Transmission coefficient (quantum tunnelling)

In summary, the conversation discusses the transmission probability (T) when the incident particle energy (E) is equal to the energy of the barrier (U). The equations for transmission and reflection probabilities are provided, and it is mentioned that the probability is not null and less than 1. The conversation also suggests taking the limit of T as E approaches U0 and solving the Schrodinger equation with E = Uo. It is advised not to find T by putting E = Uo in the equations, as the solution to the Schrodinger equation is different for positive, negative, and zero values of k. Instead, the Schrodinger equation needs to be solved again with E = Uo inside the barrier.
  • #1
Camilla
2
0

Homework Statement


What is transmission probability (T), when the incident particle energy (E) is equal to the energy of the barrier (U)?

Homework Equations


Equations to transmission and reflection probabilities.
http://i950.photobucket.com/albums/ad348/gs5720/img099.png?t=1278212132
http://i950.photobucket.com/albums/ad348/gs5720/img098.jpg?t=1278212130

The Attempt at a Solution


I know that the probability isn't null and less than 1, but i don't know how to proceed when the energy of E is equal to U.

That is all the information I have, I would appreciate any help.
 
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  • #2
Try taking the limit of T as E approaches U0. Use the fact that 1-U0/E will approach 0 and approximate sin x by x when x ~ 0.
 
  • #3
You'll need to solve the Schrodinger equation with E = Uo.

Don't try to find T by putting E = Uo in those two cases. This is because the solution to [tex]\frac{{d}^{2}\psi}{d{x}^{2}} = k[/tex] is different for the three cases where k is positive, negative or zero. You cannot solve the equation for positive k, and then put k=0 in the solution to obtain the solution for k=0.
So you'll have to solve Schrodinger equation again with E = Uo inside the barrier.
 

1. What is the transmission coefficient and what does it measure?

The transmission coefficient is a measure of the probability of a particle or wave to pass through a barrier or potential energy barrier. It represents the ratio of the transmitted wave or particle to the incident wave or particle.

2. How is the transmission coefficient calculated?

The transmission coefficient can be calculated using the Schrödinger equation, which describes the behavior of quantum particles, and the wave function of the particle both inside and outside of the barrier. It can also be calculated using the transfer matrix method or the WKB approximation.

3. What factors affect the transmission coefficient?

The transmission coefficient is affected by the height and width of the barrier, as well as the energy and mass of the particle. The shape and composition of the barrier also play a role, as well as the angle of incidence and the potential energy of the particle.

4. How does the transmission coefficient relate to the concept of quantum tunnelling?

The transmission coefficient is directly related to quantum tunnelling, as it represents the probability of a particle or wave to "tunnel" through a potential energy barrier. A higher transmission coefficient indicates a higher probability of tunnelling, while a lower coefficient indicates a lower probability.

5. What are the practical applications of the transmission coefficient and quantum tunnelling?

The transmission coefficient and quantum tunnelling have many practical applications, including in electronic devices such as transistors and tunnel diodes. They also play a role in nuclear fusion and fission processes, as well as in scanning tunneling microscopy and particle accelerators.

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