What happens to the energy of an electron after it tunnels through a barrier?

[omiros]

Hello everybody, I am a first year student and I have a question about tunneling.

If we have an electron which is bound by an atom. And that electron has a high probability to tunnel through the barrier that keeps it confined. So it manages to 'escape' from the atom.

Shouldn't its energy change after that? Classically thinking of that thing its potential energy would change dramatically as it would not be bound by the atom anymore. So the main question is, what happens to its energy and if it remains the same, what changes in the electrons 'structure' (speed, wavelength etc)?

 

[Simon Bridge]

Classically, tunnelling does not happen at all, so classical thinking does not work for this situation.

What you want to consider is a particle in an energy eigenstate of a confining potential barrier.
Since it is just a barrier, there may be a non-zero chance that it may be found beyond the classical limits.
Should a measurement of position show that result, we say that the barrier has been penetrated.
Should a measurement of position show a result farther than the far-side of the barrier, then we say that "tunnelling" has happened.

The whole point of the phenomenon of tunnelling is that the particle able to tunnel is not bound. States with a high probability of tunnelling have a low amplitude within the well.

That's for stationary states - tunnelling is usually considered in terms of plane-wave states - traveling waves.

But the answer to your question is that it's energy is unchanged by tunnelling.
The classical situation would be of a ball rolling through a tunnel that goes through a hill instead of trying to go over the top: hence the name.

 

[erst]

Total energy (potential + kinetic) is unchanged. Potential energy is lower inside the well. If it manages to tunnel outside, then there is a transfer - potential goes up and kinetic goes down. In terms of 'structure', you need to look at the form of the Schrodinger equation. Kinetic energy manifests as curvature of the wavefunction (because the kinetic term p^2/2m is a Laplacian). Thus the electron's wavefunction has sharp curves and zero-crossings inside the well but when it gets out, the wave delocalizes (spreads out) and becomes smoother. "Speed" isn't very meaningful.

 

[Khashishi]

Under normal circumstances, an electron won't tunnel out of an atom because the potential increases the farther you get from the atom. For tunneling, you need a potential hill that goes up but comes back down. If you add an electric field, then an electron can tunnel out of the atom and be spontaneously ionized.

 

[Simon Bridge]

Um - under "normal" circumstances it would be pretty unusual for a single atom to be isolated like that - there are always other potentials somewhere. Similarly it is pretty unusual to rely on tunnelling as a mechanism for ionization - normally the ionization energy is supplied, "lifting" the electron out of the well (or lowering the barrier - whichever picture you like).

OTOH: it is very normal to consider atoms in isolation as a way of learning the physics associated with them.
The standard picture for a hydrogen potential is for a single hydrogen atom all alone in the Universe - which works well for the situation that the surrounding potentials are small ... maybe the next H atom is far away.
Real Life is messy.

 

[jtbell]

According to the way we define the energy of an atomic electron, an electron that is "bound" to an atom always has E < 0. If it's "unbound", it has E ≥ 0 (E = 0 if it's at rest). Therefore something has to supply energy in order for the electron to "escape", regardless of whether there is any tunneling involved or not.

Besides, I don't know of any atoms in which one of the electrons has E > 0 and has to be confined to it by a potential barrier. Do you have an example in mind?

Are you perhaps thinking of alpha decay, in which we can think of the alpha particle as being confined inside a nucleus by a potential barrier and tunneling out?

http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/alpdec.html

 

[omiros]

Under normal circumstances, an electron won't tunnel out of an atom because the potential increases the farther you get from the atom. For tunneling, you need a potential hill that goes up but comes back down. If you add an electric field, then an electron can tunnel out of the atom and be spontaneously ionized.

I have to give a special thanks to you, because that was what I wanted to know about tunneling. I was thinking of an electron just overcoming a stupid barrier, penetrating even in the same atom( 1s to 2s)...

Besides, I don't know of any atoms in which one of the electrons has E > 0 and has to be confined to it by a potential barrier. Do you have an example in mind?

