How can a particle be in a classically prohibited region?

[SirClueless]

Hey Guys.

I'm having trouble wrapping my head around the idea of a particle being in a classically prohibited region. A particle is in a classically prohibited region if its total energy is less than the potential energy at that location. To me, this would seem to imply negative kinetic energy (and hence imaginary momentum), if we accept that total energy = kinetic energy + potential energy. I asked my instructor and he said, "I don't think you should think of total energy as kinetic energy plus potential when dealing with quantum." My TA said that the act of measurement would impart energy to the particle (changing the ψ in the process), thus allowing it to get over that barrier and be in the classically prohibited region and conserving energy in the process. This made sense to me but then if this is true, tunneling doesn't really seem as mysterious/mystifying as it was presented to be. When I googled my question, I saw some answers that said it comes down to the uncertainty principle: the more you know about the location of the particle, the less you know about the its momentum and hence kinetic energy so you can't really say anything about its kinetic energy if you measured its location in a classically prohibited region.

I would really appreciate if you guys could help me understand this as I have gotten three different answers!

 

[PeterDonis]

I'm having trouble wrapping my head around the idea of a particle being in a classically prohibited region.

The key thing to understand is that the particle will never be *detected* in a classically prohibited region. The only reason we believe that quantum mechanics allows particles to (temporarily) be in classically prohibited regions is that it's the only way we can explain phenomena like radioactive decay, where a particle "tunnels" through a potential barrier that, classically, it should not be able to get through. But when we actually detect the particle, it is always *after* the tunneling is complete and the particle is in a classically allowed region again.

A particle is in a classically prohibited region if its total energy is less than the potential energy at that location. To me, this would seem to imply negative kinetic energy (and hence imaginary momentum), if we accept that total energy = kinetic energy + potential energy. I asked my instructor and he said, "I don't think you should think of total energy as kinetic energy plus potential when dealing with quantum."

More precisely, I would say that you can't think of total energy as kinetic energy plus potential when a quantum object is "tunneling" through a region that is classically prohibited. One way of thinking about this is that the particle is taking advantage of the uncertainty principle to temporarily violate the rule that total energy is kinetic plus potential, so that it can have enough energy to get over the barrier. (Another way to think about it is that the particle is taking advantage of the uncertainty principle to temporarily violate the classical equations of motion, so that it can have more kinetic energy than it "ought" to have given its state of motion. *Still* another way to think about it is that the particle is taking advantage of the uncertainty principle to violate the classical energy-momentum relation, so that it can have negative kinetic energy but still have real momentum. The bottom line here is that what is going on can't be described in ordinary classical terms; and since our ordinary, non-technical language is based on classical terms, there is no single description of what's happening in ordinary language. This is why we use math in physics to describe these things precisely.)

My TA said that the act of measurement would impart energy to the particle (changing the ψ in the process), thus allowing it to get over that barrier and be in the classically prohibited region and conserving energy in the process.

First, note that the region outside the barrier is not classically prohibited, as I noted above. Only the region "under" the barrier is prohibited; but, as I noted above, the particle is never actually detected in that region.

That said, if you measure the particle to be outside the barrier, you are measuring it *after* it has already tunneled through the barrier, so the measurement itself can't be what caused the particle to get through (or over) the barrier.

 

[SirClueless]

That's interesting. Seeing that ψ^2 in not nonzero inside classically prohibited regions, could we theoretically detect a particle in a classically prohibited region? If so, why do we always detect it after tunneling. If not, isn't that inconsistent with the idea that ψ(x)^2dx gives us the probability of finding a particle in the region of x-x+dx? Or am I thinking about this wrong?

Thanks for responding!

 

[PeterDonis]

Seeing that ψ^2 in not nonzero inside classically prohibited regions, could we theoretically detect a particle in a classically prohibited region?

To the best of my knowledge, no.

isn't that inconsistent with the idea that ψ(x)^2dx gives us the probability of finding a particle in the region of x-x+dx?

More precisely, it's an example of the fact that that idea has limitations; you can't *always* think of the wave function as giving the probability of finding a particle.

