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All Answers (33)

11th Nov, 2014

Rajat Gupta

National Physical Laboratory - India

good question. I have wondered the same so many time. 

Tunneling occurs when the two conducting materials are very close and separated by a insulating barrier (eg. air) of width of few nanometer at very low temperatures. 

I believe that at such low temperatures, the conducting materials on either side of the barrier forms an external layer on their surface thus overcoming the barrier width. For example if the barrier width is 4nm and if the new formed outer layer on both conducting materials is of ~2-3nm then a virtual bridge would be created which will overcome the barrier and thus conduction or tunneling takes place. 

Deleted profile

If you believe that the Schrodinger equation is a good description of physics in the appropriate regime, then it is because a finite height potential barrier does not drive the probability density function to zero on the far side: see

http://en.wikipedia.org/wiki/Rectangular_potential_barrier

for a standard exposition, which can also be found in just about any introductory text.

A cheap but useful resource with lots of worked examples is the Schaum's outline of Quantum Mechanics, by Y Peleg et al. It contains detailed calculations of this situation in Chapter 3.

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11th Nov, 2014

Harry ten Brink

ECN part of TNO

Indeed

the first entry when you would have googled for

"tunneling effect quantum mechanics Wikipedia"

11th Nov, 2014

H Chris Ransford

Karlsruhe Institute of Technology

The bottom line is that the tunneling particles are borrowing energy from the quantum foam - from its virtual particles -  enough to put them over the required energy top (over the relevant energy barrier) and then relinquishing this virtual energy back to the quantum foam, before the Heisenberg relation would be violated.

Incidentally, a similar energy borrowing mechanism triggers radioactive decay at the individual atom level. Whenever some radioactive element's half life is not long, that means that the amount of virtual energy to be borrowed is low. When however the half-life is long, this in turn means that a lot of energy must be borrowed - with big helpings of virtual energy both lasting for much shorter time spans and coming about less frequently, which accounts for the longer half-life

4 Recommendations

11th Nov, 2014

Stam Nicolis

University of Tours

One way to say this  is that the potential barrier is a (semi)classical approximation-the true potential, once quantum effects have been properly taken into account is completely different. Furthermore,the statement that the total energy is less than the potential energy can't be checked to arbitrary precision. This is, of course standard material of the introductory course in quantum mechanics-that, however, stresses the departure from the classical description, rather than the final result.  Essentially, one proves that the overlap between states localized on each side of a barrier of finite width and height remains finite-which expresses tunneling. Only if the barrier is infnitely high, but of finite width, will tunneling be eliminated (as, of course, will occur for an inifnitely wide barrier of finite heght.). This is the result that is usefully generalized to infinitely many degrees of freedom, where tunneling can be suppressed.

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11th Nov, 2014

Jürg Martin Fröhlich

ETH Zurich

Assuming that standard Quantum Mechanics is correct (an assumption supported by an enormous amount of evidence), the tunneling effect is a phenomenon that is well understood and neatly described by the theory. Tunneling is encountered in quantum-mechanical scattering theory and in the theory of quantum-mechanical resonances. There are many decent text books where it is described reasonably precisely. Since, in Quantum Mechanics, a particle in a state of finite (kinetic) energy is never sharply localized in space, one should not be surprised that it can overcome a potential barrier of finite width that may be higher than its total energy.

Some fairly recent papers on resonances and tunneling are:

1. P. Pfeifer and J. Fröhlich, "Generalized Time-Energy Uncertainty Relations and Bounds on Life Times of Resonances", Rev. Mod. Phys. Vol. 67, No. 4, 759{779 (1995)

2.V. Bach, J. Fröhlich, and I.M. Sigal, "Spectral Analysis for Systems of Atoms and Molecules Coupled to the Quantized Radiation Field", Commun. Math. Phys. 207, 249{290 (1999)

and references given therein.

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11th Nov, 2014

Matts Roos

University of Helsinki

It seems to me that the question posed is: why is quantum mechanics not classical?

