Potential Step Barrier and Quantum Tunnelling Simple Notes

A potential step is a sudden rise in potential energy at a point, like a particle rolling toward a hill of fixed height. A potential barrier is a raised region of finite width between two lower regions. Classically, a particle with energy below the height can never cross. Quantum mechanics predicts a surprising exception called tunnelling.

Tunnelling matters enormously. It lets particles leak through barriers they classically cannot climb, and this single effect powers flash memory, scanning tunnelling microscopes, and even the nuclear fusion that makes the Sun shine. Understanding steps and barriers is the key to this strange but very real phenomenon.

Key Concepts

• Potential step: V rises from 0 to V₀ at x = 0 and stays there. If particle energy E < V₀, classical reflection is total, but quantum reflection is partial with a decaying wave beyond the step.
• Potential barrier: V = V₀ between x = 0 and x = a, and zero outside. Width a and height V₀ control tunnelling.
• Evanescent wave: inside a classically forbidden region the wave function decays exponentially instead of oscillating and never quite reaches zero for a thin barrier.
• Transmission coefficient T: the probability that the particle emerges on the far side; reflection probability R satisfies R + T = 1.

Formulas

Decay constant in barrier: κ = √[2m(V₀ − E)] / ħ, in units of m⁻¹, valid for E < V₀. The wave inside falls roughly as e^(−κx).

Approximate tunnelling probability (thick barrier): T ≈ e^(−2κa). This assumes a rectangular barrier and κa much larger than one. Doubling the width a makes T fall exponentially, so tunnelling is extremely sensitive to thickness.

Worked Example

An electron approaches a barrier with V₀ − E = 1.0 eV and width a = 0.50 nm. Estimate T.

1. Convert energy: 1.0 eV = 1.602 × 10⁻¹⁹ J.
2. κ = √[2 × 9.11 × 10⁻³¹ × 1.602 × 10⁻¹⁹] / 1.055 × 10⁻³⁴ = √(2.92 × 10⁻⁴⁹) / 1.055 × 10⁻³⁴ = 5.40 × 10⁻²⁵ / 1.055 × 10⁻³⁴ ≈ 5.1 × 10⁹ m⁻¹.
3. 2κa = 2 × 5.1 × 10⁹ × 0.50 × 10⁻⁹ = 5.1.
4. T ≈ e^(−5.1) ≈ 0.006, about 0.6 percent. Roughly six electrons in a thousand tunnel through, while classically none would cross.

Common Misconceptions

• Tunnelling means the particle borrows energy to climb over. Wrong. The particle never has energy above the barrier; its wave simply extends through and emerges with the same energy.
• Tunnelling works through any thickness. Wrong. Probability falls exponentially with width, so a barrier twice as thick can reduce tunnelling by thousands of times.
• A step and a barrier are identical. Wrong. A step is semi-infinite, so no transmission to the far side occurs for E < V₀, whereas a finite barrier allows tunnelling through.

Real-World Applications

Flash memory stores data by tunnelling electrons onto a floating gate. Scanning tunnelling microscopes measure tunnelling current to map single atoms. Tunnel diodes exploit tunnelling for fast switching, and alpha decay of heavy nuclei is explained by the alpha particle tunnelling out through the nuclear barrier.

In short, finite barriers always leak a little, and that exponential leak is one of the most useful effects in all of quantum technology.