No, I didn't have any of this in mind (I am quite sure that there isn't any). I was trying to think of what happens to the whole energy (lowering of kinetic, increase of potential).


So potentially, can an electron, in a 'big' molecule tunnel from one 'end' to the other, if its energy remains the same?

Also, in the Potassium-Bromine reaction, in which tunneling occurs, does the 4s¹ electron of Potassium and the 4p empty orbital energy are equal?

 

[Simon Bridge]

I still feel that you are misunderstanding the concept of "tunnelling" here.
However: the electron shared in a covalent bond has the same energy level "across both atoms" - it will not, in general, be the same energy as the single-atomic eigenstates.

Each molecular energy level will come from a solution to the schrodinger equation for the combined potential after all.

If you look at jtbell's link for alpha tunnelling, you will see how the energy level of the alpha particle is the same inside and outside the barrier, and the curvature of the wavefunction is generally higher outside the barrier than inside?

I hope you'll excuse me: you sound like you are approaching this from a chemistry background?

 

[omiros]

I hope you'll excuse me: you sound like you are approaching this from a chemistry background?

I will start with that. It is true. I am studying chemistry and molecular physics :)

However: the electron shared in a covalent bond has the same energy level "across both atoms" - it will not, in general, be the same energy as the single-atomic eigenstates.

I was thinking, how does that tunneling happen. I mean that is an Ionic bond (KBr), which mean the Br atom, takes the electron.

And also I was taking about a large molecule, so that in different places, you get many orbitals and strange situations. Like ionizing one end and then from the other, one tunnels and takes its position (if they have the same energy)

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[Simon Bridge]

Undergrad Chemistry class tends to use a blockey rule-of-thumb form of quantum mechanics ... taking the basic results as a way to illustrate the way atoms can interact. You should be wary of drawing deeper conclusions from it.

The energy levels are the eigenvalue solutions to the schrodinger equation using the entire combined potential from every source. So they are the same level throughout something that can be said to be a molecule.

Although, for large complex molecules they can be tricky to calculate and hard to describe. The simpler cases are helpful to highlight the important points.

Like I said before, tunnelling is said to happen when a particle prepared on one side of a potential barrier is detected on the other side - and there was no classical way around the barrier. The particles wavefunction exists, in principle, through all space - though we usually approximate all space to something smaller like "inside the lab".

Something to think about - considering 1D only for ease of maths:
... when you bring two potential wells together (easier to see this with square wells btw) some of the energy eigenstates will be shared across the wells, while some (the lower ones in the deeper well) will not be.

The particle from the shallow well gets shared in the shared energy level - this will make for a covalent bond.

However, if there is an unoccupied unshared state, then there is a chance the particle will decay into it ... which breaks the bond. In this way, a particle can be said to escape one well to be captured by another.

The situation is clearer in a model where only the shallow well is occupied by a single particle and the deeper well is empty, initially the deeper well is "switched off" and at some time t=t₀ the deep well is switched on. In that case, the ground-state wavefunction of the shallow well alone will be represented as a sum of the wavefunctions of the combined 2-well system ... making the energy level uncertain at any t > t₀.
The time evolution of the states will have the particle's mean position "slosh" back and forth between the wells.

If the particles are electrons, then the atom with the deep electron potential becomes negatively charged and the other one becomes positive ... an ionic bond is possible.

An ionic bond would be modeled with the two ions as the particles moving in the potential well that results from their mutual attraction. If one is a lot less massive than the other, then we can approximate it as the light ion moving in the well of the heavy one.

There are lots of ways two atoms may be ionized - the, somewhat simplistic, treatment above was because you specifically asked about how neutral atoms could end up in an ionic bond all by themselves. The situation for your specific example would be more complicated than what I described - that was just to give you a bit of a feel for what sort of thing happens.

Notice that in none of the description did we think of the bond formation as a tunneling situation.
The tunneling model of alpha decay is a much better example.