by matching wave functions at the boundaries at , we find again transcendental equations for two bound state energies. The ground state energy is more negative than that for one delta function, indicating that the molecule would be bound. A look at the wavefunction shows that the 2 delta function state can lower the kinetic energy compared to the state for one delta function, by reducing the curvature of the wavefunction. The excited state has more curvature than the atomic state so we would not expect molecular binding in that state.
Our final 1D potential, is a
model of a solid.
Jim Branson 2013-04-22