QNT-06

Rectangular-barrier tunneling

T ≈ exp(−2 κ L), κ = √(2m(V−E))/ħ.

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TunnelingGamow / WKB

Governing equation

T\approx\exp(-2\kappa L),\quad \kappa=\dfrac{\sqrt{2m(V-E)}}{\hbar}

where

V: Barrier height (eV)
E: Energy (eV)
L: Width (nm)
m: Mass (m_e)
\kappa: Decay constant (1/nm)
T: Transmission (—)

Lecture brief

Historical brief

Planck (1900), Einstein’s photoelectric law, Bohr, de Broglie, Heisenberg and Schrödinger’s 1926 equation rebuilt matter as amplitude. These sheets are the first solvable models: wells, spin, tunneling, uncertainty. This sheet (QNT-06 — Rectangular-barrier tunneling) is the form associated with Gamow / WKB. Working symbols: $V$ , $E$ , $L$ , $m$ $\rightarrow$ $\kappa$ , $T$ . The WKB (and exact thick-barrier) transmission is exponentially small in the product of decay constant and width, the origin of α-decay lifetimes.

Purpose

Purpose: compute

\kappa

T

from

V

E

L

m

in Quantum mechanics via

T\approx\exp(-2\kappa L),\quad \kappa=\dfrac{\sqrt{2m(V-E)}}{\hbar}

T ≈ exp(−2 κ L), κ = √(2m(V−E))/ħ. Use it when a real quantum mechanics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

V = 5.000\,\mathrm{eV}

E = 2.000\,\mathrm{eV}

L = 0.500\,\mathrm{nm}

m = 1.000\,\mathrm{m_e}

, the governing relation

T\approx\exp(-2\kappa L),\quad \kappa=\dfrac{\sqrt{2m(V-E)}}{\hbar}

yields

\kappa = 8.874\,\mathrm{1/nm}

T = 1.400e-4\,\mathrm{—}

. E < V, thick barrier, electron mass. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Barrier height V5 eVEnergy E2 eVWidth L0.5 nmMass m1 m_e

Outputs

Decay constant \kappa8.874 1/nm
Transmission T0.000140 —

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Narration of this film

E < V, thick barrier, electron mass.

The WKB (and exact thick-barrier) transmission is exponentially small in the product of decay constant and width, the origin of α-decay lifetimes.

Reading speed

Watch on YouTube