Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

Barrier Penetration

屏障穿透

According to classical physics, a particle of energy E less than the height U0 of a barrier could not penetrate - the region inside the barrier is classically forbidden. But the wavefunction associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. The wavefunction must also be continuous on the far side of the barrier, so there is a finite probability that the particle will tunnel through the barrier.

根据经典物理，能量为E的粒子，如果小于势垒高度U₀，就无法穿透——势垒内部的区域在经典物理中是被禁止的。但与自由粒子相关的波函数必须在势垒处连续，并在势垒内部表现出指数衰减。波函数还必须在势垒的另一侧保持连续，因此粒子有有限的概率穿透势垒。

As a particle approaches the barrier, it is described by a free particle wavefunction. When it reaches the barrier, it must satisfy the Schrodinger equation in the form

当一个粒子接近势垒时，它被描述为自由粒子的波函数。当它到达势垒时，必须满足如下形式的薛定谔方程
which has the solution

Note that in addition to the mass and energy of the particle, there is a dependence on the fundamental physical constant Planck's constant h. Planck's constant appears in the Planck hypothesis where it scales the quantum energy of photons, and it appears in atomic energy levels which are calculated using the Schrodinger equation.

需要注意的是，除了粒子的质量和能量外，还取决于基本物理常量普朗克常数h。普朗克常数出现在普朗克假设中，它对光子的量子能量进行尺度调整，并且出现在通过薛定谔方程计算的原子能级中。

There are circumstances where a reasonable approximation of the transmission coefficient through the barrier may be obtained by just taking the square of the ratio of the amplitudes of the wavefunctions at the exit and entrance to the barrier. That approximation to the transmission coefficient can be expressed as:

在某些情况下，可以通过仅仅取波函数在屏障出口和入口处振幅的比值的平方来合理地近似穿过屏障的传输系数。这个传输系数的近似表达式可以表示为：

This approximation will be evaluated below and compared to a more rigorous quantum mechanical treatment.

If a particle of energy E =x10^ joules = eV =MeV =GeV

and mass

和质量
m = x 10^ kg = me = mp = MeV/c2,

approaches a barrier of height

接近高度障碍
U0 = x 10^ joule = eV= MeV,

and thickness

和厚度
d = x 10^ m = nm= fermi,

then the attenuation factor is

则衰减因子为
α = x10^ 1/m .

and the ratio of the exit and incident amplitudes is x10^ .

出口振幅与入射振幅的比值为x10^。
中文译文中的待填/计算数值依次对应：1：arb 2：arp。实际数值以上方原输入框为准。

Since the probability is proportional to the square of the amplitude, the approximate tunneling probability is T = x10^ .

由于概率与振幅的平方成正比，近似隧道概率为 T = x10^ 。
中文译文中的待填/计算数值依次对应：1：prb 2：prp。实际数值以上方原输入框为准。
若一个粒子的能量 E = 10^ joules = eV = MeV = GeV，m = x 10^ kg = m e = m p = MeV/c²，U₀ = x 10^ joule = eV= MeV，d = x 10^ m = nm= fermi，α = x10^ 1/m 。
中文译文中的待填/计算数值依次对应：1：eb 2：ep 3：ev 4：mev 5：gev 6：mb 7：mp 8：em 9：pm 10：mmev 11：ub 12：up 13：uev 14：umev 15：lb 16：lp 17：nm 18：fm 19：alpb 20：alpp。实际数值以上方原输入框为准。

This approximate approach to the transmission coefficient through the barrier was used in the introduction to alpha decay, which proceeds by quantum mechanical tunneling through the Coulomb barrier, and was used to approximate the decay half-life of Polonium.

