In this problem we shall consider the scattering from repulsive δ-function potentials. We have already considered a single such potential. If the strength of the potential is V0, the transmitted and reflected fluxes may be represented by the transmission and reflection coefficients
T=k^2/(1+k^2) and R=1/(1 +k^2) where k =sqrt(2EV0), and we have used so-called natural units defined by the replacement hbar=m= 1. The potentials of strength V0are shown in figure 1. One might assume that the transmission probability through two barriers should always be smaller than the transmission through a single one. In the following, you will show that this is not necessarily the case and provide an explanation for this
effect. Assume the potential V(x) =V0δ(x−q0) +V0δ(x+q0).
Assume further that a particle arrives at the first potential barrier from the left with the energy E, and divide the system into the 3 regions shown in figure1. Determine the scattered wave functions by combining the wave function in each separate region, allowing for the discontinuity in the first derivative of the functions at the location of each δ-function.
b) Find an expression for the total transmission and reflection probabilities as functions of E, q0 and V0. Check that the overall flux is conserved.
c)Show that for some values of E,q0 and V0, the transmitted flux will be larger
than the case for a single delta-potential. Find or derive an explanation for
this phenomenon. It may be useful to draw or plot the real part of the wave function er some selected values of E,q0 and V0 where the phenomenon occurs.If you select E,V0= 1, what separation of the δ-function q0 gives maximum transmission?
T = abs(F/A)^2 where F is the amplitude of the transmitted wave and A is the amplitude of the incoming wave
The Attempt at a Solution
a) and b) went fine. The challenge is c).
By setting E = 1 and V_0 = 1 in the expression for transmission, you get
for the double-delta-function. For the single-delta-function from the problemtext I insert k=sqrt(2) and got:
T = 2/3
So if I can find a q0 that makes the first expression for T greater than 2/3 I have proven that the transmission sometimes is greater for a double delta-function than a single delta-function.
By setting q0 to be 0.7 I find that T is 0.89>2/3.
However I can't seem to find a good explanation for this phenomena