 

[Jilang]

One idea that you can never find it in the classically forbidden region is that it does not spend any real time there.
It was claimed by the Keller group in Switzerland that particle tunneling does indeed occur in zero real time. Their tests involved tunneling electrons, where the group argued a relativistic prediction for tunneling time should be 500-600 attoseconds (an attosecond is one quintillionth (10−18) of a second). All that could be measured was 24 attoseconds, which is the limit of the test accuracy.

 

[Orodruin]

To the best of my knowledge, no.

I disagree with this. It would be perfectly possible to detect the system in that position. Anything else would violate basic principles of QM, including how we normalize wave functions and take expectation values.

 

[Ravi Mohan]

My TA said that the act of measurement would impart energy to the particle (changing the ψ in the process), thus allowing it to get over that barrier and be in the classically prohibited region and conserving energy in the process.

He is right. The (position) measurement localizes the wavefunction. The momentum (and thus energy) gains uncertainty that allows particle to exist in the region which was "calssically prohibited" for the pre-measurement state.

(After measurement, when you detect the particle at a point, you can't claim that it has 2 units or 5 units of energy.)

This made sense to me but then if this is true, tunneling doesn't really seem as mysterious/mystifying as it was presented to be.

It is not.

 

[Jilang]

I don't think it would be possible to detect a particle in the barrier even in principle. If the measurement disturbs the particle it knocks it's energy up so it is over the barrier. I am not sure you could even describe it as being a particle when it's inside the barrier, the wavefunction is evanescent (decaying). At best is could be described as a virtual particle. Stahlhofen and Günter Nimtz developed a mathematical approach and interpretation of the nature of evanescent modes as virtual particles, which confirms the theory of the Hartmann effect (transit times through the barrier being independent of the width of the barrier).

 

[ZapperZ]

That's interesting. Seeing that ψ^2 in not nonzero inside classically prohibited regions, could we theoretically detect a particle in a classically prohibited region? If so, why do we always detect it after tunneling. If not, isn't that inconsistent with the idea that ψ(x)^2dx gives us the probability of finding a particle in the region of x-x+dx? Or am I thinking about this wrong?

Thanks for responding!

Read my post here:

https://www.physicsforums.com/showpost.php?p=3063909&postcount=13

Zz.

 

[Vanadium 50]

I'm not really happy with some of the answers here.

Let's start with the basics. No theory is good at answering the question "What is the value of X when I am not measuring it?". That is by construction an unanswerable question, and it is particularly troublesome in QM. So your concern that if I have a particle and I choose not to measure it's energy that it's energy might be some crazy number had I chosen to measure it is something in the land of the forever untestable.

Onto Peter's comment. A particle absolutely can be in the classically forbidden region. Consider the hydrogen atom. The classically forbidden region is where the energy is lower than the potential energy, which means r > 2a. The probability of that is calculable, and works out to 13e^-4, or about 1 in 4.

If you work out something that depends on the hydrogen electron doing this, for example, the polarizability of atomic hydrogen, you get the wrong answer if you truncate the probability distribution at 2a.

However, there is one point in which Peter is absolutely right. If I pick an electron in the classically forbidden region and then measure it's energy, I will find that the kinetic energy is positive - i.e. it's not forbidden to be there. By constraining its position, it is no longer in the ground eigenstate. This is why it's important to describe the system you are calculating in detail. Subtle changes in the question can make a big change to the answer.

 

[PeterDonis]

A particle absolutely can be in the classically forbidden region.

Please note that I didn't say it couldn't *be* in the classically forbidden region (i.e., the region in which the potential energy is greater than the particle's energy). I just said I didn't think it could be *detected* there. However, it is also possible that it could be detected there, but not with a classically forbidden energy. See further comments below.

If you work out something that depends on the hydrogen electron doing this, for example, the polarizability of atomic hydrogen, you get the wrong answer if you truncate the probability distribution at 2a.

Agreed; there is certainly a nonzero amplitude for the particle to be in a classically forbidden region. Even in the original scenario in this thread, tunnelling, you certainly won't get the right answer if you assume the wavefunction is zero under the potential barrier.

If I pick an electron in the classically forbidden region and then measure it's energy, I will find that the kinetic energy is positive - i.e. it's not forbidden to be there. By constraining its position, it is no longer in the ground eigenstate.

