Tunneling: Can a Particle Appear in the Barrier?

[nonequilibrium]

Hello, I'm following an introductory course in modern physics.

So I understand there's a chance the particle appears on the right side of the barrier, and this has been experimentally verified.

Now I was wondering: can the particle also appear in the barrier?

If so: is this directly measurable? If not, is this indirectly measurable? (for the latter, I'm thinking of the virtual bosons interacting for the fundamental forces: we can't directly measure them, but their effects are measurable)

If not: then what does it mean that the probability function is non-zero in the barrier? Shouldn't the blue part be made zero then?

Thank you!

 

[soothsayer]

Great question! I would think the particle would need to travel through the barrier to get to the other side, and so, necessarily be inside the barrier at some point in time. The wave function does seem to imply that the particle can be found inside the barrier. The above diagram implies that it is classically forbidden for an object to be found anywhere in the E<V region. Speaking from the point of view of QM, if a particle were found inside the barrier, it would simply mean that the particle was only able to tunnel partially through the barrier, but not all the way. If you look at equations for tunneling, they imply that this actually happens more than not, as the tunneling probability depends greatly on the "width" of the potential barrier. As you can see from the waveform, the particle is actually more likely to be found inside the barrier than all the way on the other side, as the tunneling distance required is less.

EDIT: Forgot the other part of your question...This could be easily directly measured. Imagine you have a Scanning Tunneling Microscope, where electrons are given an energy less than that required to jump the potential barrier between it and the surface being studied (the barrier typically being a vacuum), if we were to observe the classically forbidden area between the STM and the surface, we should be able to detect the presence of electrons.

 

[nonequilibrium]

But imagine we are in the situation of the image I included in my first post, and say we measure it in the barrier at some time, as you believe: then it has an energy E > U, because that x-location has a high potential. Can we extract this energy from that particle? Probably not, but why not?

 

[soothsayer]

E<U for a particle in the above barrier. The particle is literally traveling through a barrier it shouldn't be able to, it's strange but true. The particle does not spontaneously pick up enough energy to cross the barrier, so we cannot generate any energy through tunneling, that would be a violation of energy conservation.

 

[nonequilibrium]

I don't believe you can just say that a particle can be in the barier and have "E < U", that's not just intuitive or weird, but simply logically impossible.

Let me try to make my point clear:

Imagine a gravitational body floating in space, a nice big sphere.
And on top of that sphere lies an electron.
Now just like in the picture in the initial post, the wave function is spread out and the probability that the electron appears one meter above the gravitational body (call it earth) is not zero. But if we were to measure it there, one meter higher than its original location, we could grab it (the grabber could actually be the measuring device at the same time), and then extract the potential energy mgh (with h = 1 meter).

Where do you disagree in the following thought experiment?

 

[JesseM]

E<U for a particle in the above barrier. The particle is literally traveling through a barrier it shouldn't be able to, it's strange but true. The particle does not spontaneously pick up enough energy to cross the barrier, so we cannot generate any energy through tunneling, that would be a violation of energy conservation.

But in QM energy conservation is expressed in terms of the expectation values (average over many trials), so I don't think it would violate QM energy conservation if on a small fraction of trials the particle was found inside the barrier.

 

[soothsayer]

I don't believe you can just say that a particle can be in the barier and have "E < U", that's not just intuitive or weird, but simply logically impossible.

Let me try to make my point clear:

Imagine a gravitational body floating in space, a nice big sphere.
And on top of that sphere lies an electron.
Now just like in the picture in the initial post, the wave function is spread out and the probability that the electron appears one meter above the gravitational body (call it earth) is not zero. But if we were to measure it there, one meter higher than its original location, we could grab it (the grabber could actually be the measuring device at the same time), and then extract the potential energy mgh (with h = 1 meter).

Where do you disagree in the following thought experiment?

Ok, you have a point there, sorry about the confusion, the concept of Tunneling and the mathematics behind it are a concept I've only learned somewhat recently, and I think I got confused by looking at your diagram and trying to figure out how to think about it physically, but your example made a lot of sense to me. The only problem being that in your example here, 1 meter is not a quantum scale, so the probability of the electron being found there is practically infinitesimal if it didn't have practically all of the energy classically required to get it there. On a scale where quantum tunneling could occur, the amount of energy we could salvage from the tunneling particle, if at all possible, would be minuscule.

