What Are the Limits of Quantum Tunneling Theory?

[ice109]

so i was showing my friend quantum tunneling in 1D i.e. the solution of the shroedinger for a finite potential barrier and i derived the transmission coefficient and showed that it's non-zero and all that.

at the end he asks me a completely sane question: how does the electron do it? and i responded to him that that is not the right question to be asking, that it is not within the scope of the theory and even more so not within the realm of physics. am i correct in making this statement?

i understand that this is akin to the orthodox interpretation and that's what i stressed to him; that the point of physics is not to answer such questions but produce useful things such as theories with testifiable predictions. i know a lot of people will agree with this latter statement but my question is whether that is the statement i should be making in response to this specific question?

forgive me if i don't respond for a while as i'll be away most of tomorrow. please still post.

 

[leviadam]

You are correct my friend, physics is all about the creation of mathematical models that represents nature best, as long as the the theory predicts the tunneling and the electron tunnels there is no problem, the theory do not have to answer questions like why.

Of course one might find a batter theory that has a smaller scale for example and might give a batter understanding on why does the electron tunnels but there is no enforce that such a theory exists.

Adam.

 

[sokrates]

I think you can do better than that.

I don't mean to exasperate the experienced QM community here, but my attempt will be more less mathematical and more intuitive.

Let's say I generate a sound wave by screaming inside a room that's bounded in all three dimensions with doors shut and everything being closed.

Would you hear my voice? Or would you expect it to be dead silent by the time sound waves reach the doors and the walls.

My friend, tunneling is basically a WAVE phenomena, and electrons (like any other entity in the universe as far as we know) display WAVE properties, such as interference, transmission through barriers and etc...

Electron is not a small marble as you imagine it, because nobody has evidence to claim so!


You would not be surprised at all if someone said,

-Hey, you know what x-rays penetrate human body, how the heck do they do that?

OF course, there are gross differences, exclusion principle, dualities, all those complicated arguments, but my objective is not to draw it to that side.

Just because physicists can solve some differential equation describing the dynamics of the electron doesn't mean that they really understand it. You don't know what an electron is. You haven't seen it. But similarly you don't know what gamma rays are, also. Remember, they can behave like particles too...

So, there's something wrong in our every-day understanding, we have our own prejudice for the universe. That's why it sounds SO COUNTER-INTUITIVE, and unbelievable, maybe it's very simple after all.

Just my take on the issue --

 

[ice109]

i've thought about it like that, as attenuation of a signal...

there's still something odd about climbing out of a potential valley without any help from anything.

 

[zenith8]

In the Bohmian/pilot-wave interpretation the explanation for the tunnelling effect in QM is perfectly clear. Remember in pilot-wave theory it is stated that actual particles with trajectories exist in addition to the wave function, and the wave pushes the particles (in addition to the particles repelling.each other or whatever). So from this perspective, there are two influences which affect the electron trajectory. As well as the 'classical force' arising from the classical potential $$V$$, there is a 'quantum force' arising from the wave function, and this latter happens to be greatest where the curvature of the wave amplitude is greatest. In appropriate circumstances this extra 'quantum force' is sufficient to boost the particle over the barrier (it effectively lowers the barrier height).

To illustrate the point I've attached a diagram showing single-electron trajectories tunnelling through a square 1D potential barrier (see below).

There is a supposed to be a Gaussian wave packet headed towards the barrier (and in equilibrium the electrons will be distributed over an ensemble as the absolute square of this). Given that the evolving shape of the wave is known (solution to time-dependent Schrodinger equation) the trajectory of an electron heading towards the barrier depends only on its starting point x (since - according to the theory - the electron velocity vector at any point is given by $$\nabla S / m$$ where $$S$$ is the phase of the wave function). The diagram shows a selection of initial x which is uniformly distributed and does not reflect the actual Gaussian distribution.

For the given choice of energy, tunnelling occurs only if x lies in the front of the packet. Particles in the forward part of the packet are not significantly affected by the reflected wave in front of the barrier which is not yet appreciable and they maintain substantially uniform (classical) motion. These particles enter the barrier and initially slow down. Those in the extreme front are eventually accelerated and transmitted to pursue uniform motion again - these are the particles that 'tunnel' (exactly the right number of them so that the QM transmission probabilities come out right). But notice that many other trajectories enter the barrier and then turn round and join the reflected beam - they also can be said to 'tunnel' inside the barrier. The motion of some of the electrons can be quite complicated and to show the generic oscillations extra trajectories for particles 'trapped in the barrier' are plotted.

Particles in the rear of the packet and some in the front never actually reach the barrier. They are brought to rest and turned round when they encounter the effective force induced by the interference of the incident and reflected waves, where the wave function propagates the influence of $$V$$ far from where $$V$$ is finite. At points where the effective force is small trajectories congregate and give rise to fringes - a matter-wave analogue of Wiener's optical fringes...!

So, I know it's tedious to keep repeating this point (it's been done in lots of recent threads) but all known phenomena involving electrons can be given a simple quasi-mechanical explanation providing one accepts that both electrons and the wave function objectively exist and that the particles follow the obvious dynamics implied by the Schrodinger probability current.

Compare and contrast this simple qualitative explanation to the (with all due respect) waffle of the previous replies to your question. However, in their defence if one insists that 'the wave function is all there is' (or we can't know what there is) in QM then your other correspondents have no option but to waffle. It is impossible to explain the tunnelling effect consistently using an interpretation involving the wave function alone.

 

[sokrates]

Great reply thank you for sharing that

 

[Naty1]

that it is not within the scope of the theory and even more so not within the realm of physics. am i correct in making this statement?

Of course not...It would be insanely myopic if all physicsts did was to match mathematics to physical phenomena without understanding WHY something happens...Just because so far physics has been generally superior at explaining WHAT happens and rather less successful explaining WHY things happen is no reason to cop out...

A theory that fails to answer common sense questions such as your friends is a theory in need of further development...tunneling is a statistical phenomena and so is QM; anything that can happen is likely to happen.