Quantum tunneling is one of the most counterintuitive phenomena in modern physics. It describes how a particle can pass through a barrier that, according to classical physics, it should never be able to cross. This behavior is not a hypothetical curiosity. It drives nuclear fusion in stars, enables radioactive decay, and powers scanning tunneling microscopes that image individual atoms. Understanding how does quantum tunneling work requires examining the wave nature of matter and the probabilistic foundation of quantum mechanics.
Wave-Particle Duality and the Probability Wave
To grasp quantum tunneling, one must first accept that particles are not tiny billiard balls with fixed positions and velocities. At the quantum scale, every particle behaves as both a particle and a wave. This dual nature is not a metaphor. It is a measurable property confirmed by experiments such as the double slit experiment.
In quantum mechanics, the state of a particle is described by a wavefunction, often denoted by the Greek letter psi (ψ). The wavefunction encodes everything that can be known about the particle. Its square, multiplied by a small volume, gives the probability of finding the particle in that region of space. This is the Born rule, named after physicist Max Born. The wavefunction spreads out over space like a ripple on a pond. It does not collapse to a single point until a measurement occurs.
This probabilistic wave is the key to understanding how does quantum tunneling work. Even when a particle encounters a barrier that it lacks the energy to surmount in classical terms, its wavefunction does not instantly drop to zero at the barrier's edge. Instead, it penetrates into and through the barrier, decaying exponentially in amplitude. If the barrier is thin enough, the wavefunction remains nonzero on the far side. That means there is a real, calculable probability that the particle will emerge on the other side.
Crossing an Impossible Barrier
In classical physics, a particle with total energy E cannot cross a barrier of height V if E is less than V. The particle would have to gain energy it does not have. This is analogous to a ball rolling up a hill: if the ball does not have enough kinetic energy to reach the top, it rolls back.
In quantum mechanics, the barrier is a region where the potential energy exceeds the particle's total energy. Inside this classically forbidden zone, the wavefunction decays rather than oscillates. The rate of decay is determined by the barrier's height and width, as well as the particle's mass. The relevant equation is the time independent Schrödinger equation, which governs the wavefunction's spatial behavior.
For a rectangular barrier of height V₀ and width L, the transmission probability T (the chance of tunneling) is approximately:
T ≈ e-2L√(2m(V₀-E))/ħ
Here, m is the particle's mass, ħ is the reduced Planck constant (1.0545718 × 10-34 J·s), and e is the base of natural logarithms. The formula shows that transmission drops exponentially with barrier width and with the square root of the mass. This explains why tunneling is most noticeable for light particles like electrons and less so for heavier particles like protons.
Why Particles Appear on the Other Side
The quantum explanation is straightforward once the wavefunction picture is accepted. The wavefunction of the particle extends into the barrier and, if the barrier is narrow, continues beyond it. The particle does not "dig through" or "borrow energy" in the sense of a temporary violation of conservation laws. That interpretation, sometimes called the "imaginary time" picture, is misleading.
What actually happens is this: the wavefunction inside the barrier is a mathematical function that describes a real physical state. The particle's presence on the far side is a consequence of the wavefunction's continuity and the boundary conditions at the barrier interfaces. When a measurement detects the particle on the far side, the wavefunction collapses there. The particle did not take a specific path through the barrier. Quantum mechanics does not assign a trajectory in the classical sense.
The time it takes for tunneling to occur is still a subject of research. Some experiments suggest it is nearly instantaneous, while others indicate a finite delay. The Larmor clock method and attosecond spectroscopy have provided insights, but no consensus exists as of 2025. What is certain is that tunneling does not involve faster than light travel. The transmitted wavefunction does not carry information at superluminal speed.
