

Quantum particles can penetrate into regions that are forbidden classically, leading to the phenomenon of tunneling. Content
Potential barrier
Consider a classical particle (for example a ball) of mass m with a kinetic energy
K moving toward a hill of height H. Consider now a particle with energy E approaching a potential step of height
V > E. Classically the particle would be reflected by the step. A
quantum particle incident from
the left has a nonzero probability for being found to the right of the step. V = 2E, d = l
, V = 2E, d = l
, V = 2E, d = 2l
, V = 2E, d = l
, The shape of the transmitted wave packet depends on the width of the initial wave packet. The probability for tunneling vanishes exponentially with the thickness of the barrier. For V
= 2E and d = 2l the tunneling probability is less than 0.0000000001. Qualitatively the tunnel effect does not depend on the shape of the potential barrier, as illustrated in the next animation for the case of a triangular barrier. Thought experimentConsider the thought experiment in which two identical wave packets start moving at the same time. The blue colored wave packet will hit a potential barrier ( V > E ) whereas the red colored one (initially invisible because hidden under the blue wave) will not feel the step.
The wave packet that tunneled through the potential barrier (i.e. the blue wave to
the right of the strip) runs ahead of the redcolored wave packet that did not feel any
potential. The initial Gaussian wave packet can be viewed as a superposition of plane
waves with different momentum. A tunnel barrier acts as a highpass
momentum filter: For increasing momentum perpendicular to the step, the probability for tunneling
increases. This filtering effect also occurs if the motion of the particle is
onedimensional. It is a direct consequence of the fact that the tunnel probability is a
function of the energy. In free space the energy is proportional to the momentum
squared. Potential wellConsider a free particle with energy E approaching a potential well of depth V < E. In the absence of friction a classical particle would enter the well on one side and leave the well on the other side. In contrast, a quantum particle has a nonzero probability of being reflected by the well, as shown in the animations that follow.
V = 2E, d = l
, V = 2E, d = l
, V = 2E, d = 2l
, The graph shows the transmissivity of the square well as a function of the energy E. If E corresponds to a maximum in the transmissivity the potential looks perfectly transparent; there is no reflection, just like in the classical case. Otherwise part of the wave will be reflected. The maxima and minima in the transmissivity are called resonances.
Slit+barrierA wave passing through a narrow slit is scattered in many directions. Both the momentum parallel and perpendicular to the slit are no longer conserved. Placing a tunnel barrier to the right of the slit can strongly modify the properties of the transmitted wave. The next animation shows the effect of a triangular potential barrier on the transmitted wave. The properties of the transmitted wave packet strongly depends on the properties of the tunnel barrier and, to less extent, on the width of the slit whereas the length of the slit has little effect. A slit followed by a potential barrier provides a simple model to describe the electron emission from nanoscale sources. 
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