Yes, agreed. A better answer to the question I was answering about detection in the classically forbidden region is that the particle can never be measured to have a classically forbidden energy. If it is measured to have an energy less than the potential barrier, it must be either inside or outside the barrier; it can't be in the barrier region. If, OTOH, it is measured to be in the barrier region, its energy must be greater than the barrier potential.

My only question is *how*, in practice, you would actually measure the particle to have a position inside the barrier region. You can't just arbitrarily "pick" it to be there, at least not in any "ordinary" cases of tunneling, because you don't control the particle's motion. Are there any experiments that have actually tried to do this? (ZapperZ's post that he linked to describes experiments with superconductors that show that interactions can take place within the barrier region, but they still don't actually measure the particle's position to be within the barrier region.)

 

[ZapperZ]

My only question is *how*, in practice, you would actually measure the particle to have a position inside the barrier region. You can't just arbitrarily "pick" it to be there, at least not in any "ordinary" cases of tunneling, because you don't control the particle's motion. Are there any experiments that have actually tried to do this? (ZapperZ's post that he linked to describes experiments with superconductors that show that interactions can take place within the barrier region, but they still don't actually measure the particle's position to be within the barrier region.)

But this is now a different topic, i.e. the measurement of the position inside the barrier. The original topic of this thread was simply to ask how one would verify that the particle DID enter the barrier and did actually pass through the barrier, instead of simply appearing at the other side without physically being in the barrier.

I do not see how, based on the inelastic tunneling experiments, one can still have doubts that the particle did, in fact, physically traveled through the barrier, rather than simply appearing at the other side.

And before we talk about measuring the position of something like an electron is in the tunnel barrier, let's point out that we also do not have an experimental measurement of the position of an electron in a conductor! Sure, we have the mathematical QM description of it, but we have such a thing as well for the tunnel barrier. Yet, no one is questioning if such electrons exist inside the conductor.

Zz.

 

[PeterDonis]

I do not see how, based on the inelastic tunneling experiments, one can still have doubts that the particle did, in fact, physically traveled through the barrier, rather than simply appearing at the other side.

Agreed. One can show that the particle must traverse the barrier region, without actually measuring its position to be there.

 

[atyy]

Here's a paper which seems to reflect what some of what the OP's TA was saying (and I think Vanadium 50 too).

http://arxiv.org/abs/hep-th/9305075
http://dx.doi.org/10.1103/PhysRevA.48.4084
Measurements, errors, and negative kinetic energy
Y. Aharonov, S. Popescu, D. Rohrlich, L. Vaidman

" ... Quantum particles can be found in regions where a classical particle could never go, since it would have negative kinetic energy. But in quantum theory, too, the eigenvalues of kinetic energy cannot be negative. How, then, can a quantum particle "tunnel"? The apparent paradox is resolved by noting that the wave function of a tunnelling particle only partly overlaps the forbidden region, while a particle found within the forbidden region may have taken enough energy from the measuring probe to offset any energy deficit. There is no wave function that represents a particle restricted to a region where its potential energy is larger than its total energy. Nevertheless, we will show that actual measurements of kinetic energy can yield negative values, and that, under proper conditions, a remarkable consistency appears in these apparent errors. ..."

But there are a couple of differences from what the TA said. Here the emphasis is on positive energy versus negative energy. The positive energy results because the position measurement collapses the wave function into a position eigenstate. Can one talk about energy conservation across a measurement, since there is non-unitary evolution when the wave function collapses? Usually I think of energy conservation as being due to a time-independent Hamiltonian, and unitary evolution.

I'm also not sure I agree with the idea that in quantum mechanics, the total energy cannot be thought of as potential and kinetic. To me the unquantum part of the OP's question is asking about energy and position. The result of first measuring position followed by measuring energy is not the same as first measuring energy followed by measuring position, because the the energy and position operators do not commute. So the OP's question seems to assume that well-localized particles have a well defined energy, and that particles with a well-defined energy have a well-defined position, neither of which is true in quantum mechanics.