A Quantitative Example: Calculating Transmission Probability
To illustrate how the transmission probability is computed, consider a simple rectangular barrier. Suppose an electron with energy E = 2 eV encounters a barrier of height V₀ = 5 eV and width L = 0.5 nm. The particle mass is m = 9.109 × 10-31 kg, and ħ = 1.0545718 × 10-34 J·s. First, convert energies to joules: V₀ – E = 3 eV = 4.806 × 10-19 J. The exponent is 2L√(2m(V₀-E))/ħ, which evaluates to approximately 2.8, so T ≈ e-2.8 ≈ 0.061, or about 6.1%. This shows that even for a thin barrier, a significant fraction of electrons can tunnel.
This example demonstrates the exponential sensitivity. Doubling the width to 1.0 nm increases the exponent to 5.6, yielding T ≈ 0.0037, or about 0.37%. Such calculations are foundational for understanding tunneling in electronics and nuclear physics.
Real Examples: Radioactive Decay
Alpha decay is a classic example of quantum tunneling. In an unstable atomic nucleus, an alpha particle (two protons and two neutrons) is bound by the strong nuclear force. To escape, the alpha particle must overcome the Coulomb barrier, the electrostatic repulsion between the alpha particle and the remaining nucleus. In classical physics, the alpha particle lacks sufficient energy to surmount this barrier.
Quantum tunneling allows the alpha particle to appear outside the nucleus. The barrier width in alpha decay is on the order of 10-15 meters. The transmission probability is extremely small but not zero. For a typical alpha emitter like Uranium-238, the half life is 4.468 billion years because the tunneling probability is so low. For Polonium-212, the half life is 0.3 microseconds because the barrier is different.
This exponential sensitivity to barrier parameters explains the vast range of half lives observed in radioactive decay. It also underpins the Geiger Nuttall law, which relates decay energy to half life. The law was empirically known before quantum mechanics. Tunneling theory provided its first rigorous explanation.
Nuclear Fusion in Stars
Stars generate energy through nuclear fusion. In the Sun's core, hydrogen nuclei (protons) must overcome the Coulomb barrier to fuse into deuterium, then helium. The Coulomb barrier for two protons is approximately 1 MeV, requiring a substantial energy to surmount classically. The Sun's core temperature of about 15 million Kelvin corresponds to a mean kinetic energy of roughly 1.94 keV per particle (calculated as (3/2)kT, where k is Boltzmann's constant).
Classically, fusion would be impossible because almost no particles have enough energy. Quantum tunneling changes this. The Coulomb barrier is wide and high, but the tails of the proton wavefunctions overlap at energies far below the barrier height. The transmission probability is extremely small, but the Sun contains approximately 1057 protons, and each second roughly 1038 fusion reactions occur. This tunneling phenomenon is essential for stellar nucleosynthesis and, by extension, for the existence of heavy elements in the universe. Without it, stars would not shine as they do. This is yet another context where understanding how does quantum tunneling work is crucial for explaining natural phenomena.
The Gamow Factor and WKB Approximation
A more rigorous treatment of tunneling in nuclear and quantum contexts involves the Gamow factor and the Wentzel-Kramers-Brillouin (WKB) approximation. These methods provide a quantitative framework for understanding how does quantum tunneling work for barriers of arbitrary shape.
The Gamow factor, derived by George Gamow in 1928, describes the tunneling probability for alpha decay. For a Coulomb barrier, the transmission probability is proportional to exp(-2G), where G is the Gamow factor given by G = (π/2) × (Z₁Z₂e²/ħv) for two charged particles with speeds v and atomic numbers Z₁ and Z₂. This factor accounts for the barrier's curvature and the particle's energy.
The WKB approximation extends this to any barrier shape. It approximates the wavefunction inside the barrier as an exponential decay with a position-dependent decay constant. The transmission probability is T ≈ exp(-2 ∫ √(2m(V(x)-E))/ħ dx), integrated over the classically forbidden region. For a rectangular barrier, this reduces to the earlier formula. For a Coulomb barrier, it yields the Gamow factor. This distinction is essential for handling real-world potentials where barriers are neither flat nor uniform. These mathematical tools help clarify how does quantum tunneling work in diverse physical scenarios.