But in QM energy conservation is expressed in terms of the expectation values (average over many trials), so I don't think it would violate QM energy conservation if on a small fraction of trials the particle was found inside the barrier.

So, the energy E of the particle in the above diagram is an expectation value, or average, and is not necessarily the energy of the particle at any given moment or location? Would this would mean, sticking with mr. vodka's above thought experiment, there would also need to be an non zero probability for an electron not being able to reach a height above Earth that it would classically be expected to reach (E>U)? This would be the reason we could not simply harvest extra energy from the tunneling electrons, because on an average, we would lose that extra energy in electrons that did not even reach the classically forbidden region. Is this a correct assumption?

 

[kof9595995]

Imagine a gravitational body floating in space, a nice big sphere.
And on top of that sphere lies an electron.
Now just like in the picture in the initial post, the wave function is spread out and the probability that the electron appears one meter above the gravitational body (call it earth) is not zero. But if we were to measure it there, one meter higher than its original location, we could grab it (the grabber could actually be the measuring device at the same time), and then extract the potential energy mgh (with h = 1 meter).

Where do you disagree in the following thought experiment?

This analogy can't be applied to QM exactly, in the barrier potential problem, once you measure the position of the electron, energy becomes unknown(more precisely, ill-defined in any classical sense), so when you find the election inside the barrier, you no longer have E<V.

 

[JesseM]

So, the energy E of the particle in the above diagram is an expectation value, or average, and is not necessarily the energy of the particle at any given moment or location?

I realized there was an oversight in what I said (been a while since I studied QM)--if the system is measured in an energy eigenstate, and you don't measure any variables that fail to commute with the energy operator (the Hamiltonian), and if the Hamiltonian is time-independent (which I think would be the case whenever the system isn't interacting with anything external that could change its energy), then subsequent measurements should show the exact same energy. I'm not sure if the Hamiltonian for a simple model of a tunneling particle like the one above is time-independent (I think it probably is but I'd have to look it up), but I think in any case it wouldn't commute with the position operator (see the discussion on this thread for example), so that if you measure the energy, then measure the position, then measure the energy again, the energy may have changed. The graph is presumably showing the probability of finding the probability of finding the particle in different positions given that its wavefunction has some particular state vector, not sure if the state vector is supposed to be an energy eigenstate with precise energy E or just some non-eigenstate with an expectation value of E, but even if it's an eigenstate, the very act of measuring the position should change the state vector so the energy would become uncertain.

 

[Phrak]

It may be helpful to know that for a constant potential barrier, the wave equation is Ψ(x) = exp(-κx), where κ is real valued. The unnormalized probability density Ψ*(x) Ψ(x), is therefore complex valued. The probability density is not merely zero it's ill defined.

 

[JesseM]

It may be helpful to know that for a constant potential barrier, the wave equation is Ψ(x) = exp(-κx), where κ is real valued. The unnormalized probability density Ψ*(x) Ψ(x), is therefore complex valued. The probability density is not merely zero it's ill defined.

If κ is real-valued as is x, doesn't that mean exp(-κx) is real-valued too? The complex conjugate of a real number is just the same number, so I don't understand why you say Ψ*(x) Ψ(x) would be complex-valued...

 

[Phrak]

If κ is real-valued as is x, doesn't that mean exp(-κx) is real-valued too? The complex conjugate of a real number is just the same number, so I don't understand why you say Ψ*(x) Ψ(x) would be complex-valued...

Good grief. What was I thinking?

 

[ZapperZ]

Hello, I'm following an introductory course in modern physics.

So I understand there's a chance the particle appears on the right side of the barrier, and this has been experimentally verified.

Now I was wondering: can the particle also appear in the barrier?

If so: is this directly measurable? If not, is this indirectly measurable? (for the latter, I'm thinking of the virtual bosons interacting for the fundamental forces: we can't directly measure them, but their effects are measurable)

If not: then what does it mean that the probability function is non-zero in the barrier? Shouldn't the blue part be made zero then?

