248 SOLUTIONSWe use the first equation to eliminateAin the secondequation. Then each
term has a factor of which is canceled:
Multiplying bothsides of(S.5.4.5) by sinh gives
Thislastequationdetermines whichdetermines theboundstateenergy.
There is only one solution for sufficientlylarge values of
b) The minimumvalue of forcreating a boundstate iscalled It is
found byassumingthat the bindingenergy whichmeans
We examine(S.5.4.7) for smallvalues of and findthat
5.5 Two Delta Function Potentials (Rutgers)
There are two deltafunctionsingularities, one at and one at
The potential can bewritten in anequivalent way as
At each deltafunction wematch the amplitudesof the eigenfunctions as
well as the slopes,using a relationsuch as(S.5.3.3). Asingle, isolated,
attractive,professional,deltafunctionpotential hasa singleboundstate.
We expectthat apair of deltafunctionpotentials will generallyhave one
or two boundstates.
The lowestenergy state, for symmetric potentials, isa symmetriceigen-
function. Theeigenvalue has theform where is thedecay
constant of theeigenfunction. Themostgeneral symmetric eigenfunction
for a bound state is