In accordance with the classical limit, the probability per attempt of tunneling decreases towards zero as the mass m of a particle, the deficit V − E between its energy E and the barrier height V (E < V), and/or the width W of a barrier becomes large. The probability P per attempt of tunneling of a particle of mass m and energy E through a barrier of height V (E < V) and width W in the classical limit (in the limit of ever-smaller P) is
(1) P = 16[E(V – E)/V2]exp{−[8m(V − E)]1/2W/h-bar}.
In accordance with the classical limit, P approaches zero as m, V − E, and/or W become large.
But, by contrast, there seems to be a paradox if E > V. For the probability per attempt of traversing the barrier is then
(2) P = {1 + V2sin2[2m(E – V)]1/2W/h-bar]/[4E(E – V)]}-1.
The average of sin²x over one or more complete oscillations of any argument x is 1/2, so if E > V a typical value of the smoothed-out probability per attempt corresponding to a given E is
(3) P = {1 + V2/[8E(E − V)]}-1.
Letting E = NV (N > 1), Eq. (3) can be rewritten as
(4) P = [1 + (8N2 − 8N)-1]-1.
Inverting and solving Eq. (4) for N in terms of P (with the help of the quadratic formula),
(5) N = {1 + {1 + [2(P-1 − 1)]-1/2}1/2}/2.
Corresponding to P = 1/2, N = 1.15. Corresponding to P = 0.99, N = 1.92. This is reasonable for an electron traversing a barrier of atomic dimensions. But this is not reasonable for a baseball thrown over a fence of width W = 1 cm and height H = 10 meters in Earth's gravitational field g, which has P = 1 of clearing the fence if its energy E even marginally exceeds V = mgH, not merely P = 1/2 if E = 1.15 mgH and not merely P = 0.99 if E = 1.92 mgH. But if E > V the formulas (2) through (5) take no account of a classical limit: they are identical for an electron and a baseball. [Although g acts vertically, E and V for the baseball can still be construed as one-dimensional, as functions of its horizontal position directly below its path. The same lack of taking into account of a classical limit by the formulas (2) through (5) obtains for any system, however macroscopic.]
Perhaps this paradox is resolved because a PERFECTLY square potential barrier is physically UNrealistic. At the edges of any REAL, PHYSICAL, barrier, the potential increases from zero to V over a FINITE distance greater than zero. If this is taken into account, the probability that a baseball thrown over a fence with E > V = mgH being reflected, i.e., not traversing the fence, is reduced to zero. See Quantum Theory by David Bohm, Sections 3.9, 11.3, 11.4, and 12.1−12.4.
Hi.,
Kindly Check this link:
https://arxiv.org/pdf/quant-ph/0403010
Best wishes..
The paradox is resolved if one recognizes that the prediction of anomalously high reflection probabilities {or, equivalently, anomalously high [1 - (traversal probabilities)]} for macroscopic particles such as baseballs approaching a barrier with incident kinetic energy E > V or a well with incident kinetic energy E > 0 is an ARTIFACT of the assumption of PERFECTLY square barriers or wells. But PERFECTLY square barriers (or wells) canNOT exist. The potential cannot change from zero to +V if a barrier or from zero to -V if a well over an infinitely short distance. The minimum possible distance is one atomic diameter, about 10^-10 meter. If the de Broglie wavelength of an incoming particle is much smaller than this, say, 10^-12 meter or less, the probability of anomalous reflection is negligible.
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