Tunneling through step potential

In summary, the conversation discusses the calculation of transmission probability in a tunneling problem where the energy of the approaching particle is equal to the height of the potential barrier. The equations used are the same as in other cases for x<0, and the solutions for both parts of the problem are Φ_1=Aexp(ikx)+Bexp(-ikx) and Φ_2=Cexp(iqx)+Dexp(-iqx). The conversation also addresses the question of why the wave function must be zero in the square potential region, and concludes that it is due to the possibility of transmission occurring for E=V.
  • #1
frankcastle
10
0

Homework Statement


Hey, I found this interesting case in the tunneling problem.
How do we calculate the transmission probability when the
energy of the approaching particle is equal to the height of the potential barrier?

I.e E=Vo.


Homework Equations



Same equations as in other cases for x<0.

The Attempt at a Solution


How does the particle behave in the potential step region?
 
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  • #2
can we calculate like the other problem of the potential barrier .E=V so after calculating Hamiltonian operator on wave function in part 1 we have Φ_1=Aexp(ikx)+Bexp(-ikx) and part 2 :Φ_2=0? and we can calculate A and B.
 
  • #3
The solution for part 1 makes sense, but why must the wave function be zero in the square potential region? If tunneling is possible for E<V then transmission must occur for E=V too I think
 
  • #4
I got what was my mistake! :-)
General solutions are Φ_1=Aexp(ikx)+Bexp(-ikx) and Φ_2=Cexp(iqx)+Dexp(-iqx)
as we know k^2=2mE(4pi^2)/h^2 and q=2mE(E-V)(4pi^2)/h^2
so Φ_2=C+D (transmission is constant)it makes sense,doesn't it?;-)
 

Related to Tunneling through step potential

1. What is tunneling through a step potential?

Tunneling through a step potential is a quantum mechanical phenomenon in which a particle is able to pass through a potential barrier that it does not have enough energy to overcome. This is possible because of the wave-like behavior of particles at the quantum level.

2. What is a step potential?

A step potential is a sudden change in the potential energy of a particle. It can be visualized as a "step" in a potential energy graph, where the particle encounters a higher or lower potential energy as it moves along its path.

3. How does tunneling through a step potential occur?

Tunneling through a step potential occurs when a particle approaches a potential barrier and has a probability of being on the other side, even though it does not have enough energy to overcome the barrier. This is due to the wave-like nature of particles, which allows them to "tunnel" through the barrier.

4. What are some real-world applications of tunneling through a step potential?

Tunneling through a step potential has many real-world applications, including scanning tunneling microscopy, which uses the tunneling effect to image surfaces at the atomic level. It is also important in the design and operation of electronic devices such as transistors and tunnel diodes.

5. Are there any limitations to tunneling through a step potential?

Yes, there are limitations to tunneling through a step potential. The probability of tunneling decreases as the barrier height increases, and it becomes more difficult for particles to tunnel through wider barriers. Additionally, tunneling is a probabilistic phenomenon, so it is not guaranteed that a particle will always tunnel through a barrier.

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