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11th Nov, 2014

Stam Nicolis

University of Tours

Which is strange, since it's the other way around: quantum mechanics is the fundamental description, classical mechanics an approximation to it.  So it might be more appropriate to ask, why in the classical limit is tunneling suppressed. And the answer to that may be that if one examines the overlap between different states of the model, if there exists a subset, such that their overlap vanishes in the classical limit, it may be possible to deduce that these are separated by a potential barrier, that can be identified in that limit. 

Deleted profile

Maybe a simpler answer is that the probability of finding a particle on the 'other' side of the potential barrier depends on the difference between the particle's energy and the potential, its width, and Planck's constant in such a way that the smaller h is, the lower the probability (for given potential and width), and for normal values of the other variables, h is so small that the transmission probability is very small indeed.

11th Nov, 2014

F. Leyvraz

Universidad Nacional Autónoma de México

The answers given above are of course true and appropriate. Still, maybe the question of meaning, of intuitive significance, can be briefly adressed. For a classical particle, it is impossible to be in a position where the potential energy is larger than the total energy of the particle. That is because the total energy is equal to kinetic plus potential energy, and kinetic energy cannot be negative.

The two last points still hold in QM: what fails is the very first assumption. It cannot be said that the particle is somewhere. It is distributed over a certain limited region. This, of course, means that we cannot say that the particle has a given potential energy. The total energy is calculated as a kind of average potential energy, plus a kinetic energy which depends on the rate of spatial variation of the wave function. None can be calculated without looking at the whole extent of the wave function. So the wave function can perfectly well have a low, but non-zero, amplitude of being in the region where the potential is large. There must be enough of the wave function in regions where the potential energy is significantly less than the total energy. You are certainly correct in supposing that a wave function cannot be concentrated in the region where the potential energy is larger than the total energy.

You might then ask: how does the particle go from the inside region to the outside region, without ever being localized in the barrier region? To this the answer is, that you cannot think in terms of particles, but only of the wave function. The wave function continuously evolves from being large inside and nearly zero outside, to being just the oposite, but it always remains quite small in the barrier region.

What then ``is the particle really doing''? That is a question which arises naturally from classical mechanics, but it is probably not meaningful.

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11th Nov, 2014

Stam Nicolis

University of Tours

The answer is that the ``barrier region'' is a semiclassical notion-the quantum particle doesn't see the barrier region that way. What it does see can be described  through its spectrum: if the spectrum has spacing that is non-analytic in the coupling constant (thus in Planck's constant), then, one can deduce that, in the semi-classical limit, the particle moves in a potential with a barrier; if the spectrum has spacing that is analytic in the coupling constant (thus in Planck's constant) then the particle moves in a potential without a barrier, in the semi-classical limit.

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12th Dec, 2014

Alexei Voronin

P.N.Lebedev Physical

Tunneling effect assumes coordinate  description of a wave-particle, i.e. statement that particle is under barrier assumes its coordinate description. This   means that  kinetic energy is negative and velocity is complex, which seems to be a paradox. One could say that when coordinate description for the wave-particle is used , velocity of the particle does not play a role of physical variable any more. It is a complex valued function of coordinate, which has no physical meaning. Thus there is no contradiction with physcial sense. Indeed, velocity is no longer classical "dx/dt" , there is no physcial restrictions for the values of "v" any more, and hence kinetic energy is no longer positively defined function, instead it is a real valued function of coordinates T=E-U(x). 

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12th Dec, 2014

Jürg Martin Fröhlich

ETH Zurich

Well, well, this discussion, too, appears to go astray!

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12th Dec, 2014

Patrick Das Gupta

University of Delhi

Tunneling is an wave aspect of matter. If the potential barrier is higher than its kinetic energy, but its extent is smaller than the particle's de Broglie wavelength, then the particle has a large tunneling probability. Pure waves show tunneling. 

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12th Dec, 2014

Matts Roos

University of Helsinki

What has been said here recently by Stam, Jürg, Alexei, Sanjay, Patrick is all correct but let me state my own attitude to the dichotomy of classical vs. quantum.

