Tunneling
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Quantum particles can penetrate into regions that are forbidden classically, leading to the phenomenon of tunneling.

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bulletPotential barrier
bulletThought experiment
bulletPotential well
bulletSlit +barrier

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.
The ball rolls up the hill, transforming kinetic energy into potential energy given by V = mgy, where g is the acceleration due to gravitation.
If K0 < mgH the ball will not reach the top. At the point where V = K0 the ball reverses its direction and rolls back from the slope. 
If K0 > mgH the ball will roll over the top of the hill and will run down from the hill on the other side.

          

    

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. 
The animations show the tunneling of Gaussian wave packets through rectangular barriers of height V and width d (central strip). For visualization purposes, the probability of the transmitted wave (i.e. the wave appearing on the r.h.s. of the strip) has been rescaled. 

wpe2B.gif (5083 bytes)      wpe29.gif (6662 bytes)      wpe26.gif (6654 bytes)      wpe25.gif (12747 bytes)        

V = 2E, d = l ,       V = 2E, d = l ,       V = 2E, d = 2l ,     V = 2E, d = l ,
(sx,sy) = (2l,2l)    (sx,sy) = (l,2l)       (sx,sy) = (8l,8l)    (sx,sy) = (10l,10l)

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.
For the potential barrier depicted at the right, the momentum of the particle in the direction parallel to the barrier is conserved. Therefore the tunneling process does not change the direction of the transmitted wave, as shown in the next animation.

wpe8.gif (5651 bytes)       

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.

wpe24.gif (5746 bytes)       

Thought experiment

Consider 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. 

wpe1.gif (5602 bytes)       

The wave packet that tunneled through the potential barrier (i.e. the blue wave to the right of the strip) runs ahead of the red-colored 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 high-pass 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 one-dimensional. 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 well

Consider 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. 

 

wpe23.gif (5646 bytes)         wpe25.gif (5021 bytes)         wpe8.gif (8602 bytes)   

V = 2E, d = l ,          V = 2E, d = l ,          V = 2E, d = 2l ,
(sx,sy) = (l,2l)          (sx,sy) = (2l,2l)       (sx,sy) = (8l,8l)

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. 


As in the case of the potential barrier, changing the shape of the potential well does not affect the salient features of the scattering process. The next animation shows a wave packet scattered by a potential well of triangular shape. 

wpe27.gif (4839 bytes)       

Slit+barrier

A wave passing through a narrow slit is scattered in many directions. Both the momentum parallel and perpendicular to the slit are no longer conserved. 

wpe4A.gif (9221 bytes)       

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. 

wpeE.gif (8309 bytes)       

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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