Tunneling and quantum reflection and transmission via a potential barrier are not just a quantum mechanical phenomenon but rather a statistical phenomenon that precisely obeys modern statistical techniques.

Statistics from modern physics techniques (classical physics plus transition probability) show that quantum tunneling exists and that its description formula by the classical Schrödinger PDE is also correct.

The difference is that the description of classical physics techniques is understandable, while Schrödinger's classical PDE is not.

This is indeed the case.

It is quite striking that Schrödinger's revolutionary PDE, which led to the quantitative explanation of the phenomena of reflection and transmission (sometimes called tunneling) can be completely ignored and replaced with B-matrix string statistics.

However, it is rarely mentioned that the author, among other physicists, intuitively ignored Schrödinger's PDE on energy quantization and proposed a brilliant alternative to derive a reflection and transmission formula on the coefficients of a quantum energy barrier (R, T), as shown in Figure 1.

(PDF) Useless Quantum mechanics-The complex untold story.

Here is a stationary solution [1]:

Zone I:U(x) = E eV - R E eV

Zone II: U(x) = E Exp (-a x)eV

from x=0 to x=dx

Note that the exponential decay in zone II comes from the fact that this is the case for this initial value problem imposed by the statistics of the matrix B.

Zone III:

U(x) = E Exp (-a dx)eV

from x=dx to x=infinity

If we apply the continuity equation to the limit between zones I,II then we apply the continuity equation to the limit between zones II,III then the Bohr formula E=N h f we arrive at the same formula for the reflection coefficient (R) and transmission coefficient (T) as those obtained from the classical Schrödinger equation. NB: at the two boundaries I-II and II-III correspond respectively y=0 and y=dx.

1-Useless Quantum mechanics-The complex untold story.