7.7 Quantum Tunneling and Applications

Quantum Tunneling and its Probabilistic Nature

Quantum tunneling is the phenomenon where a particle passes through a potential energy barrier even when its kinetic energy is less than the barrier height. Classical physics says this is impossible, but quantum mechanics allows it because particles behave as waves, and those waves don't abruptly stop at a barrier's edge.

Understanding tunneling matters because it underpins real technology (flash memory, scanning tunneling microscopes) and explains natural processes like radioactive decay and stellar fusion. This section covers the physics behind tunneling, how to calculate tunneling probabilities, and the major applications you need to know.

Wave-Particle Duality and Barrier Penetration

In classical mechanics, a ball rolling toward a hill with insufficient energy simply bounces back. In quantum mechanics, a particle's wavefunction extends into and beyond the barrier region, giving it a non-zero probability of appearing on the other side.

This happens because the wavefunction doesn't drop to zero at the barrier wall. Instead, it decays exponentially inside the barrier. If the barrier is thin enough or low enough, a measurable portion of the wavefunction makes it through.

Several factors control how likely tunneling is:

• Barrier width (L): Wider barriers mean more exponential decay, so tunneling probability drops sharply.
• Barrier height relative to particle energy ($V_0 - E$): The bigger this difference, the faster the wavefunction decays inside the barrier.
• Particle mass (m): Heavier particles have shorter wavelengths and decay faster inside the barrier. This is why tunneling is significant for electrons and protons but negligible for baseballs.
• Barrier shape: Smoother, gradually varying barriers generally allow higher tunneling probabilities than sharp rectangular ones.

The Heisenberg uncertainty principle is connected here: confining a particle's position (trapping it near a barrier) increases the uncertainty in its momentum, which means there's always some probability of the particle having enough momentum to penetrate the barrier.

Mathematical Description and the Quantum-Classical Divide

You find the tunneling probability by solving the time-independent Schrödinger equation in three regions: before the barrier, inside it, and after it. Inside the barrier, where $E < V_0$, the wavefunction doesn't oscillate. It decays exponentially instead. Matching boundary conditions at each interface gives you the transmission coefficient.

Tunneling has no classical analog whatsoever. A classical particle with energy $E < V_0$ is always reflected. The fact that quantum particles can tunnel is one of the starkest demonstrations of how quantum and classical physics diverge.

One clarification: tunneling itself is a property of the wavefunction's evolution. Wavefunction collapse occurs when you make a measurement and find the particle either on one side or the other. Before measurement, the particle exists in a superposition of "reflected" and "transmitted" states.

Calculating Transmission Probability and Tunneling Current

Transmission Probability for Rectangular Barriers

For a rectangular barrier of height $V_0$ and width $L$, with a particle of mass $m$ and energy $E < V_0$, the approximate transmission probability is:

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

where $\kappa$ is the decay constant inside the barrier:

$\kappa = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}}$

Here, $\hbar$ is the reduced Planck constant ($\hbar = h / 2\pi$).

How to use this in practice:

1. Identify the barrier height $V_0$, the particle's energy $E$, its mass $m$, and the barrier width $L$.

2. Calculate $V_0 - E$ (the energy deficit).

3. Plug into the formula for $\kappa$.

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

Because of the exponential, even small changes in $L$ or $V_0 - E$ cause large changes in $T$. Doubling the barrier width doesn't halve the probability; it squares it (on a log scale). This extreme sensitivity is what makes tunneling useful in devices.

For non-rectangular barriers (barriers whose height varies with position), the WKB (Wentzel-Kramers-Brillouin) approximation generalizes the formula:

$T \approx e^{-2\int_{x_1}^{x_2} \kappa(x)\, dx}$

where the integral runs across the classically forbidden region and $\kappa(x)$ varies with the local barrier height.

Tunneling Current and Advanced Calculations

When many particles hit a barrier per unit time, the tunneling current is:

$I = I_0 \times T$

where $I_0$ is the incident current (proportional to the flux of incoming particles).

In scanning tunneling microscopy, the tunneling current between the tip and sample depends exponentially on the gap distance $d$:

$I \propto e^{-2\kappa d}$

A change in $d$ of just ~1 Å (about one atomic diameter) can change the current by roughly an order of magnitude. That's what gives STM its extraordinary sensitivity.

