Quantum Tunneling Calculator

Calculate tunneling probabilities and visualize quantum particles passing through barriers

Tunneling Parameters

Enter particle and barrier properties to calculate tunneling probability

eV
eV
nm
Tunneling Probability (T)
0.002
Probability of tunneling
Reflection Probability (R)
0.998
Probability of reflection
Transmission Coefficient
0.002
Ratio of transmitted particles
Wave Number (k)
1.15 × 10¹⁰ m⁻¹
Wave number inside barrier

Understanding Quantum Tunneling

Quantum tunneling is a fundamental quantum mechanical phenomenon where particles pass through potential barriers they classically shouldn't be able to overcome.

  • Occurs due to the wave nature of particles
  • Probability depends on barrier height and width
  • Exponential decay of wave function inside barrier
  • Important in nuclear fusion, semiconductor devices, and scanning tunneling microscopes
  • Explains alpha decay in radioactive nuclei
1

Wave-Particle Duality: Particles exhibit wave-like behavior

Wavefunctions extend into classically forbidden regions

2

Exponential Decay: Wavefunction decays exponentially inside barrier

ψ(x) ∝ e-κx where κ = √(2m(V₀ - E))/ℏ

3

Transmission Probability: Depends on barrier properties

T ≈ e-2κL for thick barriers

4

Applications: Nuclear fusion, STM, tunnel diodes, flash memory

Key Concepts

Transmission Probability
T
Probability of tunneling
Reflection Probability
R
Probability of reflection
Wavevector
κ
Decay constant in barrier
Penetration Depth
1/κ
Characteristic tunneling distance
Tunneling Formulas
T ≈ exp(-2κL)
κ = √[2m(V₀ - E)] / ħ
Where:
T = Tunneling probability
κ = Decay constant
m = Particle mass
V₀ = Barrier height
E = Particle energy
L = Barrier width
ħ = Reduced Planck's constant (1.055 × 10⁻³⁴ J·s)
Tunneling Examples
Scenario Barrier Height Barrier Width Tunneling Probability
Electron in STM 4 eV 0.5 nm 0.1
Alpha decay 30 MeV 30 fm 10⁻³⁸
Diode tunneling 0.3 eV 5 nm 0.05
Nuclear fusion 400 keV 3 fm 10⁻⁶
Chemical reactions 1 eV 0.1 nm 0.01

Nobel Prize in Physics

Scanning Tunneling Microscope

Awarded to Gerd Binnig and Heinrich Rohrer in 1986 for their design of the scanning tunneling microscope.

This revolutionary instrument uses quantum tunneling to image surfaces at the atomic level.

Tunneling Applications

Nuclear Fusion

Protons tunnel through Coulomb barrier to fuse in stars

Scanning Tunneling Microscope

Images surfaces at atomic resolution using electron tunneling

Flash Memory

Uses tunneling to store charge in floating gate transistors

Quantum tunneling occurs due to:

  • Wave-Particle Duality: Particles have wave-like properties
  • Wavefunction Extent: Quantum waves extend into classically forbidden regions
  • Probability Distribution: Non-zero probability of finding particle beyond barrier
  • Heisenberg Uncertainty: Energy uncertainty allows temporary barrier crossing

Mathematically, the wavefunction decays exponentially inside the barrier but doesn't immediately drop to zero, allowing a finite probability of transmission.

Tunneling probability depends on:

  • Barrier Height (V₀): Higher barriers reduce tunneling
  • Barrier Width (L): Wider barriers reduce tunneling exponentially
  • Particle Energy (E): Higher energy increases tunneling
  • Particle Mass (m): Heavier particles tunnel less easily

The approximate transmission probability for a rectangular barrier is:

T ≈ e-2κL where κ = √(2m(V₀ - E))/ℏ

This shows the exponential dependence on barrier width and the square root of barrier height and particle mass.

Quantum tunneling has several important applications:

  • Nuclear Fusion: Protons tunnel through Coulomb barrier in stars
  • Radioactive Decay: Alpha particles tunnel out of nuclei
  • Scanning Tunneling Microscope (STM): Images surfaces at atomic resolution
  • Tunnel Diodes: High-speed electronic components
  • Flash Memory: Stores data using charge tunneling
  • Quantum Computing: Qubits can tunnel between states
  • Enzyme Catalysis: Proton tunneling accelerates reactions

In theory yes, but in practice no:

  • Tunneling probability decreases exponentially with mass
  • For macroscopic objects, tunneling probability is vanishingly small
  • Example: Human (m≈70kg) through wall (V₀≈10eV, L≈0.2m)
  • κ ≈ √(2×70×10×1.6e-19)/1.05e-34 ≈ 1.5e36 m⁻¹
  • T ≈ e-2×1.5e36×0.2 ≈ e-6e35 (essentially zero)

While quantum mechanics allows for the possibility, the probabilities are so small that we never observe macroscopic tunneling.