Quantum Tunneling

Quantum Tunneling: Penetrating the Impenetrable Barrier

By Md Irfan | 2nd Year Physics (Hons)
Presented on National Science Day – 28th February 2023

Introduction

Quantum mechanics has revolutionized our understanding of nature at its most fundamental level. Among its most intriguing consequences is the phenomenon of quantum tunneling—a process that defies classical intuition and permits particles to traverse energy barriers that should be insurmountable. This principle underlies a broad spectrum of physical phenomena, from nuclear fusion in stars to the operation of nanoscale devices, and has far-reaching implications in both theoretical physics and practical technology.

Theoretical Foundation: Mathematical Framework

Let us examine the situation of a non-relativistic particle of mass m, approaching a potential barrier of height V_0 and width a. Classically, if the energy E < V_0, the particle cannot cross the barrier. However, quantum mechanics allows for a non-zero probability of transmission.

Potential Profile

\[ V(x) = \begin{cases} 0 & \text{for } x < 0 \\ V_0 & \text{for } 0 \le x \le a \\ 0 & \text{for } x > a \end{cases} \]

Schrödinger Equation in Each Region

We solve the time-independent Schrödinger equation:

\[ -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x)\psi = E\psi \]

Solutions in the three regions are:

• Region I (x < 0): \psi_I(x) = Ae^{ikx} + Be^{-ikx}, with k = \frac{\sqrt{2mE}}{\hbar}
• Region II (0 \le x \le a): \psi_{II}(x) = Ce^{\kappa x} + De^{-\kappa x}, with \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}
• Region III (x > a): \psi_{III}(x) = Fe^{ikx}

Continuity of the wavefunction and its derivative at x = 0 and x = a yields a system of equations. Solving this system gives us the transmission coefficient:

Tunneling Probability (Transmission Coefficient)

\[ T = |F|^2/|A|^2 \approx e^{-2\kappa a} = \exp\left(-2a\cdot \frac{\sqrt{2m(V_0 - E)}}{\hbar}\right) \]

This shows that the transmission probability decreases exponentially with the width and height of the barrier, and increases with the energy of the particle. This expression is a cornerstone in the semiclassical approximation known as the WKB (Wentzel-Kramers-Brillouin) method.

Deep Dive into Applications

1. Nuclear Fusion in Stars

The sun and other stars rely on nuclear fusion to generate energy. Protons within the star’s core have thermal energies much lower than the Coulomb barrier between them. However, quantum tunneling allows them to fuse:

\[ p + p \xrightarrow{\text{tunneling}} D + e^+ + \nu_e \]

This reaction initiates the proton-proton chain. Without tunneling, fusion would not occur at observable stellar temperatures. The tunneling probability, known as the Gamow factor, governs the reaction rate:

\[ P \propto \exp\left(-\sqrt{\frac{E_G}{E}}\right),\quad E_G = 2\mu \left(\frac{\pi Z_1 Z_2 e^2}{\hbar}\right)^2 \]

Here \mu is the reduced mass and Z_1, Z_2 are the charges of the nuclei.

2. Alpha Decay in Nuclear Physics

Alpha particles within heavy nuclei are confined by a potential well and a Coulomb barrier. Although classically trapped, they can quantum tunnel out, resulting in radioactive decay. George Gamow first applied quantum tunneling to explain alpha decay, leading to the Geiger–Nuttall law:

\[ \log_{10}(T_{1/2}) \propto \frac{Z}{\sqrt{E_\alpha}} \]

Where T_{1/2} is the half-life and E_\alpha is the energy of the emitted alpha particle. The theory elegantly matches experimental data.

3. Quantum Biology and DNA Mutations

Proton tunneling plays a crucial role in the tautomeric shifts of base pairs in DNA. A proton can tunnel across hydrogen bonds between adenine-thymine or guanine-cytosine, leading to mispairing during replication. This may result in point mutations, which are fundamental to biological evolution and diseases like cancer. Quantum tunneling thus connects the atomic scale to evolutionary biology.

4. Scanning Tunneling Microscope (STM)

Invented in 1981, the STM uses the principle of electron tunneling to image surfaces at the atomic level. A sharp conducting tip is brought close to a conductive sample. When a bias voltage is applied, electrons tunnel across the vacuum gap. The current depends exponentially on distance:

\[ I \propto e^{-2\kappa d}, \quad \kappa = \frac{\sqrt{2m \phi}}{\hbar} \]

Where \phi is the work function and d is the tip-sample separation. Tiny changes in d cause large changes in current, enabling atomic-resolution imaging and the manipulation of individual atoms.

Conclusion

Quantum tunneling defies classical determinism and unlocks new domains of understanding. From the nuclear heart of stars to the frontier of nanotechnology, from the mysteries of life to the decay of elements—it is an omnipresent quantum whisper shaping the fabric of reality. As our control over the quantum world advances, so too will our capacity to harness tunneling for quantum computing, nanoscale engineering, and perhaps interstellar energy generation.

“Where classical mechanics sees a wall, quantum mechanics sees a door—though a narrow one.”
—Scientificirfan