Thank you!

We can ask this question in another way. If the particle DID actually go through the barrier, then the nature of the barrier will affect the tunneling rate. If not, then the barrier is merely a separation between the two left and right states.

So without changing the barrier height and width/shape (which is all the parameters that we care about in this tunneling phenomena), can we do something inside the barrier to see if the particle does interact with something inside the barrier? We can. In a superconductor tunneling experiment, we can include, say, magnetic barrier in which the barrier topology doesn't change, but magnetic fluctuation inside the barrier can be present. And when we do that, it definitely changes the tunneling current and spectrum[1]. In fact, certain type of barriers will induce inelastic scattering inside the barrier itself.

Thus, this clearly shows that the tunneling particles DO interact with whatever is inside the barrier. This means that they actually PHYSICALLY go through the barrier and not simply pop out of existence as they enter the barrier and pop back into existence when they exit the barrier. Without having to design some experiment to detect the existence of the particle inside the barrier, this type of experiment can already answer that question.

Zz.

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

 

[JesseM]

but I think in any case it wouldn't commute with the position operator (see the discussion on this thread for example), so that if you measure the energy, then measure the position, then measure the energy again, the energy may have changed. The graph is presumably showing the probability of finding the probability of finding the particle in different positions given that its wavefunction has some particular state vector, not sure if the state vector is supposed to be an energy eigenstate with precise energy E or just some non-eigenstate with an expectation value of E, but even if it's an eigenstate, the very act of measuring the position should change the state vector so the energy would become uncertain.

Incidentally I have a question related to this last part. The chart in the OP seems to be showing the probability for finding the particle at different positions if a position measurement is made, with some nonzero probability of finding it inside the finite potential barrier. So theoretically, if a position measurement is made and the wave function collapses to a position eigenstate which is inside the potential barrier, would it always be true that the expectation value for the energy with this state vector would now be greater than or equal to V? Or is it possible that despite the fact that you have just measured the particle to be inside the barrier, the expectation value for its energy could still be less than V?

 

[Phrak]

Incidentally I have a question related to this last part. The chart in the OP seems to be showing the probability for finding the particle at different positions if a position measurement is made, with some nonzero probability of finding it inside the finite potential barrier. So theoretically, if a position measurement is made and the wave function collapses to a position eigenstate which is inside the potential barrier, would it always be true that the expectation value for the energy with this state vector would now be greater than or equal to V? Or is it possible that despite the fact that you have just measured the particle to be inside the barrier, the expectation value for its energy could still be less than V?

Since showing me the error of my ways, you've stimulated me to ask a similar question--or maybe the same question, really--expectation value assumes measurement.

The chart shows the electron field amplitude, sans the complex phase. The probability amplitude is positive in the potential barrier, and so therefore is the probability, as you've so kindly led me to notice.

So how can a particle with positive probability density be sensed when it's energy is less than zero? Electromagnetic interactions will require only consideration of charge, spin, energy and momentum. Someone, has already mentioned that an electron passing through the barrier to the right has a momentum vector pointing to the left. This seems right. I haven't checked it.

However, I don't see any problems with sensing the electron within the barrier.

To make a long story short, if the Schrödinger model works, and I don't see why it shouldn't, a recoil reaction with another particle within the barrier should be possible. Assume this second particle has positive and fairly well known energy and momentum, to keep it simple. The result is that both the electron and particle after interaction should enter coincidence counters on trajectories such that energy and momentum are conserved.

A rough search of the internet hasn't helped me find if a similar or better conceived experimentally confirming this, but I don't think my key-words are very good.

 

[Avodyne]

So theoretically, if a position measurement is made and the wave function collapses to a position eigenstate which is inside the potential barrier, would it always be true that the expectation value for the energy with this state vector would now be greater than or equal to V?

Yes. For any wave function that vanishes outside the barrier, the expectation value of the potential energy is V, and the expectation value of the kinetic energy is greater than zero.

 

[soothsayer]

What does the wave in the top diagram represent, probability density distribution? Or is it simply modeling the wave function? (Actually, I don't know if you can model the wave function like that, isn't it complex?)