7.7 Quantum Tunneling and Applications

Quantum tunneling is a mind-bending concept where particles can pass through barriers they shouldn't. It's like a magic trick in the quantum world, defying our everyday logic and showing how weird things get at the tiniest scales.

This topic dives into the math behind tunneling, its applications in cool tech like scanning microscopes, and even how it powers stars. It's a perfect example of how quantum mechanics breaks the rules of classical physics.

Quantum Tunneling and its Probabilistic Nature

Wave-Particle Duality and Barrier Penetration

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• Quantum tunneling allows particles to penetrate and pass through potential energy barriers they classically should not overcome
• Wave-like nature of particles in quantum mechanics creates non-zero probability of finding particles beyond potential barriers, even with insufficient energy
• Tunneling stems from Heisenberg uncertainty principle limiting precision of simultaneous position and momentum measurements
• Probability of tunneling decreases exponentially with increasing barrier width, height, and particle mass
• Barrier shape affects tunneling probability (smoother barriers generally allow higher probabilities)

Mathematical Description and Quantum-Classical Divide

• Time-independent Schrödinger equation calculates particle wavefunction near potential barrier yielding tunneling probability
• Tunneling has no classical analog highlighting fundamental differences between classical and quantum physics
• Wavefunction collapse upon measurement explains probabilistic nature of tunneling outcomes

Calculating Transmission Probability and Tunneling Current

Transmission Probability for Simple Barriers

• Transmission probability T for rectangular potential barrier given by $T ≈ e^{-2kL}$
  • k represents wave number inside barrier
  • L denotes barrier width
• Wave number k relates to particle mass m, energy E, and barrier height V0: $k = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}}$
  • ℏ symbolizes reduced Planck constant
• WKB (Wentzel-Kramers-Brillouin) approximation calculates transmission probability for non-rectangular barriers

Tunneling Current and Advanced Calculations

• Tunneling current I proportional to transmission probability: $I = I_0 \times T$
  • I0 represents incident current
• Scanning tunneling microscopy current exponentially depends on tip-sample distance d: $I \propto e^{-2\kappa d}$
  • κ denotes decay constant
• Gamow factor calculates tunneling probabilities in alpha decay and nuclear fusion reactions
• Numerical methods (transfer matrix method) accurately calculate transmission probabilities for complex barrier shapes

Quantum Tunneling in Devices

Scanning Tunneling Microscopy (STM)

• STM images surfaces at atomic scale by measuring tunneling current between sharp tip and sample surface
• Exponential dependence of tunneling current on tip-sample distance enables precise control and measurement of tip position
• Atomic-resolution imaging achieved through quantum tunneling phenomenon

Tunnel Diodes and Resonant Tunneling Devices

• Tunnel diodes exploit quantum tunneling to achieve negative differential resistance
• Current-voltage characteristics exhibit region where increased voltage decreases current due to energy state alignment across junction
• Resonant tunneling diodes (RTDs) use quantum wells to create discrete energy levels for selective tunneling
• Josephson junctions rely on Cooper pair tunneling to generate and detect high-frequency electromagnetic radiation

Memory and Superconducting Devices

• Flash memory devices use quantum tunneling for writing and erasing data by moving electrons through potential barrier to/from floating gate
• Superconducting quantum interference devices (SQUIDs) utilize Josephson tunneling for ultra-sensitive magnetic field detection

Quantum Tunneling in Radioactive Decay and Nuclear Fusion

Alpha Decay and Nuclear Stability

• Alpha particles tunnel through potential barrier created by strong nuclear force and Coulomb repulsion during alpha decay
• Gamow theory explains relationship between decay constant and emitted alpha particle energy using quantum tunneling
• Half-lives of radioactive nuclei directly relate to tunneling probability (higher barriers generally yield longer half-lives)

Stellar Nucleosynthesis and Fusion Processes

• Quantum tunneling facilitates nuclear fusion in stars, allowing protons to overcome Coulomb barrier at lower temperatures than classically predicted
• Tunneling probability in nuclear fusion strongly depends on colliding nuclei kinetic energy, explaining temperature dependence of stellar core fusion rates
• Proton-proton chain reaction, primary energy source in main sequence stars like Sun, relies crucially on quantum tunneling
• Cold fusion theories propose enhanced tunneling probabilities to explain hypothetical room-temperature fusion reactions (controversial and unproven)