Wave Propagation in Linear Media | re-examined

Chapter 4
One-dimensional quantum
tunnelling
4 One-dimensional quantum tunnelling
Das Bestreben, der Dispersion Herr zu werden, leuchtet mir fur langsame
Wellen ein, das Ubrige ist mir aus den Andeutungen nicht klar geworden.
Das Au allende ist, wie viel man mit klassischer Mechanik befriedigend
machen kann. Wenn es nur einmal gelange, das Prinzipielle an den Quanten
einigermassen zu erklaren! Aber meine Ho nung, das zu erleben, wird
immer kleiner.
Albert Einstein in a letter to Arnold Sommerfeld (1. 2. 1918 [103])
Having exhausted the question of electromagnetic wave propagation we now turn to the
second area where tunnelling occurs and where, after all, the term `tunnelling' was coined.
To start with, however, we shall restrict our investigation to the comparatively simple step
potential barrier, where we have to cope with only one interface. In a second step, we regard
tunnelling in its original meaning as the motion of a particle through a classically impenetrable
barrier. In the simplest form of a rectangular potential wall, such an investigation requires
the inclusion of two interfaces, which renders the treatment more complicated albeit by no
means impossible. In both cases, we study how the particle evolves in time and space upon
impinging on the barrier.
The chapter begins with the well-known formulation of the scattering process consisting of an
incident free particle being partly re ected and partly penetrating the obstacle. The solutions
will be given as Fourier integrals and primarily be independent of the actual initial condition.
We shall then explore several possibilities of initial wave forms or particle shapes that nally
are used to obtain numerical results of the wave propagation both inside and outside the
barrier. The square potential barrier will be considered next, and we shall brie y examine the
variety of tunnelling time de nitions known from the literature for monochromatic waves. The
concluding numerical evaluation of tunnelling events is restricted to Gaussian wave packets.
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