Transmission for double delta-potential vs single delta

In summary, the problem discusses the scattering from repulsive δ-function potentials and the transmission and reflection coefficients for a single potential with strength V0. The transmission probability through two barriers is not always smaller than that of a single one, as shown in the case of E=1 and V0=1, where a separation of q0=0.7 results in a transmission probability of 0.89. This phenomenon can be explained by analyzing the real part of the wave function, and the separation of the two deltas may play a role in determining the maximum transmission. Additionally, in the region between the two delta functions, two sinusoidals moving in opposite directions must be added, hinting at a possible relationship between the separation and wavelength of
  • #1
UiOStud
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Homework Statement


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).
a)
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?

Homework Equations


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

T=2/(5+cos(sqrt(8)q0)+2sqrt(2)sin(sqrt(8)q0)

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
[/B]
 
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  • #2
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?

I suspect this is a strong hint. Is there a relation between the separation of the two deltas and the wavelength of the wave-function?
 
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  • #3
In the region between the two delta function you have to add two sinusoidals moving in opposite directions. Does this ring a bell?
 
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Related to Transmission for double delta-potential vs single delta

1. What is the difference between double and single delta-potential?

The difference between double and single delta-potential lies in the number of potential barriers present in the system. In a single delta-potential, there is only one potential barrier, while in a double delta-potential, there are two potential barriers.

2. How does the transmission differ between double and single delta-potential?

The transmission for double delta-potential is higher compared to single delta-potential. This is because the presence of two potential barriers in the double delta-potential allows for more tunnelling paths, increasing the chances of particles passing through the barrier.

3. Does the energy of the particles affect the transmission in double delta-potential?

Yes, the energy of the particles does have an impact on the transmission in double delta-potential. Higher energy particles have a higher probability of tunnelling through the potential barriers compared to lower energy particles.

4. How does the width of the potential barrier affect the transmission in double delta-potential?

The width of the potential barrier has a direct impact on the transmission in double delta-potential. A wider barrier leads to a lower transmission, as there is a larger distance for particles to tunnel through. Conversely, a narrower barrier results in a higher transmission.

5. Are there any real-world applications for studying the transmission in double delta-potential vs single delta?

Yes, understanding the differences in transmission between double and single delta-potential has applications in fields such as semiconductor technology and quantum mechanics. It can also aid in the development of more efficient energy barriers in electronic devices.

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