31.7 Tunneling

Quantum Tunneling

Quantum tunneling describes how particles can pass through energy barriers that classical physics says they cannot cross. Understanding tunneling is essential for explaining nuclear phenomena like alpha decay and stellar fusion, where particles overcome barriers despite not having "enough" energy in the classical sense.

Quantum Tunneling

Concept of Quantum Tunneling

Quantum tunneling is a purely quantum mechanical effect with no classical analog. A particle encounters a potential energy barrier higher than its own kinetic energy, yet still has a chance of appearing on the other side. This happens because quantum particles don't behave like tiny billiard balls; they behave like waves, and those waves don't stop abruptly at a barrier.

Two nuclear processes depend directly on tunneling:

• Alpha decay: An alpha particle (a helium-4 nucleus) is trapped inside a larger nucleus by the nuclear potential barrier. Classically it lacks the energy to escape, but tunneling gives it a small probability of passing through. That's why alpha decay happens spontaneously but often with very long half-lives.
• Nuclear fusion in stars: Protons in a star's core repel each other through the Coulomb (electrostatic) barrier. Their thermal energies alone aren't high enough to overcome that repulsion. Tunneling allows protons to "leak" through the barrier and fuse, which is how stars like the Sun produce energy.

Particle Penetration of Potential Barriers

In classical mechanics, if a particle's kinetic energy $E$ is less than the barrier height $V$, the particle bounces back every time. End of story.

Quantum mechanics tells a different story. A particle's position is described by a wave function $\Psi(x)$, which gives the probability amplitude of finding the particle at position $x$. When this wave function reaches a potential barrier, it doesn't drop to zero at the boundary. Instead, it decays exponentially inside the barrier. This exponentially decaying portion is sometimes called an evanescent wave.

If the barrier is thin enough, the wave function still has a non-zero value on the far side. That means there's a real (though often small) probability of detecting the particle beyond the barrier. The probability depends on three things:

• The height of the barrier ($V - E$, how much the barrier exceeds the particle's energy)
• The width of the barrier (thinner barriers mean higher tunneling probability)
• The mass of the particle (lighter particles tunnel more easily)

Classical vs. Quantum Predictions

Classical prediction:

1. A particle with energy $E$ hitting a barrier of height $V$ where $E < V$ is always reflected.
2. There is zero probability of the particle appearing on the other side.

Quantum prediction:

1. A particle with energy $E < V$ has a non-zero probability of passing through the barrier.
2. That probability is quantified by the transmission coefficient $T$.
3. Using the WKB approximation, $T$ is:

$T = e^{-2\int_{x_1}^{x_2} \sqrt{\frac{2m}{\hbar^2}(V(x) - E)} \, dx}$

where $m$ is the particle's mass, $\hbar$ is the reduced Planck's constant, and $x_1$ and $x_2$ are the classical turning points (the positions where $V(x) = E$).

For a simple rectangular barrier of constant height $V$ and width $L$, this simplifies to:

$T \approx e^{-2L\sqrt{\frac{2m(V - E)}{\hbar^2}}}$

Notice the key relationships in this expression: $T$ drops sharply as the barrier gets wider, taller, or as the particle gets heavier. That exponential dependence is why tunneling is significant for lightweight particles (electrons, protons, alpha particles) through thin barriers, but negligible for everyday objects.

Fundamental Principles Behind Tunneling

Three core ideas from quantum mechanics make tunneling possible:

• Wave-particle duality: Particles have wave-like properties described by $\Psi(x)$. It's the wave nature that allows the wave function to extend into and through a barrier, rather than simply reflecting off it.
• Heisenberg uncertainty principle: You cannot know a particle's position and momentum with perfect precision simultaneously. This built-in uncertainty means you can't definitively confine a particle to one side of a thin barrier.
• Quantum superposition: Until a measurement is made, a particle can exist in a combination of "reflected" and "transmitted" states. The probability of each outcome is determined by the wave function's behavior at the barrier.

Together, these principles explain why tunneling is not just a mathematical curiosity but a real, measurable effect that drives processes from nuclear decay to the operation of tunnel diodes and scanning tunneling microscopes.