Quantum Tunneling and Barrier Penetration
Quantum tunneling is the ability of a particle to pass through a potential barrier that classical mechanics says it cannot surmount; the wavefunction decays but does not vanish inside the barrier, leaving a small probability of emerging on the other side.
Definition
Quantum tunneling is the penetration of a quantum particle through a potential energy barrier higher than its total energy, a phenomenon with no classical analogue arising because the wavefunction decays exponentially rather than terminating inside the barrier.
Scope
The topic covers scattering from rectangular and arbitrary one-dimensional barriers, the transmission and reflection coefficients, the exponential dependence of tunneling probability on barrier width and height, the wavefunction's evanescent decay in the forbidden region, resonant tunneling through double barriers, and the WKB estimate of tunneling rates for smooth barriers.
Core questions
- How can a particle cross a barrier higher than its energy?
- What determines the probability that tunneling occurs?
- How does the tunneling rate depend on the width and height of the barrier?
- When does tunneling become resonant and approach certainty?
Key concepts
- potential barrier
- transmission coefficient
- evanescent wave
- exponential suppression
- resonant tunneling
- WKB approximation
Key theories
- Transmission through a barrier
- Matching the oscillating wavefunctions outside a barrier to the exponentially decaying solution inside gives a transmission coefficient that is small but nonzero, falling off exponentially with the product of barrier width and the decay rate set by its height.
- WKB tunneling estimate
- For a smooth, slowly varying barrier the tunneling probability is approximated by an exponential of minus twice the integral of the local decay rate across the forbidden region, the formula Gamow used to explain the enormous range of nuclear decay lifetimes.
Clinical relevance
Tunneling is the operating principle behind major technologies and natural processes: the scanning tunneling microscope images atoms by measuring a tunneling current, tunnel and resonant-tunneling diodes exploit it for fast electronics, flash memory relies on it, and it governs nuclear alpha decay and fusion in stars.
History
Tunneling was recognized soon after the Schrodinger equation; Hund found it in molecular models and Gamow used it in 1928 to explain alpha decay, while Binnig and Rohrer turned it into the scanning tunneling microscope in 1981, earning the Nobel Prize.
Key figures
- George Gamow
- Friedrich Hund
- Gerd Binnig
- Heinrich Rohrer
Related topics
Seminal works
- griffiths2018
- landau1977
Frequently asked questions
- Does tunneling violate conservation of energy?
- No; the particle has the same energy before and after, and energy is never measured to exceed the barrier height inside it. The effect arises because a quantum particle does not have a definite trajectory or a sharply defined energy localized in the barrier region.
- Why is tunneling so sensitive to barrier width?
- The wavefunction decays exponentially inside the barrier, so the transmitted amplitude falls off exponentially with width; even a small increase in barrier thickness can reduce the tunneling probability by orders of magnitude, which is why the scanning tunneling microscope is so precise.