Barrier Tunneling of electron

In summary, barrier tunneling is a phenomenon in quantum mechanics where an electron can pass through a potential barrier with lower energy. This is different from classical mechanics, where the electron would not be able to penetrate the barrier. The probability of this happening is calculated using a specific formula. When the electron has higher energy compared to the barrier, a different formula must be used to calculate the probability. This is due to the statistical nature of quantum mechanics, which is explained by the Schrodinger equation.
  • #1
songoku
2,266
319
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:

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

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
 
Physics news on Phys.org
  • #2
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?
 
  • #3
No, I haven't learned about scattering and schrodinger equation. I use college physics 7th edition by Serway.

Thanks a lot for your explanation
 
  • #4
songoku said:
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.
 
  • #5
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.
 

1. What is barrier tunneling of electrons?

Barrier tunneling of electrons is a quantum mechanical phenomenon in which electrons are able to pass through energy barriers that would typically be too high for them to overcome. This is possible due to the wave-like nature of electrons, which allows them to exhibit tunneling behavior.

2. What causes barrier tunneling to occur?

Barrier tunneling occurs when there is a potential energy barrier that an electron encounters. This barrier can be a physical barrier, such as a thin insulating layer, or a potential energy barrier created by an external electric field.

3. How is barrier tunneling related to quantum mechanics?

Barrier tunneling is a key concept in quantum mechanics, as it demonstrates the wave-particle duality of electrons. This phenomenon cannot be explained by classical mechanics, but rather requires a quantum mechanical understanding of particles as both waves and particles.

4. What are some real-world applications of barrier tunneling?

Barrier tunneling has a wide range of applications, including in electronic devices such as transistors and tunnel diodes. It is also used in scanning tunneling microscopy, a technique that allows for precise imaging of surfaces at the atomic level.

5. Can barrier tunneling be observed in other particles besides electrons?

Yes, barrier tunneling can also occur with other particles such as protons, neutrons, and atoms. However, the likelihood of tunneling decreases as the mass of the particle increases, so it is most commonly observed with electrons due to their small mass.

Similar threads

Replies
7
Views
938
Replies
14
Views
1K
  • Quantum Physics
Replies
2
Views
682
Replies
4
Views
616
Replies
27
Views
2K
  • Quantum Physics
Replies
3
Views
1K
  • Quantum Physics
Replies
4
Views
712
  • Introductory Physics Homework Help
Replies
9
Views
785
  • Quantum Physics
Replies
6
Views
1K
Replies
3
Views
863
Back
Top