Consider a particle of mass m subject to an attractive delta potential V(x)=−V0δ(x), where V0>0 (V0 has the dimensions of Energy×Distance).
(a) In the case of negative energies, show that this particle has only one bound state; find the binding energy and the wave function.
(b) Calculate the probability of finding the particle in the interval −a≤x≤a.
(c) What is the probability that the particle remains bound when V0 is (i) halved suddenly, (ii) quadrupled suddenly?
(d) Study the scattering case (i.e., E>0) and calculate the reflection and transmission coefficients as a function of the wave number k.