这种近似方法用于估算穿过势垒的透射系数，曾在α衰变的介绍中被使用。α衰变通过量子力学隧道效应穿过库仑势垒进行，该方法也被用来近似钋的半衰期。

The accuracy of this approximate approach is not sufficient for tunneling applications in solid state electronics, where one may deal with the tunneling of electron flow through very thin insulating layers. The more rigorous treatment of the transmission coefficient gives the expression

这种近似方法的准确性对于固态电子学中的隧穿应用来说是不够的，因为在固态电子学中，人们可能需要处理电子通过非常薄的绝缘层进行隧穿流动的情况。更严谨的传输系数处理给出了以下表达式：

Using the more rigorous requirement of continuity of both the wavefunction and its first derivative at both boundaries, the transmission coefficient for the barrier described above becomes T = x10^ .

使用波函数及其一阶导数在两个边界处均连续的更严格条件，上述势垒的透射系数变为 T = x10^ 。
中文译文中的待填/计算数值依次对应：1：tteb 2：ttep。实际数值以上方原输入框为准。
Alpha decay barrier
α衰变势垒
Details about the transmission coefficient calculation
关于传输系数计算的细节
这一近似将在下文进行评估，并与更严谨的量子力学处理方法进行比较。
其有解
 Index

Schrodinger equation concepts

Beiser
Concepts of Modern Physics,
Ch 5
索引 斯涅尔方程概念 Beiser 《现代物理概念》，第5章
 
HyperPhysics***** Quantum Physics
HyperPhysics ***** 量子物理
R Nave
 Go Back
返回





Quantum Mechanical Barrier Penetration

量子力学中的势垒穿透

According to classical physics, a particle of energy E less than the height U0 of a barrier could not penetrate - the region inside the barrier is classically forbidden. But the wavefunction associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. The wavefunction must also be continuous on the far side of the barrier, so there is a finite probability that the particle will tunnel through the barrier. In addition to being continuous at the entrance and exit from the barrier, the first derivative of the wavefunction must also be continuous upon entrance and exit from the barrier. Since the probability of detecting the particle is proportional to the square of the wavefunction, the transmission coefficient through the barrier is proportional to the square of the transmitted wavefunction divided by the square of the incident wavefunction. But since the wavefunction can reflect from the first and second surfaces of the barrier, deriving the expression for the transmission coefficient is a non-trivial mathematical exercise.

根据经典物理，能量为E的粒子，如果小于势垒的高度U₀，就无法穿透——势垒内部的区域在经典物理中是被禁止的。但是，与自由粒子相关的波函数必须在势垒处连续，并且在势垒内部会表现出指数衰减。波函数也必须在势垒的另一侧保持连续，因此粒子有有限的概率穿透势垒。除了在进入和离开势垒处保持连续外，波函数的第一导数在进入和离开势垒时也必须连续。由于检测到粒子的概率与波函数的平方成正比，因此穿过势垒的透射系数与透射波函数平方除以入射波函数平方成正比。但是由于波函数可以从势垒的第一和第二表面反射，推导透射系数的表达式是一项非平凡的数学工作。

Solving for the transmission coefficient requires evaluation of the boundary conditions at the barrier surface.

求解透射系数需要评估势垒表面的边界条件。

The equations can be divided through by A since it sets the scale for the other amplitudes. That gives four equations with four unknowns which can be solved by repeated substitutions. The resulting single equation for the transmission coefficient can be expressed in terms of the known parameters of the barrier.

这些方程可以被A除，因为A决定了其他振幅的尺度。这样就得到了四个方程和四个未知数，可以通过反复代入求解。最终的透射系数单个方程可以表示为已知参数的函数。

The transmission coefficient can be calculated in the Javascript calculation above.

Alpha decay barrier
α衰变势垒
透射系数可以在上述JavaScript计算中计算
 Index

Schrodinger equation concepts

Beiser
Concepts of Modern Physics,
Ch 5

Serway, Moses, Moyer
Modern Physics, 2nd Ed
Ch 6
索引 斯涅尔方程概念 Beiser 《现代物理概念》，第5章 Serway, Moses, Moyer 《现代物理》，第二版 第6章
 
HyperPhysics***** Quantum Physics
HyperPhysics ***** 量子物理
R Nave
 Go Back
返回