Quantum Tunneling Probability Calculator

What is Quantum Tunneling: Penetrating Classically Forbidden Barriers?

Quantum tunneling is a phenomenon where a particle passes through a potential energy barrier that it classically could not surmount. In classical physics, a ball rolling toward a hill with insufficient energy simply bounces back. In quantum mechanics, the particle's wavefunction extends through the barrier, giving it a non-zero probability of appearing on the other side. This effect — impossible in classical physics — is the operating principle behind tunnel diodes, scanning tunneling microscopes, nuclear fusion in stars, and flash memory storage.

Mathematical Foundation

Transmission Coefficient (Rectangular Barrier)

T \approx e^{-2\kappa L}, \quad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}

T

= Transmission coefficient — probability (0 to 1) that the particle tunnels through the barrier.

\kappa

= Decay constant inside the barrier (inverse meters). Larger κ = faster wavefunction decay.

V_0

= Barrier height — potential energy of the barrier (eV or Joules).

E

= Particle kinetic energy (must be less than V₀ for tunneling to apply).

m

= Particle mass (kg). Electron: 9.109×10⁻³¹ kg. Proton: 1.673×10⁻²⁷ kg.

L

= Barrier width (meters or nanometers).

\hbar

= Reduced Planck constant: 1.055×10⁻³⁴ J·s.

Laws & Principles

Exponential Sensitivity: The tunneling probability drops exponentially with barrier width L and the square root of mass × barrier excess (V₀ − E). Doubling the barrier width squares the exponent, reducing transmission by many orders of magnitude. This is why electrons tunnel easily but protons rarely do at the same barrier.
E Must Be Less Than V₀: Tunneling only applies when the particle energy E is below the barrier height V₀. When E ≥ V₀, the particle passes over classically with T approaching 1 (though quantum reflection still occurs at interfaces). The WKB approximation breaks down near E ≈ V₀.
Mass Determines Feasibility: The decay constant κ scales as √m. Electrons (9.1×10⁻³¹ kg) tunnel through barriers that are completely opaque to protons (1836× heavier). Alpha particles (4 amu) tunnel out of nuclei in radioactive decay, but the same barrier is impassable for heavier nuclei.

Step-by-Step Example Walkthrough

" An electron (m = 9.109×10⁻³¹ kg) with 3 eV kinetic energy encounters a 5 eV barrier that is 0.5 nm wide. "

1. Barrier excess: V₀ − E = 5 − 3 = 2 eV = 3.204×10⁻¹⁹ J.
2. κ = √(2 × 9.109×10⁻³¹ × 3.204×10⁻¹⁹) / (1.055×10⁻³⁴) = 7.25×10⁹ m⁻¹.
3. 2κL = 2 × 7.25×10⁹ × 0.5×10⁻⁹ = 7.25.
4. T = e⁻⁷·²⁵ ≈ 7.1×10⁻⁴.

Final Result: The electron has a 0.071% chance of tunneling through — roughly 1 in 1,400 attempts. At 10¹⁵ attempts per second (typical in solid-state devices), tunneling events occur ~7×10¹¹ times per second.

Real-World Tunneling

Scanning Tunneling Microscope (STM)

The STM uses tunneling current between a sharp tip and a surface to image individual atoms. The current depends exponentially on the tip-surface gap (~0.1 nm changes cause 10× current variation). This extreme sensitivity enables sub-angstrom resolution — earning its inventors the 1986 Nobel Prize in Physics.

Nuclear Fusion in Stars

Protons in the Sun's core have only ~1 keV kinetic energy but face a ~1 MeV Coulomb barrier. Classical physics says fusion is impossible at solar temperatures. Quantum tunneling allows protons to penetrate this barrier with probability ~10⁻²⁸ per collision — but with 10³⁸ collisions per second, fusion sustains stellar energy output.

Particle	Mass (kg)	T (1 nm, 1 eV barrier)	Application
Electron	9.109×10⁻³¹	~13%	Tunnel diodes, STM
Proton	1.673×10⁻²⁷	~10⁻¹⁹	Nuclear fusion
Alpha particle	6.645×10⁻²⁷	~10⁻³⁸	Radioactive decay
Muon	1.884×10⁻²⁸	~0.3%	Muon-catalyzed fusion

Particle

Mass (kg)

T (1 nm, 1 eV barrier)

Application

Electron

9.109×10⁻³¹

~13%

Tunnel diodes, STM

Proton

1.673×10⁻²⁷

~10⁻¹⁹

Nuclear fusion

Alpha particle

6.645×10⁻²⁷

~10⁻³⁸

Radioactive decay

Muon

1.884×10⁻²⁸

~0.3%

Muon-catalyzed fusion

Quantum Physics Best Practices (Pro Tips)

Do This

✓Use eV and nanometers for atomic-scale problems. SI units (Joules, meters) produce unwieldy exponents. The natural units for tunneling are eV for energy and nm for distance. Convert: 1 eV = 1.602×10⁻¹⁹ J, 1 nm = 10⁻⁹ m.

Avoid This

✗Don't use the WKB approximation when E ≈ V₀. The approximation breaks down when particle energy approaches the barrier height. Near E = V₀, use the exact solution with transmission and reflection coefficients derived from boundary conditions. The WKB is accurate when 2κL >> 1.

Frequently Asked Questions

Can a person tunnel through a wall?

Technically yes, but the probability is ~10⁻³⁸ⁿ where n is astronomically large. For a 70 kg person and a 10 cm wall, T ≈ 10⁻¹⁰³⁰. You would need to run into the wall more times than there are atoms in the observable universe before tunneling becomes likely. It's technically non-zero but effectively impossible.

Why does tunneling probability depend on mass?

The de Broglie wavelength λ = h/(mv) is inversely proportional to mass. Heavier particles have shorter wavelengths and their wavefunctions decay faster inside the barrier (larger κ). Electrons are light enough that their wavefunctions extend significantly through nanometer-scale barriers. Protons, 1836× heavier, have wavefunctions that decay 43× faster.

How does tunneling enable flash memory?

Flash memory stores data by trapping electrons on a floating gate surrounded by a thin oxide barrier (~8-10 nm). To program: a high voltage makes electrons tunnel through the oxide onto the gate (Fowler-Nordheim tunneling). To erase: reverse voltage tunnels them back. The barrier is thick enough that trapped electrons persist for 10+ years without applied voltage.

What is the WKB approximation?

The Wentzel-Kramers-Brillouin (WKB) approximation simplifies the Schrödinger equation for slowly varying potentials. For a rectangular barrier, it gives T ≈ e⁻²ᵏᴸ. This is accurate when 2κL >> 1 (thick barrier limit). For very thin barriers or E near V₀, the exact solution with matching boundary conditions is needed.

Quantum Tunneling Engine

Quantum Mechanics: Particle Tunneling Probability

Successful Quantum Transmission (T)