Quantum Tunneling Examples

Why This Matters

Quantum tunneling sits at the heart of modern physics—it's the phenomenon that makes stars shine, enables atomic-scale imaging, and powers cutting-edge quantum technologies. When you study tunneling, you're exploring one of the most counterintuitive predictions of quantum mechanics: particles can pass through energy barriers they classically shouldn't be able to cross. This principle connects directly to wave-particle duality, the uncertainty principle, and probability amplitudes—all core concepts you'll be tested on.

Don't just memorize that "tunneling happens in alpha decay" or "STMs use tunneling." You need to understand why tunneling occurs in each case—what's the barrier, what's tunneling through it, and what physical consequence results. The exam will ask you to apply tunneling concepts to unfamiliar scenarios, so focus on the underlying mechanism: the exponential decay of the wavefunction inside a classically forbidden region allows a nonzero probability of transmission.

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Overcoming Nuclear Barriers

These examples demonstrate tunneling through the Coulomb barrier—the electrostatic repulsion between positively charged nuclei. Without tunneling, nuclear reactions would require impossibly high energies or temperatures.

Alpha Decay in Radioactive Nuclei

• Alpha particles tunnel out of the nucleus—despite being bound by the strong nuclear force, the particle's wavefunction extends beyond the Coulomb barrier
• Barrier width determines decay rate—the Geiger-Nuttall law relates half-life to alpha particle energy through tunneling probability
• Transforms the parent nucleus—emission reduces atomic number by 2 and mass number by 4, creating a different element

Nuclear Fusion in Stars

• Protons tunnel through Coulomb repulsion—enabling fusion at temperatures (~15 million K) far below classical requirements (~billions K)
• Powers stellar nucleosynthesis—the pp-chain and CNO cycle both depend critically on tunneling for their first steps
• Gamow peak determines reaction rates—fusion probability balances increasing tunneling probability against decreasing thermal energy distribution

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Electron Tunneling in Materials

These applications exploit electrons tunneling through thin potential barriers in solids or across vacuum gaps. The tunneling current depends exponentially on barrier width, making these systems extraordinarily sensitive.

Scanning Tunneling Microscope (STM)

• Tunneling current maps surface topography—current varies exponentially with tip-surface distance, achieving sub-angstrom vertical resolution
• Probes electronic density of states—the tunneling spectrum reveals information about surface electronic structure, not just geometry
• Enables atomic manipulation—by adjusting tip voltage and position, individual atoms can be moved, enabling nanotechnology breakthroughs

Field Emission

• Electric fields narrow the barrier—applying $\sim 10^9$ V/m reduces the effective barrier width, dramatically increasing tunneling probability
• Fowler-Nordheim equation governs current—tunneling current depends on field strength as $J \propto E^2 \exp(-B/E)$ where $B$ is a material constant
• Enables bright electron sources—used in electron microscopes and flat-panel displays for high-brightness, coherent electron beams

Cold Emission

• Tunneling without thermal excitation—electrons escape at low temperatures purely through field-assisted tunneling, unlike thermionic emission
• Sharp tips enhance local fields—field enhancement at nanoscale features makes cold emission practical at moderate applied voltages
• Critical for vacuum electronics—provides electron sources for devices requiring minimal thermal noise or rapid switching

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Tunneling in Semiconductor and Superconductor Devices

These technologies harness tunneling across engineered thin barriers to achieve unique electronic properties. Device performance depends on precise control of barrier thickness at the nanometer scale.

Tunnel Diodes

• Negative differential resistance from tunneling—current initially increases with voltage, then decreases as tunneling probability drops when band alignment shifts
• Ultrafast switching speeds—tunneling is nearly instantaneous, enabling operation at frequencies exceeding 100 GHz
• Narrow, heavily doped junctions—the depletion region must be thin enough ($\sim$ 10 nm) for significant tunneling probability

Josephson Junctions

• Cooper pairs tunnel coherently—paired electrons maintain phase coherence while tunneling through an insulating barrier ($\sim$ 1-2 nm thick)
• DC Josephson effect—supercurrent flows without applied voltage when phase difference exists across the junction
• AC Josephson effect—applied voltage $V$ produces oscillating current at frequency $f = 2eV/h$, enabling precision voltage standards

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Tunneling in Molecular and Biological Systems

Tunneling influences chemistry and biology at the molecular scale, where barrier widths are naturally on the order of atomic dimensions. These examples show quantum mechanics isn't just for physics labs—it shapes everyday processes.

Ammonia Inversion

• Nitrogen tunnels through the hydrogen plane—the $\text{NH}_3$ molecule oscillates between two pyramidal configurations at ~24 GHz
• Inversion splitting observable spectroscopically—the tunneling rate creates a characteristic doublet in microwave spectra
• First molecular maser—ammonia's inversion transition was used in the first maser (1954), demonstrating quantum coherence in molecules

Quantum Biological Processes

• Exciton tunneling enhances photosynthesis—energy transfers between chlorophyll molecules faster than classical hopping would allow
• Near-unity quantum efficiency—tunneling enables photosynthetic complexes to achieve >95% energy transfer efficiency
• Enzyme catalysis may involve tunneling—proton and hydrogen tunneling could explain anomalously fast reaction rates in some enzymes

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Tunneling at Cosmological Scales

Even gravity and spacetime curvature don't escape quantum tunneling effects—though these remain theoretical frontiers.

Hawking Radiation from Black Holes

• Virtual pairs separate at the event horizon—one particle tunnels outward while its partner falls inward, converting vacuum energy to real radiation
• Black holes have temperature—Hawking temperature $T_H = \frac{\hbar c^3}{8\pi G M k_B}$ decreases with mass, so smaller black holes radiate faster
• Implies black hole evaporation—over cosmological timescales, black holes lose mass and eventually disappear, raising the information paradox

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Quick Reference Table

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Self-Check Questions

1. Both alpha decay and nuclear fusion involve tunneling through Coulomb barriers. What determines whether tunneling occurs into or out of a nucleus, and how does this affect the energy requirements?

2. The STM and tunnel diode both rely on electron tunneling. Compare the role of barrier width in each device—how is it controlled, and why does exponential sensitivity matter?

3. Explain why nuclear fusion can occur in stellar cores at temperatures far below what classical physics predicts. What would happen to stellar lifetimes if tunneling didn't exist?

4. Ammonia inversion and photosynthetic energy transfer both demonstrate tunneling at room temperature. What features of these systems make quantum effects observable despite thermal noise?

5. FRQ-style: A student claims that Josephson junctions and tunnel diodes operate on the same principle. Evaluate this claim by identifying one key similarity and two key differences in their tunneling mechanisms.