Table of Contents
Overview#
Quantum tunneling is a phenomenon where particles can pass through potential barriers that would be classically forbidden. This has no classical analog and is purely quantum mechanical.
Classical vs Quantum#
Classical Mechanics#
A particle with energy \(E\) encountering barrier \(V_0 > E\):
- Result: Complete reflection
- Probability of transmission: 0
Quantum Mechanics#
The wave function can penetrate the barrier:
- Result: Finite probability of transmission
- Probability: Non-zero (depends on barrier properties)
Mathematical Description#
Setup#
Consider a rectangular barrier:
$$ V(x) = \begin{cases} 0 & x < 0 \\ V_0 & 0 \leq x \leq a \\ 0 & x > a \end{cases} $$Wave Functions#
Region I (x < 0):
$$ \psi_I = Ae^{ikx} + Be^{-ikx} $$Region II (0 ≤ x ≤ a):
$$ \psi_{II} = Ce^{\kappa x} + De^{-\kappa x} $$Region III (x > a):
$$ \psi_{III} = Fe^{ikx} $$Where:
$$ k = \frac{\sqrt{2mE}}{\hbar}, \quad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar} $$Transmission Coefficient#
For thick barriers (\(\kappa a \gg 1\)):
$$ T \approx 16\frac{E}{V_0}\left(1 - \frac{E}{V_0}\right)e^{-2\kappa a} $$General form:
$$ T = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0 - E)}} $$Key Observations#
| Factor | Effect on Tunneling |
|---|---|
| Barrier width ↑ | Transmission ↓ exponentially |
| Barrier height ↑ | Transmission ↓ |
| Particle mass ↑ | Transmission ↓ |
| Particle energy ↑ | Transmission ↑ |
Decay Length#
The wave function decays inside barrier:
$$ |\psi|^2 \propto e^{-2\kappa x} $$Decay length:
$$ \delta = \frac{1}{2\kappa} = \frac{\hbar}{2\sqrt{2m(V_0-E)}} $$WKB Approximation#
For arbitrary barrier shapes:
$$ T \approx e^{-2\gamma} $$Where:
$$ \gamma = \int_{x_1}^{x_2} \frac{\sqrt{2m(V(x) - E)}}{\hbar} dx $$Integration is over the classically forbidden region.
Applications#
1. Scanning Tunneling Microscope (STM)#
Electrons tunnel between tip and surface:
$$ I \propto e^{-2\kappa d} $$- Atomic resolution imaging
- Surface structure analysis
2. Alpha Decay#
Alpha particle tunnels out of nucleus:
$$ \lambda = f \cdot T $$Where \(f\) is attempt frequency and \(T\) is tunneling probability.
3. Tunnel Diodes#
Electrons tunnel through thin barrier:
- Negative resistance region
- High-speed switching
- Microwave applications
4. Josephson Junction#
Cooper pairs tunnel between superconductors:
$$ I = I_c \sin(\phi) $$- SQUID magnetometers
- Quantum computing (qubits)
5. Nuclear Fusion#
Protons overcome Coulomb barrier:
- Powers stars
- Enables fusion reactors
Resonant Tunneling#
For double barriers, resonance occurs at specific energies:
$$ T = 1 \text{ when } E = E_n \text{ (resonance)} $$Used in:
- Resonant tunneling diodes (RTDs)
- Quantum cascade lasers
Time Aspects#
Tunneling Time#
How long does tunneling take?
Various definitions:
- Phase time: \(\tau_\phi = \hbar \frac{\partial \phi}{\partial E}\)
- Dwell time: Time spent in barrier
- Büttiker-Landauer time
Still debated in physics community.