Question on Finding Transmission Coefficient

In summary, the problem involves finding the transmission coefficient for a potential with two delta function barriers. The approach is to solve the Schrodinger wave equation in the three regions where the potential is zero, and then use physical conditions such as continuity of the wavefunction and Schrodinger's equation to obtain relationships between the coefficients.
  • #1
arenaninja
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Homework Statement


Find the transmission coefficient for the potential [tex]V(x)=-\alpha\left[\delta\left(x+a\right)+\delta\left(x-a\right)\right][/tex], where alpha and a are positive constants.

Homework Equations


[tex]T \equiv \frac{\left|F\right|^{2}}{\left|A\right|^{2}}[/tex]

The Attempt at a Solution


I'm technically not sure on where to begin this problem. After reading the section (twice now), I noticed that it pretty much explicitly omits considering potentials which are not zero. And also, there is another formula for the transmission coefficient
[tex]T=\frac{1}{1+ \beta^{2}}[/tex]
but, again, I'm not too sure on whether I should be using this one. My first instinct is to solve the Schrodinger wave equation, but it looks to be fairly mmessy with those dirac deltas.

Am I missing something really obvious? To recap, it looks like the book's section wanted to purposely omit non-zero potentials, and yet this problem appears to have a non-zero potential, leaving me clueless as to how to deal with it short of starting from scratch.
 
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  • #2
The potential is zero in the 3 regions: [tex]x< -a, -a<x<a, a<x[/tex], so you can solve for the wavefunctions in those regions. You can obtain relations between the coefficients by demanding that certain physical conditions be satisfied. For example, the wavefunction must be continuous at [tex]x=\pm a[/tex]. However, because the potential is not continuous, we cannot demand that the derivative of the wavefunction is continuous. However, Schrodinger's equation must still be valid, so we can obtain relationships across the singular points by integrating the Schrodinger equation around [tex]x=\pm a[/tex]:

[tex]\int_{x=\pm a-\epsilon}^{x=\pm a+\epsilon} \left( - \frac{\hbar^2}{2m} \psi''(x) + (V(x)-E) \psi(x) \right) =0.[/tex]
 
  • #3
ahhh, that's right. Thanks a lot, it makes a lot more sense from your first sentence and I can make some more progress on this one now.
 

1. What is a transmission coefficient?

A transmission coefficient is a measure of the efficiency of energy transmission through a medium. It is often represented by the symbol T and is expressed as the ratio of transmitted energy to incident energy.

2. How is the transmission coefficient calculated?

The transmission coefficient is calculated by dividing the transmitted energy by the incident energy and multiplying by 100 to express it as a percentage. This calculation can be used for various forms of energy such as light, sound, or electricity.

3. What factors affect the transmission coefficient?

The transmission coefficient can be affected by several factors, including the properties of the medium, such as its density and temperature, as well as the wavelength and angle of incidence of the energy being transmitted.

4. What is the importance of the transmission coefficient in scientific research?

The transmission coefficient is an important measurement in scientific research as it helps to understand the behavior of energy as it passes through different mediums. It is particularly useful in fields such as optics, acoustics, and electrical engineering.

5. How can the transmission coefficient be used in real-world applications?

The transmission coefficient has practical applications in various industries, such as telecommunications, where it is used to determine the efficiency of signal transmission through different materials. It is also used in the design and development of materials for energy-efficient buildings and in the study of atmospheric conditions for weather forecasting.

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