31.7 Tunneling

Quantum tunneling is a mind-bending phenomenon where particles sneak through barriers they shouldn't be able to cross. This defies classical physics and plays a crucial role in nuclear processes like radioactive decay and fusion in stars.

Unlike classical mechanics, quantum mechanics allows particles to pass through potential barriers, even when their energy is lower than the barrier height. This occurs due to the wave-like nature of particles, with a non-zero probability of finding them on the other side of thin barriers.

Quantum Tunneling

Concept of quantum tunneling

• Quantum tunneling is a phenomenon where particles pass through potential barriers that they classically should not be able to overcome
• Purely quantum mechanical effect with no classical analog
• Plays a crucial role in nuclear physics:
  • Alpha decay: Alpha particles (helium nuclei) tunnel through the potential barrier of the nucleus, causing radioactive decay
  • Nuclear fusion: Protons tunnel through the Coulomb barrier to fuse with other nuclei, occurring in stars and being researched for fusion energy

Particle penetration of potential barriers

• In classical mechanics, a particle cannot pass through a potential barrier if its energy is less than the barrier height
• In quantum mechanics, particles exhibit wave-like properties, and their position is described by a wave function $\Psi(x)$ representing the probability amplitude of finding the particle at a given position
• When a particle encounters a potential barrier, its wave function does not abruptly drop to zero at the barrier but decays exponentially within the barrier region (evanescent wave)
• If the barrier is thin enough, there is a non-zero probability that the particle will be found on the other side of the barrier, depending on the barrier height, width, and the particle's energy

Classical vs quantum tunneling predictions

• Classical mechanics:
  1. A particle with energy $E$ cannot pass through a potential barrier of height $V$ if $E < V$
  2. The particle will always be reflected back from the barrier
• Quantum mechanics:
  1. A particle with energy $E$ has a non-zero probability of passing through a potential barrier even if $E < V$
  2. The tunneling probability depends on the barrier height, width, and the particle's energy
  3. The tunneling probability is given by the transmission coefficient $T$, calculated using the WKB approximation:
     • $T = e^{-2\int_{x_1}^{x_2} \sqrt{\frac{2m}{\hbar^2}(V(x) - E)} dx}$
     • $m$ is the particle's mass, $\hbar$ is the reduced Planck's constant, and $x_1$ and $x_2$ are the classical turning points where $V(x) = E$
• The existence of quantum tunneling demonstrates the limitations of classical mechanics and the need for a quantum mechanical description of particles at the atomic and subatomic scales (electrons, protons, alpha particles)

Fundamental principles of quantum mechanics related to tunneling

• Wave-particle duality: Particles can exhibit both wave-like and particle-like properties, which is essential for understanding tunneling behavior
• Uncertainty principle: The impossibility of simultaneously knowing a particle's exact position and momentum, which allows for the possibility of tunneling
• Quantum superposition: Particles can exist in multiple states simultaneously, contributing to the probability of tunneling through barriers