Quantum "tunneling" of sorts and QFT....

[asimov42]

Hi all,

Another naive question related a previous post (where the topic diverged somewhat). I'm wondering about the following thought experiment:

Consider the field associated with a single electron. Now, confine the field to a region (volume) of radius R - that is, field values outside of R are exactly zero (out to infinity, say). This would, in some sense represent a classical potential barrier of infinite height and infinite width.

Next, reduce the width of the barrier (very quickly) to some finite value, say M. Now you have a situation in which the field, which was initially exactly zero beyond R, can assume non-zero values at radius R + M, but is still confined to be zero in the 'barrier' region between R and R + M (i.e., the barrier is now a spherical shell).

The question: can anything approximating this situation happen in reality? If so, it would seem that, with the field forced to remain zero inside the 'barrier' region, there would be no way to propagate and obtain a non-zero field value outside the 'barrier' (because the field must evolve continuously)... the poor electron would remain forever bounded inside the sphere of radius R.

If the field were somehow to evolve through the barrier - this would need to occur no faster than c?

How does QFT handle the situation above (again, if possible)? You can't suddenly have a non-zero field value at radius R + M, without some type of propagation (presumably causally) through the barrier region.

Thanks.

 

[asimov42]

Or is it the case that the field strength (related to the electron) can never be zero? Or that, in reality, one cannot trap the electron in a finite volume? (presumably yes, I'm just looking for an explanation)

 

[asimov42]

As above - the field (wave) amplitude must be non-zero through the barrier, otherwise the wave could not propagate. So very similar to the standard wavefunction explanation in QM. Any comments on the correctness of this?

Apologies for being the only one posting - just hoping I can phrase the question (really just the difference between tunneling as expressed in QFT and tunneling in regular QM) is a way that's coherent. Any help appreciated as usual.

Thanks all.

 

[asimov42]

Could anyone comment on whether the field (wave packet) is non-zero throughout the barrier (just to confirm)? Or simply point to a suitable reference that describes tunneling in QFT?

 

[Jilang]

Now just how fast are you planning to change the barrier?

 

[asimov42]

Presumably the barrier cannot change faster than c. But let's forget the case of an infinite potential well - I'm also wondering about the description of tunneling in QFT for a finite barrier of finite width.

 

[Jilang]

Why do you think it would be different to regular QM?

 

[asimov42]

I was wondering if changing over to a relativistic description of field propagation would have any effects? Or does the wave packet just extend (i.e., is nonzero) through the barrier region?

 

[asimov42]

Actually, there's more to it - I clearly understand tunneling from the perspective of the wave function extending through the barrier. However, in the QFT case, I don't think I'm fully clear on the ontology of the 'field' itself, and hence what the field formulation would say about the barrier region (as @vanhees71, @PeterDonis, and others have pointed out, the wave function has limited meaning in QFT).

(I'm still struggling with QFT in general - hence any references that are relatively easily digestible would be helpful.)

 

[PeterDonis]

Consider the field associated with a single electron

What field are we talking about? The electric field?

confine the field to a region (volume) of radius R

How do you propose to do this? You can't magically make an electric field disappear.

How does QFT handle the situation above

First we need to have a situation that is possible.

 

[asimov42]

What field are we talking about? The electric field?

I was thinking of the electron field.

First we need to have a situation that is possible.

Right - so let's just consider the case of a fixed-height, fixed-width potential barrier. In the tunneling situation, what does QFT say about the (electron) field in the barrier region? (e.g., always nonzero)

 

[PeterDonis]

I was thinking of the electron field.

Ok.

In the tunneling situation, what does QFT say about the (electron) field in the barrier region?

For a finite height barrier, I believe it decays exponentially, just like the wave function does in ordinary QM.

 

[asimov42]

For a finite height barrier, I believe it decays exponentially, just like the wave function does in ordinary QM.

Ah, ok, thanks @PeterDonis! So regular (run of the mill, so to speak) traveling wave packet, hits the barrier and decays exponentially, and then shows up on the other side at a greatly reduced amplitude (if one can say such a thing about a wave packet)? (to confirm)

 

[PeterDonis]

So regular (run of the mill, so to speak) traveling wave packet, hits the barrier and decays exponentially, and then shows up on the other side at a greatly reduced amplitude (if one can say such a thing about a wave packet)?

At this level of approximation, yes, that's basically what's going on.

What you might not be grasping is that, for the case under discussion, there is no real difference between QFT and ordinary QM. Ordinary QM is an approximation to QFT; in this approximation you can start with QFT quantum fields and derive ordinary QM wave functions. There are plenty of cases where this approximation doesn't work, but quantum tunneling through a barrier is not one of them.

 

[asimov42]

Ok, thanks! When you mention the level of approximation, is there an easy way (unlikely ) to say what issues arise when one considers the full QFT description? (i.e., what complications arise, at a high level)

 

[PeterDonis]

is there an easy way (unlikely ) to say what issues arise when one considers the full QFT description?

Not really, no. I would recommend working through a QFT textbook, but I don't think anyone would say that was "easy".

 

[asimov42]

Not really, no. I would recommend working through a QFT textbook, but I don't think anyone would say that was "easy".

Great - my assumption is that, even in the full QFT description, you always have nonzero field values on both sides of, and inside, the barrier region? (hehe last question, promise, and thanks again @PeterDonis!)

I'm actually looking for a good text, if anyone has suggestions - at the moment, the one most useful is "Quantum Field Theory for the Gifted Amateur", but I'm not sure I'm in that category ).