Classical physics is a limiting case of quantum mechanics it is derivable from there. But one cannot go the other way and derive quantum mechanics from classical physics, the quantum world is not a limiting case of the classical world.

My conclusion is then that the quantum world is more fundamental, it contains the classical world and in addition quantum phenomena like tunneling.

Even many of my colleagues, experts on quantum mechanics, cling to the hope that it one day would be possible to describe quantum mechanics in classical terms. This they hope because they see themselves and the world around us as classical and as fundamental. I think this is an error: the world is fundamentally quantal although we are not living on the quantum scale.

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12th Dec, 2014

Jürg Martin Fröhlich

ETH Zurich

Dear Roos: Why are you presenting generalities which essentially everybody will agree with, but which do not have anything to do with the question that has been asked? Incidentally, a derivation of classical dynamics from quantum dynamics - while most probably possible quite generally - is not an easy thing to do! You are jumping to conclusions that, in one sense, are obvious and, in an other sense, represent mathematical challenges.

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12th Dec, 2014

Faramarz Rahmani

Institute for Research in Fundamental Sciences (IPM)

Your question has no a convincing answer in quantum mechanics. Because it is not a deterministic theory like classical mechanics. Basically, it is a theory for the interpretation of the experimental results. I think you can convince yourself by reading the book ((The Quantum Theory Of Motion)) by P.Holland.

It is a book about Bohm's deterministic and causal quantum mechanics. In chapter 5 you can study about interference and tunneling.

Best

12th Dec, 2014

Matts Roos

University of Helsinki

Jürg, I am not jumping to conclusions. I am stating my belief in a paradigm, that the world is fundamentally quantal although we are not living on the quantum scale. As a paradigm it need not be proved.

12th Dec, 2014

Jürg Martin Fröhlich

ETH Zurich

Who would doubt that your paradigm, dear Matts, is correct among people who know their physics?! However, what you are writing has NOTHING to do with the question that has been asked. Moreover, your remarks are close to being somewhat trivial.- I apologize for expressing myself openly.

However, to come up with a mathematically precise understanding of how classical physics emerges from quantum mechanics in suitable regimes is a challenge that has not been coped with adequately in all respects! It is here that you are jumping to conclusions.

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12th Dec, 2014

Matts Roos

University of Helsinki

Jûrg,

some well known people who know their physics doubt it, I've had a debate about it in Arkhimedes,

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12th Dec, 2014

Victor Christianto

University of New Mexico

I know some professors have answered here, but referring to the hope as expressed by Matts Roos that one day quantum mechanics can be described in classical terms, allow me to add that i just read an interesting result by Carl Bender and Daniel Hook. They argue that a classical particle can exhibit quantum like behavior including tunneling, given a conplex energy. Unfortunately, they do not give a detailed calculation so i cannot verify their arguments. Nonetheless, perhaps it is worth to check their result in complex classical mechanics. 

http://arxiv.org/pdf/1011.0121v2.pdf

12th Dec, 2014

Faramarz Rahmani

Institute for Research in Fundamental Sciences (IPM)

The tunneling effect has no a convincing reason in standard quantum mechanics. People usually refer to uncertainty principle. In general, standard quantum mechanics is not a causal theory so ask about the cause of the phenomenon is not in its program .You can convince yourself by deterministic quantum mechanics(Bohmian quantum mechanics). Please study the book '' The quantum theory of motion'' by P.Holland

12th Dec, 2014

Matts Roos

University of Helsinki

Victor,

as you can read in a paper by Turok (http://arXiv.org/pdf/1312.1772) the idea of Bender and Hook does not work, they actually admit reservations themselves, and their preprint it has not been published in the 4 years since it was on arXiv.

