Tunneling from Rectangular barrier - Exponential Decay ?

• svrphy
In summary: I'll take your word on the derivation (I'm not sure what 'Require' means here, and it looks like you're just applying some boundary conditions), but I guess I'm still a little confused on the actual values of A and B.On a side note, is it just me or is the formatting on this site kind of a nightmare? It seems like it's almost impossible to make a post with anything more than trivial math in it without it getting mangled.Yeah, the formatting is a bit of a nightmare. I've given up on it.
svrphy
Tunneling from Rectangular barrier - Exponential Decay ??

Consider the Rectangular Potential Barrier. If one solves bound state Problem in this case, wavefunctions of Exponentially Decaying and rising kind are found for the Region in the Barrier.
ψ = A eαx + B e-αx

Yet Most Books and internet sources state that the Wavefunction in the Region is just Exponentially Decaying. From Wikipedia :
Note that, if the energy of the particle is below the barrier height, k_1 becomes imaginary and the wave function is exponentially decaying within the barrier (en.wikipedia.org/wiki/Rectangular_potential_barrier)
From Hyperphysics:
But the wavefunction associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. (http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html)

Simulation here : http://www.st-andrews.ac.uk/physics/quvis/embed_item_3.php?anim_id=16&file_sys=index_phys

I just can't Understand these apperent Double Standards here. The wavefunction is assumed to be "Exponentially Decaying inside the Barrier". At the same time, the Exponentially Rising term is inevitably used in the derivation of tunneling probabilities.

Just what is Going on ??

The rectangular potential barrier does not have bound states, so I'm not sure what you mean by "If one solves bound state Problem in this case".

If you consider the reflection/transmission problem, it is true that, inside the barrier, both rising and falling exponentials must be included. However, in the case where ##\alpha a\gg 1##, the coefficient of the rising exponential is smaller than the coefficient of the falling exponential by a factor of approximately ##\exp(-2\alpha a)##, where ##a## is the width of the barrier. This means that the wave function inside the barrier is very well approximated by a strictly falling exponential.

Avodyne said:
The rectangular potential barrier does not have bound states, so I'm not sure what you mean by "If one solves bound state Problem in this case".

Really sorry for that wrong piece of terminology. Yes, I indeed was referring to Reflection/Transmission Problem. Could you elaborate on the consideration of αa>>1 ? Does it come about when boundary conditions are applied ??

Suppose αa = 1000. How does e1000 compare to e-1000?

For more detail, look up the actual result for the transmission probability, which you can find on Wikipedia:

http://en.wikipedia.org/wiki/Rectangular_potential_barrier

Scroll down to the section "Analysis of the obtained expressions", subsection E < V0, and look at the formula for T. The two exponentials we're talking about are buried inside the sinh(k1a) piece. (Look up the definition of sinh(x) if you need a reminder.)

For a very "thin" barrier, both exponentials do contribute significantly to the result; but in many cases we get a very good approximation by ignoring the negative exponential. In that case the exact solution reduces to the one in Hyperphysics as an approximation.

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1 person
Set up the problem:
$$\psi(x)=e^{ikx}+R e^{-ikx},\ \ x<0$$
$$\psi(x)=A e^{\alpha x}+B e^{-\alpha x},\ \ 0<x<a$$
$$\psi(x)=T e^{ikx},\ \ x>a$$
Require ##\psi(x)## and ##\psi'(x)## to be continuous at ##x=0## and ##x=a##. You will find
$${A\over B}={\alpha+i k\over \alpha-ik}\,e^{-2\alpha a}.$$

1 person
Ah ha! So both terms in the second line are actually falling exponentials?

1. What is tunneling from a rectangular barrier?

Tunneling from a rectangular barrier refers to the quantum mechanical phenomenon where a particle has a non-zero probability of passing through a potential barrier, even though it does not have enough energy to overcome the barrier. This is due to the wave-like nature of particles at the quantum level.

2. How does tunneling from a rectangular barrier occur?

Tunneling from a rectangular barrier occurs when a particle's wave function extends beyond the edges of the potential barrier. This allows the particle to have a non-zero probability of penetrating the barrier, even though classically it would not have enough energy to do so.

3. What is exponential decay in the context of tunneling from a rectangular barrier?

Exponential decay in the context of tunneling from a rectangular barrier refers to the rate at which the probability of a particle tunneling through the barrier decreases as the barrier height increases. As the barrier becomes higher, the probability of tunneling decreases exponentially.

4. What factors affect the rate of tunneling from a rectangular barrier?

The main factors that affect the rate of tunneling from a rectangular barrier include the height and width of the barrier, as well as the energy and mass of the particle. A higher and wider barrier will decrease the probability of tunneling, while a lower and narrower barrier will increase it.

5. What are some real-world applications of tunneling from a rectangular barrier?

Tunneling from a rectangular barrier has many applications in various fields, such as in electronics for creating tunneling diodes and tunneling transistors. It is also used in scanning tunneling microscopy, which allows for high-resolution imaging of surfaces. In nuclear physics, tunneling is involved in the process of alpha decay. Additionally, tunneling is a crucial concept in understanding the behavior of particles in quantum mechanics.

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