Exploring Barrier Tunneling in Quantum Mechanics

[songoku]

Barrier tunneling happens when, let say, an electron tunnels through a region when it has lower energy compared to the energy of the region (potential barrier).

What differs Quantum from classical mechanics is that CM states the electron will never be able to penetrate the potential barrier while QM states there is finite probability the electron will be observed at the other side of the barrier as if nothing happens inside the barrier.

The formula used to calculate the probability is:

$$T = e^{-2kL} ~where ~k = \sqrt{\frac{8 \pi^2 m (V_0 - E)}{h^2}}$$

I want to ask several questions:
1. Why an electron can penetrate through the barrier even though it has lower energy compared to the barrier?

2. CM states that if the energy of electron is higher than the barrier, it will definitely passes through while QM states there is finite chance that it will be reflected back. Why does QM states that? Why doesn't the electron behaves just like what CM predicts, penetrating through the barrier when it has higher energy than the barrier?

3. Can we use the same formula to calculate the probability when electron has higher energy compared to the barrier? Or because it is not tunneling (the term "tunneling" only applies when electron has lower energy with respect to the barrier) we can't use the formula (the formula is strictly limited to "tunneling")?

4. If we can't use the same formula, is there other formula used to calculate the transmission probability when electron has higher energy compared to barrier (because in QM the probability of electron passing through barrier is not 100% even though it has higher energy)?

Thanks

 

[Simon Bridge]

1&2: There is no "why" for these things - those are the properties of the wavefunction and not restricted to electrons.
3. You use a different formula for probabilities at higher energies to that used for tunnelling.
4. When the incoming particle energy is higher than the barrier energy there are more terms in the wavefunction than when the energy is lower.

By the time students usually meet these concepts they have already met the schrodinger equation and have done some calculations involved with scattering and various potentials. You questions suggest that you may not have seen this groundwork. There are lectures online that can help with this - which textbook are you working from?

 

[songoku]

No, I haven't learned about scattering and schrodinger equation. I use college physics 7th edition by Serway.

Thanks a lot for your explanation

 

[jtbell]

3. Can we use the same formula to calculate the probability when electron has higher energy compared to the barrier?

No, because the formula you gave is only an approximation to the exact formula. It applies in situations in which the electron's energy is much lower than the "height" of the barrier, and (I think) the barrier is not too "thick".

There is a single formula that can be used for all energies and thicknesses. Take the formula for $t$ listed on

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

under "Transmission and Reflection", use it to find $T = |t|^2 = t^*t$, and substitute the definitions of $k_0$ and $k_1$ that you'll find earlier on the page.

 

[Simon Bridge]

No, I haven't learned about scattering and schrodinger equation. I use college physics 7th edition by Serway.

... that looks like an introductory college text ... so Serway has just given you the equation without telling you where it comes from.
Now I think I understand where you are coming from:

Tunnelling, and other QM effects, are a result of the statistical characteristics that Nature shows on the small scale.
On large scales, the possible variations tend to average out to give the classical "laws"... the exception are quite rare so they require very sensitive equipment or special situations to set up.

To get the details though - you have to wait until you learn about the schrodinger equation ... which would be introduced in a second-year college course.