Quantum Tunneling
Quantum tunneling is the quantum mechanical phenomenon by which a particle penetrates through a potential energy barrier that it classically could not surmount. Unlike classical mechanics, where a particle must have enough energy to overcome a barrier, quantum mechanics allows a non-zero probability of the particle's wave function existing on the other side of the barrier. This effect is responsible for nuclear fusion in stars, the operation of tunnel diodes, scanning tunneling microscopes, and radioactive alpha decay.
Key Formula
T ≈ exp(−2κL), where κ = sqrt(2m(V₀ − E)) / ℏ
LaTeX: T \approx e^{-2\kappa L}, \quad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}
| Symbol | Meaning | Unit |
|---|---|---|
| T | Transmission probability (tunneling probability) | dimensionless |
| κ | Decay constant inside the barrier | m⁻¹ |
| L | Width of the potential barrier | m |
| m | Mass of the particle | kg |
| V₀ | Height of the potential barrier | J (or eV) |
| E | Total energy of the particle | J (or eV) |
| ℏ | Reduced Planck constant (h/2π) | J·s |
Worked Example
Problem
An electron (mass m = 9.11 × 10⁻³¹ kg) with kinetic energy E = 1.0 eV encounters a rectangular potential barrier of height V₀ = 2.0 eV and width L = 0.5 nm. Estimate the tunneling transmission probability T.
Solution
Step 1: Convert units. E = 1.0 eV = 1.6 × 10⁻¹⁹ J; V₀ = 2.0 eV = 3.2 × 10⁻¹⁹ J; L = 0.5 × 10⁻⁹ m; ℏ = 1.055 × 10⁻³⁴ J·s. Step 2: Calculate κ. κ = sqrt(2 × 9.11×10⁻³¹ × (3.2×10⁻¹⁹ − 1.6×10⁻¹⁹)) / (1.055×10⁻³⁴) = sqrt(2 × 9.11×10⁻³¹ × 1.6×10⁻¹⁹) / 1.055×10⁻³⁴ = sqrt(2.916×10⁻⁴⁹) / 1.055×10⁻³⁴ = 5.40×10⁻²⁵ / 1.055×10⁻³⁴ ≈ 5.12 × 10⁹ m⁻¹ Step 3: Calculate 2κL. 2κL = 2 × 5.12×10⁹ × 0.5×10⁻⁹ = 5.12 Step 4: Calculate T. T ≈ e^(−5.12) ≈ 0.006
Answer
T ≈ 0.006, meaning about 0.6% of electrons tunnel through the barrier.
Real-World Applications of Quantum Tunneling
| Application | Particle | Barrier Type | Key Effect |
|---|---|---|---|
| Alpha decay | Alpha particle | Nuclear potential well | Radioactive emission |
| Stellar fusion | Proton | Coulomb barrier | Energy generation in stars |
| Tunnel diode | Electron | Thin semiconductor junction | Negative resistance region |
| Scanning Tunneling Microscope | Electron | Vacuum gap (< 1 nm) | Atomic-resolution imaging |
| Flash memory | Electron | Oxide layer | Writing/erasing data bits |
Interactive Tools
PhET Quantum Tunneling and Wave Packets
Visualize wave packets tunneling through potential barriers.
Open ToolWolfram Alpha — Tunneling Probability
Compute tunneling transmission coefficients with custom parameters.
Open ToolBrilliant.org — Quantum Tunneling
Guided problems and intuitive explanations of tunneling phenomena.
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Quantum Superposition
Quantum superposition is the principle that a quantum system can exist in multiple distinct states simultaneously until a measurement is performed, at which point the wave function collapses to a single definite state. Mathematically, the state of a particle is described by a linear combination (superposition) of basis states, each with a complex amplitude whose squared modulus gives the probability of that outcome. The principle underpins interference phenomena, quantum computing (qubits), and famous thought experiments such as Schrödinger's cat.
Energy Level
An energy level is one of the discrete, quantized values of energy that a bound quantum system (such as an electron in an atom or a molecule) is permitted to have. Unlike classical systems where energy can take any continuous value, quantum mechanics constrains bound particles to specific allowed states, each characterized by a set of quantum numbers. Transitions between energy levels result in the absorption or emission of photons with energies exactly equal to the difference between the two levels, producing the characteristic spectral lines used in atomic spectroscopy.
Pauli Exclusion Principle
The Pauli Exclusion Principle states that no two identical fermions (particles with half-integer spin) can simultaneously occupy the same quantum state within a quantum system. This principle, formulated by Wolfgang Pauli in 1925, explains the structure of the periodic table and the stability of matter — electrons in an atom must each have a unique set of quantum numbers (n, l, m_l, m_s). It underlies the existence of distinct atomic orbitals, the hardness of solids, and the phenomenon of electron degeneracy pressure in white dwarf stars.
The term "tunneling" is a metaphor coined in the early 1920s–1930s, as the particle appears to "tunnel" through a wall it classically cannot climb over. "Quantum" derives from the Latin quantum, meaning "how much," introduced into physics by Max Planck in 1900.