Newtonian Barriers

1. Oct 31, 2005

'AQF

I am trying to figure how barrier tunneling can be reconciled with Newtonian mechanics for large objects. I know that Newton’s Law work for large objects but not for small ones. Can anyone tell me why?

2. Oct 31, 2005

James R

In general, the probability of tunnelling decreases as the mass of the object increases. For "large" objects, the probability of tunnelling is negligible. So, you never see a tennis ball quantum tunnell through a brick wall, but an electron may well do so.

3. Oct 31, 2005

Gokul43201

Staff Emeritus
The transmission coefficient goes roughly like $$e^{-l\sqrt{2m(V_0 - E)}/\hbar}$$

4. Oct 31, 2005

'AQF

Gokul, can you explain what each variable in the formula means and how it relates to the problem?

5. Oct 31, 2005

Staff: Mentor

$l$ is the thickness of the barrier.
$m$ is the mass of the particle.
$V_0$ is the "height" of the barrier in terms of energy
$E$ is the energy of the particle.
$\hbar$ is Planck's constant $h$ divided by $2\pi$.

The formula as a whole gives you the probability that the particle will get through the barrier (rather than be reflected). It's an approximation that is good for small probabilities, not so good for large ones (maybe above about 0.10 or so). I don't have a book with the exact formula handy here at home, but believe me, you don't want to deal with it if you don't have to!

Last edited: Oct 31, 2005
6. Nov 1, 2005

'AQF

This general statement makes sense. Yet how does one reconcile the fact that in Newtonian mechanics, E cannot be less than the potential energy U (because the KE is never negative), the situation here where this is completely broken?

7. Nov 1, 2005

ZapperZ

Staff Emeritus
[QUOTE='AQF]This general statement makes sense. Yet how does one reconcile the fact that in Newtonian mechanics, E cannot be less than the potential energy U (because the KE is never negative), the situation here where this is completely broken?[/QUOTE]

I think you are missing a punch line in all of this. Newtonian mechanics CANNOT be reconcilled with tunneling phenomenon. I thought that is the whole point in us having to study quantum mechanics. So what you are attempting to do is not only puzzling, but futile.

This is just one example where Newtonian mechanics fail. There are many others. If Newtonian mechanics can be "reconcilled" to explain all of these phenomena, then why bother having a separate field of physics called "quantum mechanics"?