For more complex situations:

• The Gamow factor applies the WKB approach to calculate tunneling probabilities in alpha decay and nuclear fusion, where the barrier is the Coulomb potential.
• The transfer matrix method is a numerical technique that handles arbitrary barrier shapes by breaking them into thin slices and multiplying transmission matrices together.

Quantum Tunneling in Devices

Scanning Tunneling Microscopy (STM)

The STM works by bringing an atomically sharp conducting tip extremely close to a surface (within a few angstroms) and applying a voltage. Electrons tunnel across the vacuum gap between tip and sample, producing a measurable current.

Because that current depends exponentially on the tip-sample distance, even sub-angstrom height variations on the surface produce detectable current changes. By scanning the tip across the surface and recording the current (or adjusting the tip height to keep current constant), you build up an image with true atomic resolution. STM was the first instrument to image individual atoms on a surface.

Tunnel Diodes and Resonant Tunneling Devices

Tunnel diodes are heavily doped p-n junctions where the depletion region is thin enough for electrons to tunnel through. Their key feature is negative differential resistance: there's a voltage range where increasing the voltage actually decreases the current. This happens because the energy levels on opposite sides of the junction move out of alignment as voltage increases, reducing the number of states available for tunneling.

Resonant tunneling diodes (RTDs) take this further. They sandwich a quantum well between two thin barriers, creating discrete energy levels inside the well. Electrons tunnel efficiently only when their energy matches one of these discrete levels. This selective tunneling produces sharp peaks in the current-voltage curve, useful for high-frequency oscillators and fast switching.

Josephson junctions consist of two superconductors separated by a thin insulating barrier. Cooper pairs (paired electrons in a superconductor) tunnel coherently through the barrier. This produces effects used to generate and detect high-frequency electromagnetic radiation, and it forms the basis for superconducting qubits in quantum computing.

Memory and Superconducting Devices

Flash memory uses tunneling to store data. To write or erase a bit, a voltage is applied that causes electrons to tunnel through a thin oxide layer onto (or off of) a floating gate. Once the voltage is removed, the electrons are trapped, and the stored charge persists without power.

SQUIDs (Superconducting Quantum Interference Devices) incorporate Josephson junctions in a superconducting loop. Tiny changes in magnetic flux through the loop alter the tunneling current, making SQUIDs sensitive enough to detect magnetic fields as small as $10^{-15}$ T. They're used in medical brain imaging (MEG) and geological surveys.

Quantum Tunneling in Radioactive Decay and Nuclear Fusion

Alpha Decay and Nuclear Stability

During alpha decay, an alpha particle (two protons and two neutrons) forms inside a nucleus and faces a potential barrier created by two competing forces: the strong nuclear force (attractive, short-range) pulls it in, while Coulomb repulsion (repulsive, long-range) pushes it out. The combined potential creates a barrier that the alpha particle doesn't have enough energy to classically overcome.

Gamow's theory (1928) explained alpha decay as quantum tunneling through this Coulomb barrier. The key prediction: the decay constant (and therefore the half-life) depends exponentially on the alpha particle's energy. Small differences in energy produce enormous differences in half-life. This is known as the Geiger-Nuttall law, and it matches experimental data remarkably well. Nuclei with higher barriers or lower alpha particle energies have longer half-lives, sometimes billions of years.

Stellar Nucleosynthesis and Fusion Processes

Stars fuse light nuclei into heavier ones, but the temperatures at stellar cores aren't high enough for protons to classically overcome their mutual Coulomb repulsion. The core of the Sun is about 15 million K, which gives protons a typical kinetic energy far below the Coulomb barrier height. Fusion happens anyway because of quantum tunneling.

The tunneling probability for fusion depends strongly on the kinetic energy of the colliding nuclei, which is why fusion rates are extremely sensitive to temperature. Even modest temperature increases dramatically boost the tunneling probability and thus the fusion rate.

The proton-proton chain reaction is the dominant energy source in main-sequence stars like the Sun. In the first step, two protons must tunnel through their Coulomb barrier to fuse. This step is actually very unlikely for any given pair of protons, but the sheer number of protons in a stellar core (and the billions of years available) makes it happen often enough to power the star.