Barrier Tunneling & Newtonian Mechanics for Large Objects

['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?

 

[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.

 

[Gokul43201]

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

 

['AQF]

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

 

[jtbell]

$l$ 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!

 

['AQF]

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.

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?

 

[ZapperZ]

[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"?

<scratching head>

Zz.

 

[arildno]

I would just add that there are no inherent problems with Newtonian mechanics not being able to account for the smallest scale features of reality, nor is that the reason why QM should be regarded as a superior theory.

The reason why QM is a superior theory is that it explains the smallest features of reality IN ADDITION to the fact that it gives Newtonian mechanics as a special case of itself.
If QM had not been able to do this last point, or that the limit process had given macroscopic predictions differing from NM, and hence, differed from observed macroscopic reality, QM could not have claimed any superiority to NM.