Quantum Tunneling Probability

Equivalent forms

T \approx \exp\!\left(-2\kappa L\right),\;\kappa = \sqrt{2m(V_0 - E)}/\hbar

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

An exponential of an integral — and stars shine, transistors switch, and electrons leak through gate oxides because of it.

Symbol	Name	SI unit	Dimension	Range
$T$	Transmission probabilityoutput Probability of barrier penetration	1	1	1e-30 – 1
$m$	Particle mass Mass of the tunneling particle	kg	M	9.1093837015e-31 – 1.67e-27
$V_0$	Barrier height Potential energy at the top of the barrier	J	ML^2T^-2	1e-21 – 1e-15
$E$	Particle energy Kinetic energy approaching the barrier (E < V_0)	J	ML^2T^-2	1e-22 – 1e-15
$L$	Barrier width Spatial width of the rectangular barrier	m	L	1e-12 – 1e-7

Unit systems

Symbol	SI	CGS	Imperial
$T$	1	1	1
$m$	kg	g	lb
$V_0$	J	erg	ft*lbf
$E$	J	erg	ft*lbf
$L$	m	cm	ft

Where it holds

Non-relativistic, single-particle, rectangular barrier (or WKB-smooth). Breaks down for resonant tunneling, strong fields, or relativistic particles.

Dimensional analysis

T \to

dimensionless

exp(integral of

\sqrt{M*M*L^2*T^-2}/(M*L^2*T^-1) dx) = \exp

(integral of

(1/L) dx) = \exp

(dimensionless)

Argument of exp is dimensionless.

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Discovery

George Gamow · 1928

Gamow applied Schrödinger's wave equation to alpha decay, showing nuclei escape via barrier penetration — the first quantitative success of quantum mechanics outside atoms. Hund had described molecular tunneling the year before.

Try this

How does an electron pass through a wall it doesn't have the energy to climb?

A particle with less energy than a potential barrier still has a nonzero probability to appear on the other side. What is the exact transmission probability, and why does the Sun burn because of it?

Research status: stable

Real-world applications

Scanning Tunneling Microscope (STM) imaging
Tunnel diodes and Josephson junctions
Alpha decay and proton-proton fusion in stars
Flash memory (Fowler-Nordheim tunneling through oxide)

Common misconceptions

Tunneling does not give the particle extra energy — total E is conserved
Particles do not 'pass through' instantaneously — the Hartman effect is widely misinterpreted
T is exponentially sensitive to L — a 10% change in width can change T by $100\times$

Experimental verification

Alpha decay half-lives spanning 10^29 (Gamow 1928, Geiger-Nuttall law). Tunnel diodes (Esaki 1958, Nobel). Scanning tunneling microscope (Binnig-Rohrer 1981, Nobel) — single-atom resolution from exponential dependence on gap distance. Hydrogen fusion in stars.

Derivation

Inside a rectangular barrier of height V_0 and width L, the Schrödinger equation gives

\psi \sim \exp (\pm \kappa *x)

with

\kappa = \sqrt(2*m*(V_0 - E))

/hbar.

Matching psi and psi' at both edges yields

T = 1/(1 + V_0^2*\sinh ^2(\kappa *L)/(4*E*(V_0-E)))

.

For thick barriers (kappa*L >>

1) \sinh \approx \exp /2

, giving

T \approx 16*E*(V_0-E)/V_0^2 * \exp (-2*\kappa *L)

— the WKB form.

Limiting cases

E \to V_0

⟶

T \to 1

(classical limit)Particle has nearly enough energy — barrier vanishes effectively.

kappa*L >> 1⟶

T \approx 16*E*(V_0-E)/V_0^2 * \exp (-2*\kappa *L)

Thick-barrier WKB limit — exponential suppression dominates.

L \to 0

or

m \to 0

⟶

T \to 1

Massless or zero-width barrier is transparent.

What if…

What if the Sun's protons could not tunnel?

Proton-proton fusion would require $\sim 10\times$ higher core temperature — stellar nucleosynthesis would not occur, and stars would not shine.

What if the barrier is a person walking through a wall?

$\kappa *L \sim 10^35$ — $T \sim \exp (-10^35)$ , so don't wait.

What if E > V_0?

Above-barrier scattering — T < 1 due to wave reflection (purely quantum), but no exponential suppression.

1

Electron through 1 nm, 10 eV barrier (E = 1 eV)

Given ·

m:: 9.1093837015e-31
V 0:: 1.602e-18
E:: 1.602e-19
L:: 1e-9

Find · T

Steps

$V_0 - E = 1.442e-18\,\mathrm{J}$
$\kappa = \sqrt{2 * 9.1093837015e-31 * 1.442e-18} / 1.054571817e-34 \approx 1.54e10 /m$
$2*\kappa *L = 30.8$
$T \approx \exp (-30.8) \approx 4.2e-14$

Result ·

\kappa \approx 1.54e10 /m

;

2*\kappa *L \approx 30.8 \to T \approx \exp (-30.8) \approx 4.2e-14

2

Same barrier, shorten L to 0.3 nm

Given ·

m:: 9.1093837015e-31
V 0:: 1.602e-18
E:: 1.602e-19
L:: 3e-10

Find · T

Steps

kappa unchanged $\approx 1.54e10 /m$
$2*\kappa *L = 9.24$
$T \approx 9.7e-5$ — basis of STM atomic resolution

Result ·

2*\kappa *L \approx 9.24 \to T \approx \exp (-9.24) \approx 9.7e-5 (10

orders of magnitude larger)

Time-Independent Schrödinger Equation→particle in boxHeisenberg Uncertainty Principle→