Quantum Tunnelling and Applications: A Guide for Physics Undergraduates

Quantum tunnelling is one of the most counter-intuitive yet fundamental phenomena in modern physics. It describes the process where a particle, such as an electron, passes through a potential energy barrier that it classically lacks the energy to surmount. For an undergraduate physicist, mastering this topic is essential, as it underpins everything from the nuclear fusion powering our Sun to the operation of modern flash memory.

In this article, we will move beyond the qualitative "ball rolling over a hill" analogy. We will derive the transmission probability using the Schrödinger equation and explore how this phenomenon manifests in alpha decay and scanning tunnelling microscopy (STM). Understanding these concepts is vital for your exams, as they test your ability to apply wave mechanics to constrained systems.

The Wave Nature of Matter and Barrier Penetration

In classical mechanics, if a particle with energy $E$ encounters a potential barrier of height $U_0$ where $E < U_0$, the particle is reflected with 100% probability. In quantum mechanics, however, the particle is described by a wavefunction $\psi(x)$. The time-independent Schrödinger equation is given by:

$$-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$$

Inside the barrier ($0 < x < L$), where $U(x) = U_0$, the equation becomes:

$$\frac{d^2\psi}{dx^2} = \frac{2m(U_0 - E)}{\hbar^2} \psi$$

Let $\kappa^2 = \frac{2m(U_0 - E)}{\hbar^2}$. The solution inside the barrier is an exponential decay: $\psi(x) = Ae^{-\kappa x} + Be^{\kappa x}$. Because the wavefunction does not vanish at the barrier boundary, there is a non-zero probability of finding the particle on the other side.

Calculating the Transmission Coefficient

For a rectangular barrier of width $L$, the transmission coefficient $T$ (the probability of the particle passing through) is approximately given by:

$$T \approx e^{-2\kappa L}$$

where $\kappa = \sqrt{2m(U_0 - E)}/\hbar$. This shows that $T$ is extremely sensitive to both the width of the barrier and the energy deficit $(U_0 - E)$.

Worked Example 1: Electron Tunnelling

An electron ($m = 9.11 \times 10^{-31}$ kg) with energy $E = 2.0$ eV approaches a barrier of height $U_0 = 5.0$ eV and width $L = 0.2$ nm. Calculate the transmission probability.

1. Convert units: $U_0 - E = 3.0$ eV $= 3.0 \times 1.602 \times 10^{-19}$ J $= 4.806 \times 10^{-19}$ J.
2. Calculate $\kappa$: $\kappa = \sqrt{2(9.11 \times 10^{-31})(4.806 \times 10^{-19})} / (1.055 \times 10^{-34}) \approx 8.87 \times 10^9$ m$^{-1}$.
3. Calculate $2\kappa L$: $2(8.87 \times 10^9)(0.2 \times 10^{-9}) \approx 3.55$.
4. $T \approx e^{-3.55} \approx 0.0287$ or 2.87%.

Alpha Decay and Nuclear Physics

Alpha decay is a classic application of tunnelling. An alpha particle is trapped inside a nucleus by the strong nuclear force, creating a potential well. To escape, the particle must tunnel through the Coulomb barrier created by the electrostatic repulsion of the remaining nucleus.

Worked Example 2: Alpha Particle Escape

If an alpha particle ($m \approx 6.64 \times 10^{-27}$ kg) faces a barrier where the effective $\kappa L$ product is 20, calculate the transmission probability.

1. $T \approx e^{-2(20)} = e^{-40}$.
2. $T \approx 4.25 \times 10^{-18}$.

This extremely low probability explains why some radioactive isotopes have half-lives of billions of years; the particle must attempt to tunnel many times before a successful escape occurs.

Scanning Tunnelling Microscopy (STM)

STM uses a sharp metallic tip brought very close to a conducting surface. A bias voltage is applied, and electrons tunnel across the vacuum gap. Because $T$ depends exponentially on the gap width $L$, even a tiny change in the surface topography causes a massive change in the tunnelling current. This allows for atomic-scale imaging.

Common Mistakes

1. Energy Loss: Students often incorrectly assume the particle loses energy during tunnelling. The particle's energy $E$ remains constant; only the probability amplitude changes.
2. Classical Analogy: Do not treat the particle as a classical object that "climbs" the barrier. It is a wave phenomenon.
3. Ignoring the Pre-factor: While $T \approx e^{-2\kappa L}$ is a common approximation, remember that for very thin or low barriers, the full transmission formula involving trigonometric functions is required.

FAQ

Does tunnelling violate the conservation of energy? No. The particle's total energy $E$ is conserved throughout the process.

Can macroscopic objects tunnel? In theory, yes, but the probability is so infinitesimally small (due to the large mass $m$ in the exponent) that it is effectively zero.

Why is tunnelling important for the Sun? Protons in the Sun's core do not have enough thermal energy to overcome their mutual Coulomb repulsion; they must tunnel to fuse.

Conclusion

Quantum tunnelling is a cornerstone of quantum mechanics that challenges our classical intuition. By mastering the exponential dependence of the transmission coefficient, you are well-equipped to handle advanced topics in solid-state physics and nuclear theory. To visualise these wavefunctions in motion, generate a free animated lesson on this topic at MathInstructor AI.