Quantum Tunneling: Particle location

[CAF123]

A quantum particle can tunnel through a barrier and we can describe why this is so by the uncertainty principle. However, there is something I want to clarify about the reasoning behind this.
I believe the argument goes like this: Imagine the quantum particle never entered the barrier and we know for sure it is in the classically permissible area. Then we know Δx = 0 which implies Δp →∞. (Only allowed for infinite potential wells). Hence the original assumption that the particle was never in the barrier is false and thus some of the probability wave must leak into the barrier. That is why we know quantum tunneling is possible.

What I don't understand, however, is even if we assume the particle can never enter the barrier and we know it is still outside it, then why does this imply Δx = 0?
As a friend of mine said: If I know my keys aren't in my pocket, that does not tell me exactly where they are..

I understand the above anology is obviously very different from the quantum world, but the question still remains for me: why does Δx = 0? Even if we know the particle is outside the barrier, surely it can still be anywhere so why does this automatically imply Δx =0?

Thanks for any input.

 

[jfy4]

It doesn't, $\Delta x \neq 0$. It's some finite value, same with $\Delta p$.

 

[Many_S_Theory]

This seems a little muddled to me. Perhaps it would help to point out that the "Heisenberg uncertainty principle" is not, in fact, anything special to quantum mechanics, in fact it's a general property of ALL waves. The key conceit of quantum mechanics is that the behaviour of "particles" isn't determined by a particle equation but rather described by a WAVE equation giving a probability AMPLITUDE (not the probability). Once you've established that you have a wave, the "uncertainty principle" comes for free.

The reason I bring this up is because ALL waves (water, sound, etc.) do something call Evanescence (ya, like the band). http://en.wikipedia.org/wiki/Evanescent_wave . The evanescence of a quantum wavefunction is called tunneling. Just to recap what I'm saying, the real "pill" one must swallow when moving to quantum mechanics is that reality is described by a WAVE equation which gives PROBABILITY AMPLITUDES (the key difference between a probability amplitude and a probability is that amplitudes can interfere and cancel each other). Once this is established, what we called "the uncertainty principle" and "tunneling" are in fact properties of ALL waves and come free of charge.

 

[ZapperZ]

A quantum particle can tunnel through a barrier and we can describe why this is so by the uncertainty principle. However, there is something I want to clarify about the reasoning behind this.
I believe the argument goes like this: Imagine the quantum particle never entered the barrier and we know for sure it is in the classically permissible area. Then we know Δx = 0 which implies Δp →∞. (Only allowed for infinite potential wells). Hence the original assumption that the particle was never in the barrier is false and thus some of the probability wave must leak into the barrier. That is why we know quantum tunneling is possible.

What I don't understand, however, is even if we assume the particle can never enter the barrier and we know it is still outside it, then why does this imply Δx = 0?
As a friend of mine said: If I know my keys aren't in my pocket, that does not tell me exactly where they are..

I understand the above anology is obviously very different from the quantum world, but the question still remains for me: why does Δx = 0? Even if we know the particle is outside the barrier, surely it can still be anywhere so why does this automatically imply Δx =0?

Thanks for any input.

Please read a recent thread that will correct a lot of your misunderstanding of the tunneling phenomenon.

https://www.physicsforums.com/showthread.php?t=615164

I can cite for you many experiments that will show that the particles DO go through the barrier. I do not know where people get the info that the particles don't. Are you people reading Wikipedia or something that gave such ridiculous idea?

Zz.

 

[CAF123]

Oops, sorry no wonder. What I wrote in the opening statement is complete nonsense, as I had my Δx =0 in the wrong situation.

What I meant to say was, imagining that in the barrier Δx=0 (ie the particle can never go into the barrier) and we know for sure it is the classical permissible area, then this implies by the uncertainty principle, Δp →∞ and so it's energy is infinite. However, since this is a finite potential well, with infinite energy, the particle could escape.

Hence the original assumption that Δx =0 in the barrier is false and we see by the uncertainty principle there must be a finite probability that the particle can be located inside the barrier.

Sorry about that.

 

[ZapperZ]

Oops, sorry no wonder. What I wrote in the opening statement is complete nonsense, as I had my Δx =0 in the wrong situation.

What I meant to say was, imagining that in the barrier Δx=0 (ie the particle can never go into the barrier) and we know for sure it is the classical permissible area, then this implies by the uncertainty principle, Δp →∞ and so it's energy is infinite. However, since this is a finite potential well, with infinite energy, the particle could escape.

Hence the original assumption that Δx =0 in the barrier is false and we see by the uncertainty principle there must be a finite probability that the particle can be located inside the barrier.

Sorry about that.

This is a serious misapplication of the HUP.

The Δx in the HUP is NOT the barrier width in this case. It is the location in which you have confined the particle, so that the uncertainty in the position corresponds to the confinement size.

Your particle NEVER entered the barrier. So the width of the barrier is irrelevant in the tunneling phenomenon, and certainly has no bearing on any uncertainties for the particle.

For the last time, the HUP is NOT the origin of the tunneling phenomenon. I do not know why people are stuck on this one. Have ever looked at the QM description of this phenomenon?

Zz.