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Mathematics for Physics

139 Quantum Tunneling

A process in which a particle has nonzero probability to appear beyond a classically forbidden barrier that is used to explain barrier penetration in quantum mechanics.

Exponential decay in the forbidden region. In a region where \(E<V\), a classical particle cannot enter, but the Schrödinger wavefunction decays exponentially rather than vanishing. This principle is used to assign a nonzero finding probability inside the barrier.

The decay constant in the forbidden region is

\[ \kappa = \dfrac{\sqrt{2m(V-E)}}{\hbar} \]

where

  • \(\kappa\) is the decay constant.
  • \(m\) is the mass.
  • \(V\) is the potential height.
  • \(E\) is the energy.
  • \(\hbar\) is the reduced Planck constant.

Transmission with \(E<V\). A wave incident on a barrier of finite width yields a nonzero transmitted amplitude on the other side. This principle is used to define tunneling as transmission with \(E<V\).

The exponential dependence of \(T\) on width. The transmission probability of a wide barrier falls exponentially with width and with \(\kappa\). This principle is used to estimate tunneling rates.

The wide-barrier transmission is

\[ T \sim e^{-2\kappa L} \]

where

  • \(T\) is the transmission probability.
  • \(L\) is the barrier width.
  • \(\kappa\) is the decay constant.

Note: Also called barrier penetration. Also called tunneling.

139.1 References

  1. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — tunneling as transmission with \(E<V\).
  2. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — nonzero transmission through a barrier.
  3. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — barrier penetration.