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12th Dec, 2014

Oscar Chavoya-Aceves

Glendale Community College, AZ USA

While playing with a one-dimensional Schroedinger equation with a potential energy that's independent on time may be an interesting mathematical exercise, the results are inconsistent with a fundamental principle of physics: the principle of inertia. In this scenario of "tunnel effect" the potential energy appears as a ghostly entity (with a non material origin) ignoring the fact that it makes not sense to speak of a potential energy unless there are (at least) two physical entities interacting. Assuming, for example, that we are dealing with the wave function of an electron, we cannot have a maximum of the potential energy unless (according to classical electrodynamics [Poisson equation]) around the point of the maximum there is a density of negative charge and, as a consequence, perhaps... more electrons. Thus, Newton's pendulum, for example, would give us an example of "tunnel effect" in the macroscopic world.

3rd Mar, 2016

Biswajoy Brahmachari

Vidyasagar Evening College, 39 Sankar Ghosh Lane, Kolkata 700006, India

I have read all the answers with a lot of interest because I teach this topic in my college every year. Such a discussion is quite beneficial for a teacher.

I would like to stress on the wave function of the particle. In single particle quantum mechanics every particle has a wave function. What does the wave function do ? It carries encoded information about the particle. It's position, momentum, angular momentum, energy etc. In relativistic case it may also include information about spin. Once wave function is known, the standard apparatus which is to be used is the eigenvalue equation   ô ψ = λ ψ. Where ô represents an operator (position, momentum, ...)  λ is the value which will be measured. In tunneling we will be interested in the position operator so ô ≡ x̂

Next question is how to get this wave function? The standard method is to solve Schrodinger equation. The space part of the wave function satisfies a second order differential equation. We will thus need boundary conditions to fix arbitrary constants. It turns out that in barrier penetration problems position and shape of potential barrier gives appropriate boundary conditions. After solving the differential equation one finds that wave function is non-zero in the other side of a finite potential barrier. So we see that |ψ|2 is non-zero. That is there is a small but non-zero probability of finding the particle in other side of the barrier, the place which is forbidden classically.

Ref: A classic reference is L. I. Schiff, Quantum Mechanics.

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10th Oct, 2016

Martin Klvana

Unaffiliated

Jürg Martin Fröhlich wrote: ". . . a particle in a state of finite (kinetic) energy is never sharply localized in space . . ."

---

If so, then it is not a particle.

10th Oct, 2016

Jürg Martin Fröhlich

ETH Zurich

A fundamental fact about quantum mechanics is that an object that people usually call "particle" cannot simultaneously be localized sharply in physical space and in momentum space. Furthermore, a state in which a particle is localized in physical space completely sharply has an arbitrarily large kinetic energy. These are some elementary facts about quantum mechanics! Let me add that the picture of a particle as a pointlike object whose position can be determined arbitrarily precisely strikes me as a rather nonsensical idealization.

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10th Oct, 2016

Martin Klvana

Unaffiliated

If the particles are not 'point-like objects' how can they exist "in a state in which [they are] localized in physical space completely sharply"?

Deleted profile

Some states correspond to a well-determined position, but a very poorly-determined energy; others can have well-determined energy but a poorly-determined position. The point is that a particle can be in a state in which its position is not well-determined, so the notion of 'point-like' is not appropriate.

10th Oct, 2016

Sofia D. Wechsler

Technion - Israel Institute of Technology

Something is wrong in the question. INSIDE the nuclear barrier, the TOTAL energy, not the KINETIC energy, is lower than the barrier peak. The kinetic energy is at all NEGATIVE. That means, the linear momentum is imaginary. If one expands the wave-function inside the barrier in Fourier components, the form of the components is exp(kr), not exp(ikr), and the vector k is real.

A movement with imaginary linear momentum has no counterpart in the classical mechanics. This is once more a proof that a quantum "particle" is not a point-like entity which moves like a classical particle. The quantum "particle" is an object extended in space, it occupies a volume, and it is a NONLOCAL object. The dynamics inside this volume is not the dynamics known to us from the classical mechanics.

Unfortunately, since we the humans are big creatures used to the classical reasoning, it is very hard for us to understand the quantum behavior, and even harder to understand nonlocality.

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