What happens with a tunneling particle when its momentum is imaginary in QM?

Question

In classical mechanics the motion of a particle is bounded if it is trapped in a potential well. In quantum mechanics this is no longer the case and there is a non zero probability of the particle to escape the potential through a process call quantum tunneling.

This seems extraordinary from the point of classical mechanics because it implies the particle must cross a zone where it has imaginary momentum. I understand that from the point of view of quantum mechanics there is a non zero probability for the particle to be in such zones.

What is it know about the behaviour of the particle in this zone?

Links to research experiments or papers would be appreciated.

Answer 1

I can give you one example. In a semiconductor reverse-biased p-n junction, a potential barrier exists that prevents electrons from crossing the junction. There is an energetically-forbidden region in the vicinity of the junction. The wave functions of electron states in both the valence and conduction bands are real exponential in this region. Additionally it's possible that the only spatial overlap between the valance and conduction bands occurs in the forbidden region. Yet optical absorption occurs due to valance to conduction band transitions. The interpretation is that electrons in the forbidden region are promoted from the exponential tail of the valence band to the exponential tail of the conduction band. This process is called the Franz-Keldysh effect or tunneling-assisted absorption.

Here's a nice figure from the German Wikipedia page. The English page doesn't have such a nice figure.

Answer 2

One of the strangenesses concerning the behaviour of particles crossing a tunnel barrier is that one : crossing the tunnel barrier at superluminal group velocity is theoretically possible (see Wigner time and Hartman effect).

This has been experimentally confirmed by experiments conducted by R. Chiao with the participation of P. Kwait and A. Steinberg at Berkeley University (1993-1995). These experiments were devoted to the measurement of the tunnelling time of photons at a wavelength of 702 nm.

These measurement were achieved by comparison with the travelling time of "EPR twins" photons. These EPR twins were travelling the same optical path length, but in vacuum. They were created by Parametric Down Conversion, hence at the same time than their "EPR twins" crossing the tunnel barrier. The comparison of arrival times rested on the use of a Hong, Ou and Mandel interferometer.

The tunnel barrier consisted of a dielectric mirror with eleven quarter-wavelength layers of alternately high index material (titanium oxide with n = 2.22) and low index material (fused silica with n = 1.45). The total thickness of the eleven layers was 1.1 μm. This implied an in vacuo traversal time across the structure of 3.6 fs.

The data in Fig. 3(a) implies that after traversing the tunnel barrier, the peak of a photon wave packet arrived 1.47 ± 0.21 fs earlier than it would had it traversed only vacuum.

Tunneling Times and Superluminality: a Tutorial, Raymond Y. Chiao http://arxiv.org/abs/quant-ph/9811019

Answer 3

There is a detailed description of the behaviour of quantum systems during tunnelling. They do not involve the particle travelling faster than light. The "group velocity" does not correspond to the velocity of anything in that situation. What happens instead is that the wave function undergoes interference inside the barrier in such a way that it decays exponentially and it does this slower than light. Physicists who have claimed otherwise have misinterpreted experimental results, see Herbert Winful's papers on this topic

http://sitemaker.umich.edu/herbert.winful/modest_contributions

and in particular

http://sitemaker.umich.edu/herbert.winful/files/physics_reports_review_article__2006_.pdf.