Journal of Neijiang Normal University

Probability Current DensityIs a Constant vector in the Penetration of a Rectangular Potential Barrier

Hong Zhu

Published Jan 1, 2003• 0 citations

Abstract

It is demonstrated that throughout the penetration of the rectangular potential barrier the probability current density J → remains constant

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Journal of Neijiang Normal University

Probability Current DensityIs a Constant vector in the Penetration of a Rectangular Potential Barrier

Hong Zhu

Published Jan 1, 2003• 0 citations

OpenAlex

Abstract

It is demonstrated that throughout the penetration of the rectangular potential barrier the probability current density J → remains constant

Related papers

Quantum mechanical penetration through multiple barriers

Y S T Rao, K Pappootty, C Radhakrishnan

Journal of Physics A Mathematical and GeneralMar 1, 19783 citations

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Htay Yee, Thida Win Aye Aye Myint Kyaw Kyaw Naing, Aye Than Kyae

International Journal of Trend in Scientific Research and DevelopmentAug 22, 20191 citation

A Simple Treatment of Potential Barrier Penetration

Bernard L. Cohen

American Journal of PhysicsFeb 1, 196521 citations

Quantum Tunneling: Wave-Mechanical Barrier Penetration and Relativistic Limits — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, governed by exponential decay of the wavefunction inside the barrier; "impossible" QFT measurements are possible but non-ideal, with relativistic causality constraints. MATH: Transmission coefficient $T \approx e^{-2\kappa L}$ , where $\kappa = \sqrt{2m(V_0-E)}/\hbar$ ; barrier penetration probability $P \propto e^{-2\int \kappa(x)dx}$ . For a rectangular barrier: $T = \left[1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1}$ . In the WKB limit, $T \sim \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx\right)$ . No new fundamental constants emerge; the key is the exponential suppression factor. CONNECTION: The exponential decay $e^{-\kappa L}$ has no direct golden-ratio link. However, the *critical* barrier opacity $\kappa L = \ln(2)$ (for 50% transmission) and $\kappa L = \ln(\phi) \approx 0.4812$ (where $\phi = 1.618...$ ) appear as natural thresholds — but t 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, governed by exponential decay of the wavefunction inside the barrier; "impossible" QFT measurements are possible but non-ideal, with relativistic causality constraints. MATH: Transmission coefficient $T \approx e^{-2\kappa L}$, where $\kappa = \sqrt{2m(V_0-E)}/\hbar$; barrier penetration probability $P \propto e^{-2\int \kappa(x)dx}$. For a rectangular barrier: $T = \left[1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1}$. In the WKB limit, $T \sim \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx\right)$. No new fundamental constants emerge; the key is the exponential suppression factor. CONNECTION: The exponential decay $e^{-\kappa L}$ has no direct golden-ratio link. However, the *critical* barrier opacity $\kappa L = \ln(2)$ (for 50% transmission) and $\kappa L = \ln(\phi) \approx 0.4812$ (where $\phi = 1.618...$) appear as natural thresholds — but t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 14, 20260 citations

The probability of potential barrier penetration in the generalized wkB approximation

Н. И. Жирнов

Russian Physics JournalJan 1, 19693 citations

The probability of potential barrier penetration in the generalized wkB approximation

Н. И. Жирнов

Russian Physics JournalJan 1, 19690 citations

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

, where

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

; barrier penetration probability

P \propto e^{-2\int \kappa(x)dx}

. For a rectangular barrier:

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

. In the WKB limit,

T \sim \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx\right)

. No new fundamental constants emerge; the key is the exponential suppression factor. CONNECTION: The exponential decay

e^{-\kappa L}

has no direct golden-ratio link. However, the *critical* barrier opacity

\kappa L = \ln(2)

(for 50% transmission) and

\kappa L = \ln(\phi) \approx 0.4812

(where

\phi = 1.618...

) appear as natural thresholds — but t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com