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
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.
\[ V(x) = \begin{cases} 0 & \text{for } x < 0 \quad (\text{Region I}) \\ V_0 & \text{for } 0 \le x \le L \quad (\text{Region II}) \\ 0 & \text{for } x > L \quad (\text{Region III}) \end{cases} \] -\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x) \psi_I(x) = A e^{ik_1 x} + B e^{-ik_1 x} \quad \text{where } k_1 = \frac{\sqrt{2mE}}{\hbar} \psi_{II}(x) = C e^{k_2 x} + D e^{-k_2 x} \quad \text{where } k_2 = \frac{\sqrt{2m(V_0 - E)}}{\hbar} \psi_{III}(x) = F e^{ik_1 x} T = \frac{|F|^2}{|A|^2} = \frac{1}{1 + \frac{V_0^2 \sinh^2(k_2 L)}{4E(V_0 - E)}} T \approx \frac{16 E (V_0 - E)}{V_0^2} e^{-2k_2 L}
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 in this model
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()