2.4 Quantum tunneling

Quantum tunneling is a mind-bending concept where particles can pass through barriers they shouldn't be able to. It's all thanks to the wave-like nature of particles in quantum mechanics, which gives them a tiny chance of popping up on the other side.

This weird behavior is super important in the quantum world. It explains stuff like how atoms decay and how some electronic devices work. Quantum tunneling shows us just how different things are at the tiniest scales compared to what we're used to.

Quantum Tunneling

Wave-Particle Duality and Barrier Penetration

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• Quantum tunneling enables particles to pass through potential barriers classically impossible to overcome
• Wave-like nature of particles in quantum mechanics creates non-zero probability of finding particles beyond potential barriers
• Uncertainty principle underpins tunneling phenomenon
  • Prevents simultaneous precise knowledge of particle position and momentum
• Tunneling probability decreases exponentially with:
  • Increasing barrier width
  • Increasing barrier height
  • Increasing particle mass
• Challenges classical particle behavior notions
• Demonstrates probabilistic nature of quantum mechanics

Scale and Significance

• Tunneling effects become significant at atomic and subatomic scales
• Negligible impact for macroscopic objects
• Important implications for various fields
  • Nuclear physics (radioactive decay)
  • Solid-state physics (semiconductor devices)
  • Quantum computing (quantum bits)

Transmission Probability Calculation

Mathematical Framework

• Transmission probability quantifies likelihood of successful particle tunneling through potential barrier
• Time-independent Schrödinger equation derives transmission probability for given potential barrier
• Rectangular potential barrier calculations use:
  • WKB approximation
  • Exact solution of Schrödinger equation
• Transmission coefficient T defined as ratio of transmitted wave amplitude to incident wave amplitude squared
• Reflection coefficient R relates to transmission coefficient
  • R + T = 1 (conservation of probability)

Approximations and Applications

• Thin barriers approximated using Gamow factor
  • Depends on barrier height and width
• Tunneling current in devices (tunnel diodes) calculated using:
  • Transmission probability
  • Density of states of materials involved

Factors Influencing Tunneling

Barrier Properties

• Tunneling probability inversely proportional to exponential of square root of barrier height
  • Higher barriers result in lower tunneling probabilities
• Barrier width significantly impacts tunneling probability
  • Wider barriers lead to exponentially lower transmission probabilities
• Barrier shape affects tunneling probability
  • Rectangular, triangular, or parabolic shapes require different mathematical approaches
• Multi-barrier systems can exhibit resonant tunneling
  • Energy levels in adjacent potential wells align
  • Leads to enhanced transmission probabilities

Particle Characteristics and External Factors

• Incident particle energy relative to barrier height affects tunneling probability
  • Particles closer to barrier top have higher transmission probabilities
• Particle mass influences tunneling likelihood
  • Lighter particles have higher probability of tunneling through given barrier
• External factors modify effective barrier properties
  • Applied electric fields
  • Applied magnetic fields

Applications of Quantum Tunneling

Microscopy and Electronics

• Scanning Tunneling Microscopy (STM) images surfaces at atomic scale
  • Measures tunneling current between sharp tip and sample surface
• Tunnel diodes operate based on quantum tunneling
  • Semiconductor device with heavily doped p-n junction
  • Produces negative differential resistance
• Flash memory devices use quantum tunneling for data operations
  • Writing and erasing data by moving electrons through thin insulating layers

Nuclear and Superconductor Physics

• Alpha decay in radioactive nuclei explained by quantum tunneling
  • Alpha particles tunnel through potential barrier of nucleus
• Josephson effect in superconductors relies on tunneling
  • Cooper pairs tunnel through thin insulating barrier
  • Used in SQUID magnetometers and voltage standards
• Nuclear fusion reactions in stars facilitated by quantum tunneling
  • Allows fusion to occur at lower temperatures than classically predicted

Advanced Scientific Applications

• Tunneling ionization in strong-field physics explains atomic ionization by intense laser fields
  • Applications in attosecond science
  • Applications in high-harmonic generation