Scanning Tunneling Microscopes
The scanning tunneling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer in 1981, is a direct technological application of quantum tunneling. An STM uses a sharp metal tip brought within a nanometer of a conductive sample surface. A voltage bias is applied between tip and sample. Electrons tunnel across the vacuum gap.
The tunneling current is exponentially sensitive to the tip sample distance, typically changing by about a factor of ten for every angstrom of separation. By scanning the tip across the surface and adjusting its height to maintain constant current, the STM maps the surface topography with atomic resolution. This technique earned Binnig and Rohrer the 1986 Nobel Prize in Physics.
The STM does not just image atoms. It can also manipulate individual atoms. Researchers have used STM tips to move atoms across surfaces, write letters with xenon atoms, and assemble molecular structures. The physics behind these feats is the same quantum tunneling that governs alpha decay, scaled to electrons and a vacuum barrier.
Tunnel Diodes and Electronic Applications
Tunnel diodes exploit quantum tunneling in semiconductors. A tunnel diode is a heavily doped p n junction where the depletion region (the barrier) is very thin, typically 5 to 10 nanometers. When a small forward voltage is applied, electrons can tunnel from the valence band of the p side to the conduction band of the n side.
This produces a region of negative differential resistance, where an increase in voltage leads to a decrease in current. Tunnel diodes are used in high frequency oscillators, switching circuits, and microwave amplifiers. They operate at speeds far beyond conventional diodes because tunneling is a quantum process with essentially no transit time. Explore our guide to Matter and the Physical World for more context.
The Esaki diode, named after Leo Esaki who discovered tunneling in semiconductors in 1957, is the most common type. Esaki shared the 1973 Nobel Prize in Physics for this work. Tunnel diodes are not as widespread today as they were in the 1960s and 1970s, but they remain important in specialized high speed electronics.
1. Can quantum tunneling be observed in everyday life?
Not directly. Tunneling probabilities are exponentially suppressed for large, massive objects. At human scales, the probability of tunneling through a wall is astronomically small. However, tunneling effects are essential for everyday technology. Flash memory and EEPROM chips rely on tunneling through thin oxide layers. The Sun's energy, which drives all life on Earth, depends on tunneling in its core.
2. Does quantum tunneling violate the conservation of energy?
No. The wavefunction inside the barrier does not imply that the particle possesses negative kinetic energy in a literal sense. The uncertainty principle allows the particle's energy to be ill defined over very short timescales. The overall process conserves energy: the particle emerges on the far side with the same total energy it had before encountering the barrier. No free energy is extracted.
3. How is the tunneling probability calculated?
For simple barrier shapes, the probability is derived from the time independent Schrödinger equation. For a rectangular barrier, the transmission coefficient is given by the formula involving exponential decay. For real world barriers, such as the Coulomb barrier in nuclear fusion, the Gamow factor or the Wentzel Kramers Brillouin (WKB) approximation are used.
4. Is there a limit to how thick a barrier can be for tunneling?
Yes, the barrier thickness determines the probability. For electrons, tunneling through a few nanometers is common. For alpha particles, the effective barrier width is on the order of femtometers. For macroscopic barriers, the probability drops to effectively zero. There is no absolute cutoff, but the exponential dependence means that beyond a few atomic diameters for electrons, the chance becomes negligible.
5. Can tunneling be controlled or switched off?
In practical devices, tunneling is controlled by adjusting barrier height and width. In an STM, the tip sample distance is precisely controlled. In tunnel diodes, doping levels set the barrier profile. Quantum tunneling is a fundamental physical process and cannot be "switched off" in principle. However, engineers design systems where tunneling is either harnessed or suppressed by geometric and material choices.
Sources & References
- Griffiths, David J. Introduction to Quantum Mechanics. 2nd ed., Cambridge University Press, 2017. (Standard textbook treatment of tunneling.)
- The Nobel Prize Organization. "The Scanning Tunneling Microscope." NobelPrize.org, 1986.
- Esaki, Leo. "Discovery of the Tunnel Diode." IEEE Transactions on Electron Devices, vol. 23, no. 7, 1976, pp. 644-647.