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}

$L \in \left[0.05, 5\right]$

where

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

The figure, part by part

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①

Governing law — red

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

$L \in \left[0.05, 5\right]$

$L = 0.05$ $\kappa = 0.412$

Along the curve

②

Domain of the variable — blue

L_{\min} = 0.05,\quad L_{\max} = 5

$L \in \left[0.05, 5\right]$

$L = 0.05$ $L = 0.05$

Along the curve

③

Response curve — green

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

$L \in \left[0.05, 5\right]$

$L = 0.05$ $\kappa = 0.412$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\kappa}{\mathrm{d}L}

$L \in \left[0.05, 5\right]$

$L = 0.081$ $\kappa^{\prime} = -4.423$

Along the curve

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