The transmission of electrons through a thin barrier provides an introduction to the equation that governs how a wave function changes with time.
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平均1 eVの電子が厚さ0.2 nm・高さ2 eVの壁を透過する確率は？
同じ壁・平均エネルギーで、電子の1836倍重い陽子の透過率は？
The equations explain the phenomenon
1A free electron is represented by a wave.ψ=ei(kx − ωt)，p=ħk，E=ħω……
2Differentiation reveals the connection to momentum and energy.−iħ ∂ψ∂x=p ψ，iħ ∂ψ∂t=E ψ……
3The fundamental equation of quantum mechanics is introduced.iħ ∂ψ∂t=−(ħ²2m) ∂²ψ∂x²+V ψ……
4The wave function determines the probability density for position.Probability density for position=|ψ|²……
5Superposing waves produces a moving wave packet.Speed of the wave packet's center=Mean momentumm……
6Stationary states have probability densities that do not change with time.−(ħ²2m) φ″+V φ=E φ……
7The wave function and its slope are matched at both edges of the barrier.φ and φ′ are continuous.（x=0，a）……
8The 4 equations determine the transmission amplitude t.t=e−ikacosh κa + i g sinh κa……
9The ratio of probability currents determines the transmission probability.Transmission probability T=1cosh²(κa)(Wave with definite E, V₀=2E)……
Returning to the motion in the opening example, ψ(x, t) is obtained by multiplying each energy state by e−iEt/ħ and adding the states with their respective weights. Its |ψ|² forms the moving peak in the diagram above.Once the waves are far from the barrier, the area on the right equals the average transmission probability in step 9.
Further study: An electron in a box
10A particle is confined in a box.φn=√(2L) sin(nπxL)，En=n² h²8 m L²……
11The lowest energy of an electron in a box of length 1 nm is calculated.E₁=0.376 eV……
What this chapter has shown
The time evolution of the wave function is governed by iħ ∂ψ/∂t=−(ħ²/2m) ∂²ψ/∂x²+Vψ. This equation is adopted as a fundamental law of quantum mechanics, guided by the correspondence that replaces p and E in E=p²/2m+V with operators. The probability of finding a particle can be calculated from |ψ|².
40.4 %The transmission coefficient for the initial conditions
For the same barrier, mean energy, and spread in position, a proton with 1836 times the electron mass has a transmission probability of 3 × 10−36%.
In everyday lifeFlash memory writes data by using quantum tunneling to transfer electrons through a thin film.
The time evolution of the wave function is governed by the Schrödinger equation iħ ∂ψ/∂t = −(ħ²/2m) ∂²ψ/∂x² + V ψ. Motivated by the plane-wave relations p = ħk and E = ħω, this equation is obtained by replacing p and E in the energy equation with operators, such as pop = −iħ ∂/∂x, and is postulated as a fundamental law valid for general waves. The absolute value of ψ raised to the power 2 gives the probability density for finding the particle. A peak formed by superposing waves moves like a particle. For a barrier of finite width and height, the wave function decays inside the barrier and extends beyond it, allowing transmission. When a particle is confined by infinitely high walls, its energy takes discrete values for the same reason that only certain standing waves can form on a string.