Quantum Tunnelling as Resonant Transmission

An RQP Artifact exploring the phenomenon of barrier penetration through the lens of wave mechanics and resonance

Lead Researcher: Richard Alexander Tune

Core Thesis

The phenomenon of quantum tunnelling, while often presented as one of the more counter-intuitive aspects of quantum mechanics, is fundamentally a manifestation of wave mechanics and can be understood through the lens of resonance. Our core thesis is that quantum phenomena can be demystified by seeing them as consequences of resonance within a system defined by the Schrödinger equation.

Viewing tunnelling through the lens of resonance strips away the need for explanations based on "quantum weirdness." It is not a particle magically teleporting. It is a wave phenomenon, governed by the same principles of propagation, attenuation, and interference that govern all waves.

The transmission of the wave through the barrier is a predictable outcome of the system's response to the incident wave, and this response is maximized under conditions that can be precisely defined as resonant. Tunnelling, therefore, is not an exception to the rules of physics, but a confirmation of the universal, wave-like nature of matter and the central role of resonance in its interactions.

Keywords

Quantum TunnellingResonant TransmissionWave MechanicsSchrödinger EquationPotential BarrierFabry-Pérot ResonatorMathematical Formalism

Mathematical Formalism

The Time-Independent Schrödinger Equation (TISE) for a rectangular potential barrier demonstrates that the particle's wavefunction decays exponentially within the barrier but does not vanish, allowing for a non-zero transmission probability.

Potential Function

V(x) = 0 for x < 0 (Region I)

V(x) = V₀ for 0 ≤ x ≤ L (Region II)

V(x) = 0 for x > L (Region III)

Time-Independent Schrödinger Equation

-ℏ²/(2m) d²ψ(x)/dx² + V(x)ψ(x) = Eψ(x)

Wavefunctions in Each Region

Region I:

ψ₁(x) = A e^(ik₁x) + B e^(-ik₁x)

where k₁ = √(2mE)/ℏ

Region II:

ψ₂(x) = C e^(k₂x) + D e^(-k₂x)

where k₂ = √(2m(V₀ - E))/ℏ

Region III:

ψ₃(x) = F e^(ik₁x)

Transmission Coefficient

T = |F|²/|A|² = 1 / [1 + V₀² sinh²(k₂L) / (4E(V₀ - E))]

Approximation for E ≪ V₀:

T ≈ [16E(V₀ - E)/V₀²] e^(-2k₂L)

Python Simulation

This Python script calculates and plots the transmission coefficient, clearly showing the exponential decay for energies below the barrier height and resonant behavior above it.

quantum_tunneling.py

import numpy as np
import matplotlib.pyplot as plt

def transmission_coefficient(E, V0, L, m):
    """
    Calculates the transmission coefficient for a particle
    tunnelling through a rectangular barrier.
    """
    if E == V0:
        return 1.0 # Avoid division by zero, classical limit
    hbar = 1.0 # Use atomic units for simplicity
    k2_squared = 2 * m * (V0 - E) / hbar**2
    if k2_squared < 0:
        # Energy is above the barrier, perfect transmission
        return 1.0
    k2 = np.sqrt(k2_squared)
    numerator = V0**2 * np.sinh(k2 * L)**2
    denominator = 4 * E * (V0 - E)
    T = 1.0 / (1.0 + numerator / denominator)
    return T

# --- Simulation Parameters ---
m = 1.0          # Mass of the particle (e.g., electron)
V0 = 10.0        # Height of the potential barrier
L = 1.0          # Width of the barrier

# --- Energy Range ---
energies = np.linspace(0.1, 2 * V0, 500)
transmission_probs = [transmission_coefficient(E, V0, L, m)
                      for E in energies]

# --- Plotting ---
plt.figure(figsize=(10, 6))
plt.plot(energies / V0, transmission_probs,
         label='Transmission Coefficient (T)')
plt.axvline(x=1, color='r', linestyle='--',
            label='Barrier Height (V0)')
plt.xlabel('Energy / Barrier Height (E/V0)')
plt.ylabel('Transmission Probability (T)')
plt.title('Quantum Tunnelling: Transmission Coefficient vs. Energy')
plt.grid(True)
plt.legend()
plt.show()

Wave Mechanics Perspective

Incident Wave

→

Potential Barrier

Exponential Decay

→

Transmitted Wave

"Quantum tunnelling is not quantum weirdness—it is wave mechanics in action. The particle doesn't magically teleport through the barrier; the wave propagates through it, attenuated but persistent, governed by the same resonance principles that shape all wave phenomena."

— RQP Research Archives