Quantum Tunneling: The Particle Through the Wall
How quantum particles can pass through barriers that should be impossible to penetrate according to classical physics
The Quantum Paradox
Imagine throwing a tennis ball at a concrete wall. Classically, if you don't throw it hard enough to break through, it will bounce back. But in quantum mechanics, there's a probability that the ball will simply appear on the other side without breaking the wall! This is quantum tunneling.
Classical vs. Quantum Behavior
A classical particle approaching a potential barrier behaves predictably according to Newtonian physics:
If the particle's kinetic energy is greater than the barrier height, it goes over the barrier.
If the particle's kinetic energy is less than the barrier height, it reflects completely.
The outcome is deterministic: same initial conditions always produce the same result.
No classical particle can be found inside a region where its kinetic energy would be negative.
A quantum particle approaching a potential barrier behaves according to wave mechanics:
Even if E < V₀ (kinetic energy less than barrier), there's a non-zero probability the particle will appear on the other side.
The wavefunction penetrates into the barrier with exponential decay rather than stopping abruptly.
The outcome is probabilistic: we can only calculate probabilities of transmission and reflection.
The particle can be found in "classically forbidden" regions where V(x) > E.
Visualizing Quantum Tunneling
Particle Energy: E < V₀ (Classically forbidden to transmit)
Approach
The quantum particle approaches the barrier as a wave
Penetration
The wavefunction decays exponentially inside the barrier
Emergence
With reduced amplitude, the particle emerges on the other side
Mathematical Derivation of Tunneling
Consider a rectangular potential barrier of height V₀ and width a:
A particle with energy E < V₀ approaches from the left. We solve the time-independent Schrödinger equation in three regions:
Incident wave + reflected wave
Exponentially decaying solutions (E < V₀ so κ is real)
Transmitted wave only (no wave coming from right)
Applying continuity of ψ and dψ/dx at x = 0 and x = a gives four equations. Solving these yields the transmission coefficient T.
Tunneling Probability Formula
For a rectangular barrier with E < V₀, the transmission probability (tunneling probability) is:
where:
Key insights from this formula:
The probability depends exponentially on barrier width (a) and the square root of barrier height (√(V₀ - E)).
Doubling the barrier width decreases tunneling probability by factor e² ≈ 7.4 (for same κ).
As E approaches V₀, tunneling probability increases dramatically.
Interactive Tunneling Factors
The tunneling probability depends critically on these three factors. In general:
- Electrons tunnel more easily than protons (lighter mass)
- Narrower barriers allow more tunneling
- Lower barriers (closer to particle energy) allow more tunneling
Numerical Simulation of Tunneling
Here's Python code to calculate tunneling probabilities for different parameters:
import numpy as np
import matplotlib.pyplot as plt
# Constants
hbar = 1.0545718e-34 # J·s (reduced Planck constant)
eV_to_J = 1.60217662e-19 # Conversion factor
def tunneling_probability(V0, E, a, m):
"""
Calculate tunneling probability through rectangular barrier
V0: Barrier height (in eV)
E: Particle energy (in eV)
a: Barrier width (in meters)
m: Particle mass (in kg)
"""
# Convert to joules
V0_J = V0 * eV_to_J
E_J = E * eV_to_J
if E_J >= V0_J:
# Above barrier: not technically tunneling
k1 = np.sqrt(2*m*E_J)/hbar
k2 = np.sqrt(2*m*(E_J - V0_J))/hbar
# Transmission coefficient for E > V0
T = 1/(1 + ((k1**2 - k2**2)**2)/(4*k1**2*k2**2)*np.sin(k2*a)**2)
else:
# Tunneling regime (E < V0)
k = np.sqrt(2*m*E_J)/hbar
kappa = np.sqrt(2*m*(V0_J - E_J))/hbar
# Exact formula for rectangular barrier
T = 1/(1 + ((k**2 + kappa**2)**2)/(4*k**2*kappa**2)*np.sinh(kappa*a)**2)
return T
# Example: Electron tunneling
m_electron = 9.10938356e-31 # kg
# Vary barrier width
widths = np.linspace(0.1e-9, 2e-9, 100) # 0.1 to 2 nm
V0 = 5.0 # eV
E = 3.0 # eV
probabilities = [tunneling_probability(V0, E, a, m_electron) for a in widths]
# Plot results
plt.figure(figsize=(10, 6))
plt.semilogy(widths*1e9, probabilities, 'b-', linewidth=2)
plt.xlabel('Barrier Width (nm)')
plt.ylabel('Tunneling Probability')
plt.title('Quantum Tunneling: Probability vs Barrier Width')
plt.grid(True, alpha=0.3)
plt.show()
print(f"For a={0.5e-9:.1e} m, V0={V0} eV, E={E} eV:")
print(f"Tunneling probability = {tunneling_probability(V0, E, 0.5e-9, m_electron):.2e}")
Running this simulation shows how tunneling probability decreases exponentially with barrier width, as predicted by the formula T ∝ e^{-2κa}.
Real-World Applications of Tunneling
Uses electron tunneling between a sharp tip and a surface to image individual atoms with resolution down to 0.1 nm. The tunneling current is exponentially sensitive to tip-sample distance.
Electrons tunnel through thin oxide barriers in flash memory cells to program and erase data. This enables the non-volatile memory in USB drives, SSDs, and memory cards.
Protons in the Sun's core tunnel through the Coulomb barrier to fuse into helium. Without tunneling, the Sun wouldn't shine at its current rate, as classical physics gives too low fusion probability.
Alpha particles (helium nuclei) tunnel out of radioactive nuclei like uranium. Gamow's 1928 quantum theory of alpha decay was one of the first applications of tunneling.
Quantum devices with negative differential resistance used in high-frequency oscillators and ultrafast switches for communications and radar systems.
Next-generation transistors that use band-to-band tunneling instead of thermal injection, potentially enabling lower-power electronics.
Experimental Verification
Tunneling was experimentally confirmed through several key experiments:
Fowler and Nordheim explained cold electron emission from metals under strong electric fields as quantum tunneling through a triangular potential barrier.
Leo Esaki observed negative resistance in heavily doped p-n junctions due to electron tunneling, earning him the 1973 Nobel Prize.
Gerd Binnig and Heinrich Rohrer invented the STM, providing direct visual evidence of tunneling and atomic-scale imaging (1986 Nobel Prize).
Experiments with nanostructures demonstrated tunneling of individual electrons, leading to the field of single-electron transistors.
Why Tunneling Seems Counterintuitive
Quantum tunneling seems to violate energy conservation, but it doesn't:
The uncertainty principle ΔE·Δt ≥ ħ/2 allows "borrowing" energy ΔE for short time Δt. During tunneling, energy conservation can be violated for extremely short times.
Particles are described by wavefunctions that don't have sharp boundaries. The exponential tail can extend through barriers, allowing a small probability amplitude on the other side.
Tunneling is purely quantum with no classical counterpart. It results from the wave-like description of matter in quantum mechanics.
While strange from our macroscopic perspective, tunneling is a well-established, experimentally verified phenomenon that forms the basis for many modern technologies.
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