I do agree that tunneling is not mysterious, but that's because I think of a quantum particles as a wave, and even classical waves exhibit tunneling, eg. the evanescent wave http://en.wikipedia.org/wiki/Evanescent_wave

 

[SirClueless]

Ok let me see if I understood everything correctly. We know that a particle can pass through a classically forbidden region because as Zz posted out on his previous answer on another thread, we can see that the particle interacts with stuff (like magnetic fluctuations inside a barrier) implying that the particle passed through the barrier.

The part I still get tripped up on is the whole measuring business. This is my understanding: Let's prepare a particle in an energy eigenstate with its total energy less than that of the barrier. If we make a measurement of the particle's position and find it in a classically forbidden region, the measurement changes the state of the particle from what is was before the measurement and hence we cannot definitively say anything about it's total energy because it's no longer in an energy eigenstate. So its wrong for me to say that since the particles total energy before the measurement is less than the barrier that post-measurement it's new energy is still less than the barrier which would seem to imply negative KE. And I can't say anything about KE since localization of the wave function introduces uncertainty for momentum. On the other hand, if I make a measurement of the particle's kinetic energy, I will always find it to be positive (right?) and as a result I know it's not in a classically forbidden region? Or since we know it's kinetic energy accurately because of HUP I can't say anything about its position? I'm not so sure about my reasoning about the last part... could someone clarify?

So in the end it comes down to the uncertainty principle right? Perhaps all 3 answers I got originally are the same?

But there's still the whole thing about whether or not we can measure a particle inside the barrier. Is this possible? Peter, if a particle can be in a classically forbidden region (by your own admission) why can't we measure/detect it there? Is it just hard experimentally or is it physically impossible?

 

[PeterDonis]

Peter, if a particle can be in a classically forbidden region (by your own admission) why can't we measure/detect it there?

As I responded to Vanadium 50, I think a better way of saying it is that you can't measure the particle to have a position in the "forbidden" region *and* an energy less than that of the potential barrier. As you note, once you've measured the position, the particle is no longer in an energy eigenstate, even if it was in one before the measurement; so if you measure the position to be in the "forbidden" region, you can't say that the particle has less energy than the barrier energy. And if you measure the particle's energy, then it's not in a position eigenstate, so you can't say where it is, and therefore you can't say that it's in the "forbidden" region.

Is it just hard experimentally or is it physically impossible?

I don't think it's physically impossible to set up a way of measuring position that could, in principle, detect a particle in a "potential barrier" region. But I think it would be very difficult in practice.

 

[Water nosfim]

It came from the many worlds , , you see it moves throw ananter dimension ( some kind of MWI )

 

[bhobba]

I'm not really happy with some of the answers here.

Neither am I.

I'm having trouble wrapping my head around the idea of a particle being in a classically prohibited region

The trap you are falling into is ascribing any property, other than the state, to a quantum object when not observed. It does not have the property of 'being anywhere' until its observed to have the property of 'being somewhere' ie a position.

Also the state is like probabilities. It's not real in a usual sense, it doesn't exist out there like say an electric field does. Its like the probabilities we assign to the sides of a possibly biased coin. It's only function is to tell us the likelihood of that side coming up if you flip it. Its simply something abstract that helps us in calculating the long term averages of flipping it.

Note - this is what the formalism of QM says. Interpretations have their own take.

Now when we solve the Schroedinger equation at a barrier the solution has values on both sides of the barrier meaning if you were to measure the position there you would have such and such probability of detecting it. Now inside the barrier also has values as well, but the meaning of observing it there is entirely dependant on the exact set-up eg exactly what the barrier and combined measuring apparatus is - barriers are usually just modeled as a potential function. You would need to analyse the entire set-up to figure out what the result would be.

Another way to look at it is consider a hydrogen atom. Its a standard exercise to calculate the wave-function of the electron in the atom, and it has exactly the same meaning of its square giving the probability of locating it there if observed. But you can't say its there until you observe it, and to do that would require some rather tricky observational set-up. Think about it - you would need some macroscopic arrangement that would alter the whole situation beyond recognition. That would need to be analysed in its totality to figure out what the result would be.

