A quantum mechanical particle of mass moves in one dimension in the presence of a negative delta function potential
where is a parameter with dimensions of length.
(a) Write down the time-independent Schrödinger equation for energy eigenstates , with energy . By integrating this equation across , show that the gradient of the wavefunction jumps across according to
[You may assume that is continuous across ]
(b) Show that there exists a negative energy solution and calculate its energy.
(c) Consider a double delta function potential
For sufficiently small , this potential yields a negative energy solution of odd parity, i.e. . Show that its energy is given by
[You may again assume is continuous across .]