Undergrad Understanding Electron Behavior in Quantum Tunneling through Dielectrics

[Samson4]

When an electron leaves a conductor and tunnels through a thin dielectric; how does it behave in the space occupied by the dielectric? Does it behave like a ballistic electron traveling through vacuum? Is it instead that the electron never really occupies the space in the dielectric?

 

[Nugatory]

We can measure the position of the electron and find it on one side of the dielectric barrier, and then measure its position again at some later time and find it on the other side of the barrier.

If the electron behaved like a classical object (maybe it's like a grain of sand, except way smaller?) then we would be justified in saying that at some in-between time the electron was somewhere in the middle of the space occupied by the dielectric - how could it get from one side to the other without passing through the space in between?

But the electron does not behave like a classical object. It's a quantum object (the word "particle", implying something like a small grain of sand is an unfortunate historical accident) and according to the standard mathematical formalism of quantum mechanics, it has no position between measurements. That's "has no", not "somewhere, but we're not sure where". Asking about whether it occupies the space or not is like asking about your lap when you're standing up - neither yes nor no but just plain undefined.

 

[Samson4]

What happens if we place a conductor close to the dielectric and measure for acceleration? If the electron at any point is ballistic, wouldn't it accelerate in the opposite direction of the conductor as it tunnels through the dielectric? Does the act of measuring this acceleration change the probability of the electron being found on the other side of the dielectric?

 

[Nugatory]

If the electron at any point is ballistic...

You are still speaking in classical terms here. When you say that something is "ballistic", you're talking about a trajectory. A "trajectory" is basically a claim about the position of the object as a continuous function of time and that model is altogether incompatible with the quantum mechanical model that there is no position between position measurements.

What happens if we place a conductor close to the dielectric? Does the act of measuring this acceleration change the probability of the electron being found on the other side of the dielectric?

This isn't measuring the acceleration (a completely undefined concept because acceleration is the second derivative of position, and the position is undefined) but if you arrange things properly you may be getting a position or momentum measurement out of this setup - much will depend on the exact details of the interaction between the conductor, the particle, and the dielectric. To do this right you'd have to solve Schrodinger's equation for this physical system, and the presence of the conductor will change the Hamiltonian that appears in that equation. It's safe to say that changing the Hamiltonian by introducing the conductor into the system will change the tunneling probability.

You also want to be clear about exactly what this "tunneling probability" is. It is the probability, given that position measurement found the particle on one side of the barrier at one time, that a subsequent position measurement will find the particle on the other side. You're making a logical leap to far if you try to interpret it as the probability that the particle moves from one side to the other.

 

[Samson4]

Thank you very much Nugatory.

 

[ZapperZ]

When an electron leaves a conductor and tunnels through a thin dielectric; how does it behave in the space occupied by the dielectric? Does it behave like a ballistic electron traveling through vacuum? Is it instead that the electron never really occupies the space in the dielectric?

I need to clear something up here.

In tunneling, we do use the word "ballistic" tunneling, but this is not to imply that this is a classical trajectory. Rather, "ballistic tunneling" is synonymous with "elastic tunneling", whereby the tunneling particle does not change its energy as it tunnels through the barrier.

Since we needed to distinguish that something is elastic here, it automatically means that there is a non-ballistic or inelastic tunneling. This is where the tunneling particles interact with something in the barrier, causing it to change its energy. An example of this is in superconducting tunnel junctions, such as SIN or SIS, where impurities or atoms with magnetic moments are embedded in the tunnel barrier. Here, the potential energy profile does not alter significantly than before, but the tunneling particles can interact with the magnetic moments in the barrier, resulting in an inelastic tunneling phenomenon.[1]

Zz.

[1] J.R. Kirtley and D.J. Scalapino, Phys. Rev. Lett. v.65, p.798 (1990).

 

[Samson4]

Zapperz, if the electron is not interacting with the dielectric, consider the following situation:
Two concentric conducting cylinders are separated by a very thin dielectric. The outer conducting cylinder is insulated from ground. If the central conductor is charged with a continuous supply of electrons, there will be some probability that the electrons will tunnel through the dielectric. When they do they spread out on the surface of the outer conductor. The barrier wall for quantum tunneling should not be impacted by the charge accumulation on the outer conductor. Would this lead to a voltage build up on the outer conductor until it starts ionizing the air around it? Is this a quantum diode?

 

[ZapperZ]

Zapperz, if the electron is not interacting with the dielectric, ...

Where did I say that? I don't need to justify something that I never claim.

Zz.

 

[Samson4]

Where did I say that? I don't need to justify something that I never claim.

Zz.

Didn't intend to put words in your mouth. I simply thought that was what you were implying when you made these two statements:
Rather, "ballistic tunneling" is synonymous with "elastic tunneling", whereby the tunneling particle does not change its energy as it tunnels through the barrier.

This is where the tunneling particles interact with something in the barrier, causing it to change its energy.
From what I've been reading and my comprehension; correct or not, I landed on that idea. I'm not arguing a point here, I'm generally curious if this would happen. I looked up quantum diode and it was something else entirely. Thank you for your posts.