Thanks
Bill

 

[atyy]

The part I still get tripped up on is the whole measuring business. This is my understanding: Let's prepare a particle in an energy eigenstate with its total energy less than that of the barrier. If we make a measurement of the particle's position and find it in a classically forbidden region, the measurement changes the state of the particle from what is was before the measurement and hence we cannot definitively say anything about it's total energy because it's no longer in an energy eigenstate. So its wrong for me to say that since the particles total energy before the measurement is less than the barrier that post-measurement it's new energy is still less than the barrier which would seem to imply negative KE. And I can't say anything about KE since localization of the wave function introduces uncertainty for momentum. On the other hand, if I make a measurement of the particle's kinetic energy, I will always find it to be positive (right?) and as a result I know it's not in a classically forbidden region? Or since we know it's kinetic energy accurately because of HUP I can't say anything about its position? I'm not so sure about my reasoning about the last part... could someone clarify?

More or less yes, except that I would stress that the question itself is not "quantum" since the idea of "found in a classically forbidden place" seems to require joint measurement of position ("found") and kinetic energy ("classically forbidden"). However, particles with well defined kinetic energy are not localized, and particles with well defined position don't have well defined kinetic energy.

I think the positivity of the kinetic energy is a red herring, since heuristically a projective measurement yields an eigenvalue, and kinetic energy only has positive eigenvalues. I do take back my complaint about the total energy not being the sum of KE and PE - the total energy operator (Hamiltonian) is the sum of KE and PE operators, but the KE and total energy operators don't commute, so depending on what one means, either could be argued for.

However, see the paper linked in post #14 for a devious argument that some sorts of "measurements" can yield the expected negative KE for particles found in the classically forbidden region! I'm not sure it is right, so read skeptically.

So in the end it comes down to the uncertainty principle right? Perhaps all 3 answers I got originally are the same?

Essentially, provided one is thinking really of the underlying non-commutation properties that give rise to the various uncertainty principles.

But there's still the whole thing about whether or not we can measure a particle inside the barrier. Is this possible? Peter, if a particle can be in a classically forbidden region (by your own admission) why can't we measure/detect it there? Is it just hard experimentally or is it physically impossible?

The usual answer is that any Hermitian/self-adjoint operator can be in principle measured by an experimentalist that is sufficiently ingenious and with enough resources. Here we are simply measuring position, and the localization is simply due to the squared wave function giving the probability of a particle being found at a certain position.

 

[atyy]

http://arxiv.org/abs/hep-th/9404163
http://dx.doi.org/10.1103/PhysRevD.50.5409
Detection of Particles Under Potential Barrier
Alexander Vilenkin, Serge Winitzki

 

[anorlunda]

Professor Leonard Susskind in his video lectures mentioned two things that sound relevant to tunneling.

Think of energy-time uncertainty. If the time to pass through the barrier is very small, then the particle can violate conservation of energy temporarily enough to lift it over the barrier.

To observe a particle inside the barrier, you would have to hit it with a photon of sufficiently short wavelength to resolve the width of the barrier. If you calculate the minimum energy of such a photon, it will be just enough to boost the particle over the barrier. Very frustrating; all your attempts to observe it in the barrier will fail.

Either way, you can observe a particle inside the barrier and later outside the barrier but you can not observe whether it tunneled through or jumped over.

 

[SirClueless]

Do you have a link to this video lecture?

 

[Nugatory]

Either way, you can observe a particle inside the barrier and later outside the barrier but you can not observe whether it tunneled through or jumped over.

Which makes sense, as the distinction between "tunneled through" and "jumped over" implicitly assumes some sort of trajectory between the points of the two observations - and that notion doesn't exist in QM.

 

[bhobba]

Do you have a link to this video lecture?

It would likely be somewhere in the following:
http://theoreticalminimum.com/courses

Thanks
Bill

 

[anorlunda]

I view the lectures from iTunesU which does not provide me with a URL. (That might tbecome a serious problem if the trend continues to provide content with no URLs)

But here is a site that claims to link to all of Susskind's online lectures. You want the ones about Quantum Mechanics. I'm not sure exactly which lecture, but if you watch them all, I wager you'll be very pleased. Susskind is an exceptional teacher.

The link is glenmartin.wordpress.com

You can also simply search on youtube.com for "Susskind quantum"