Formula for penetration coefficient in barrier penetration when ∪0-E is presented, and the question in the article 《The reserch on barrier of quantum mechanics from a piece of exercise》(《College Physics》 1993, 12(10)) is explained using this formula.

College PhysicsJan 1, 20050 citations

American Journal of Physics

Optical Analog for Quantum-Mechanical Barrier Penetration

Harry F. Meiners

Published May 1, 1965• 3 citations

Publisher

DOI: 10.1119/1.1971660

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FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, not a classical "jump"; its probability depends exponentially on barrier width and height, with no known teleportation mechanism. | MATH: Transmission coefficient $T \approx e^{-2\kappa L}$ , where $\kappa = \sqrt{2m(V_0-E)}/\hbar$ ; for a rectangular barrier, exact $T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}$ . No new constants or ratios emerge beyond $\hbar$ , $m$ , $V_0$ , $E$ , $L$ . | CONNECTION: The exponential decay $e^{-2\kappa L}$ has no direct ratio to 0.382/0.618/1.618; however, the *critical* condition $E = V_0$ yields $\kappa = 0$ and $T \to 1$ (resonant transparency), which is a phase-transition-like threshold — analogous to a golden-ratio bifurcation point in some dynamical systems, but not evidenced here. No crystallographic or base-60 link. | DEPTH: 3 — Standard quantum mechanics, well-understood; the "impossible measurements" QFT paper (arXiv:2311.13 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, not a classical "jump"; its probability depends exponentially on barrier width and height, with no known teleportation mechanism. | MATH: Transmission coefficient $T \approx e^{-2\kappa L}$, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$; for a rectangular barrier, exact $T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}$. No new constants or ratios emerge beyond $\hbar$, $m$, $V_0$, $E$, $L$. | CONNECTION: The exponential decay $e^{-2\kappa L}$ has no direct ratio to 0.382/0.618/1.618; however, the *critical* condition $E = V_0$ yields $\kappa = 0$ and $T \to 1$ (resonant transparency), which is a phase-transition-like threshold — analogous to a golden-ratio bifurcation point in some dynamical systems, but not evidenced here. No crystallographic or base-60 link. | DEPTH: 3 — Standard quantum mechanics, well-understood; the "impossible measurements" QFT paper (arXiv:2311.13 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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When we dealt with the finite potential well in the last chapter, it was observed that the wave function penetrated into the barriers. This process does not occur in classical mechanics, since a particle will actually bounce off the barrier. However, when we treat the particle as a wave, then the wave nature of barrier penetration can occur. This is familiar in electromagnetic waves, where the decaying wave (as opposed to a propagating wave) is termed an evanescent wave. For energies below the top of the barrier, the wave is attenuated, and it decays exponentially. Yet, it takes a significant distance for this decay to eliminate the wave completely. If the barrier is thinner than this critical distance, the evanescent wave can excite a propagating wave in the region beyond the barrier. In optics, this effect is termed frustrated total internal reflection . We are familiar with total internal reflection just from a swimming pool. Water has a higher dielectric constant than free space, so when we are under water, there is critical angle beyond which we can no longer see out (from under the water) of the pool. In optics, we can use a prism, and there will be reflection from the back surface for angles greater than a critical angle. The wave in back of the prism is attenuated and decaying. But, if we place another high dielectric constant material close to the prism, the decaying wave can excite a wave in this new material. 1 Thus, the wave can penetrate the barrier and continue to propagate, with an attenuated amplitude, in the trans-barrier region. For particles, this process is termed tunneling , with analogy to the miners who burrow through a mountain in order to get to the other side! In fact, it is known that the nuclear fusion process by which our sun produces its power would not occur were there no tunneling. The intra-nuclear repulsion is just too strong and tunneling is required for the nuclei to find one another.When we dealt with the finite potential well in the last chapter, it was observed that the wave function penetrated into the barriers. This process does not occur in classical mechanics, since a particle will actually bounce off the barrier. However, when we treat the particle as a wave, then the wave nature of barrier penetration can occur. This is familiar in electromagnetic waves, where the decaying wave (as opposed to a propagating wave) is termed an evanescent wave. For energies below the top of the barrier, the wave is attenuated, and it decays exponentially. Yet, it takes a significant distance for this decay to eliminate the wave completely. If the barrier is thinner than this critical distance, the evanescent wave can excite a propagating wave in the region beyond the barrier. In optics, this effect is termed frustrated total internal reflection . We are familiar with total internal reflection just from a swimming pool. Water has a higher dielectric constant than free space, so when we are under water, there is critical angle beyond which we can no longer see out (from under the water) of the pool. In optics, we can use a prism, and there will be reflection from the back surface for angles greater than a critical angle. The wave in back of the prism is attenuated and decaying. But, if we place another high dielectric constant material close to the prism, the decaying wave can excite a wave in this new material. 1 Thus, the wave can penetrate the barrier and continue to propagate, with an attenuated amplitude, in the trans-barrier region. For particles, this process is termed tunneling , with analogy to the miners who burrow through a mountain in order to get to the other side! In fact, it is known that the nuclear fusion process by which our sun produces its power would not occur were there no tunneling. The intra-nuclear repulsion is just too strong and tunneling is required for the nuclei to find one another.

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The resarch on penetration coefficient in barrier penetration when ∪_0 =E

Yue Liu

College PhysicsJan 1, 20050 citations

T \approx e^{-2\kappa L}

, where

\kappa = \sqrt{2m(V_0-E)}/\hbar

; for a rectangular barrier, exact

T = [1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}]^{-1}

. No new constants or ratios emerge beyond

\hbar

m

V_0

E

L

. | CONNECTION: The exponential decay

e^{-2\kappa L}

has no direct ratio to 0.382/0.618/1.618; however, the *critical* condition

E = V_0

yields

\kappa = 0

and

T \to 1

(resonant transparency), which is a phase-transition-like threshold — analogous to a golden-ratio bifurcation point in some dynamical systems, but not evidenced here. No crystallographic or base-60 link. | DEPTH: 3 — Standard quantum mechanics, well-understood; the "impossible measurements" QFT paper (arXiv:2311.13 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com