The Quantum tunneling effect is the first quantum effect encountered when learning quantum physics (not discussing the uncertainty principle and quantization assumption). It does not occur under classical conditions. The quantum tunneling effect means that the quantum can traverse a potential barrier with higher energy than its own and appear on the other side of the barrier. A common metaphor is that someone can pass through a wall unscathed without breaking the wall.

The quantum tunneling effect is everywhere and plays a critical role in the functioning of the world—for example, the decay of atomic nuclei and the nuclear fusion in the sun. Diodes and Josephson junctions in electronic technology are also inseparable from the quantum tunneling effect.

The quantum mechanical understanding of tunneling is that the wave function does not vanish in the barrier but penetrates and declines. As long as the barrier is not infinitely high, the wave function does not diminish to zero, so there is a certain chance of appearing outside the barrier.

Another way of understanding is the principle of uncertainty between energy and time. The shorter the time, the greater the energy uncertainty, and there is a certain probability of crossing the potential barrier.

The mathematical formulation of quantum tunneling is the WKB approximation: the wave function in and outside the potential barrier is fitted by the wave and decay modes, making the wave function continuous at the border. Thus, we have two wave amplitudes inside and outside the potential barrier and conclude that quantum can pass through the barrier.

There is a slight problem here: First, we must assume that the quantum is inside the barrier, but this solution works both inside and outside. Moreover, if the space outside is large enough, like in free space, then quantum can actually only appear outside the barrier. In other words, the odds of finding the quantum inside and in the barrier are close to zero. Suppose the external space is limited, such as dividing an infinitely deep potential well into two parts and placing a potential barrier in the middle. In that case, the final result should be that the amplitudes of the wave functions on both sides are the same because tunneling occurs in both directions, and eventually, the two sides must be balanced. In this case, the WKB method is not self-consistent. The reason here is that the solution of the Schrödinger equation is a steady-state problem (stationary state) or an equilibrium problem, and the solution wave function is stationary. However, quantum tunneling is a dynamic process that is an initial value evolution problem.

Of course, the tunneling effect can still describe problems such as nuclear decay well and give quantitative analysis.

The full name of the WKB method is WKB quasi-classical approximation. The three letters come from the three scientists who proposed the method (Wenzel, Kramers, Brillouin). It is a quasi-classical method, so the purity of the quantum is a little flawed.

How does the global approximation interpretation (GAI) make sense of this "pure" quantum phenomenon?

Let us start with a quasi-classical example: the evaporation of a liquid. Suppose there is some liquid in a sealed container, such as a water bottle. The water will evaporate into the space, and the water vapor will also condense on the water surface or other surfaces to form a gas-liquid equilibrium. At this time, the pressure formed by the gas molecules is called the saturated vapor pressure, which is generally only related to the system temperature. The higher the temperature, the higher the saturated vapor pressure.

There are intermolecular forces between water molecules, which can be van der Waals forces or hydrogen bonds (whether the force or bond are from quantum effects is not discussed here), so the water molecules need some energy to escape from the surface and evaporate into the space. The molecules have to cross the surface energy barrier. Since the system is a thermodynamic one with a temperature, the velocity or energy of gas molecules satisfies the Boltzmann distribution at the temperature. There must be a certain amount of water molecules on the surface of the liquid whose energy exceeds the constraint of intermolecular binding and can escape away from the surface. Of course, the water molecules in the air will also be captured by other water molecules on the water surface again to achieve equilibrium. The system's temperature determines the density (or pressure) of water molecules in the air. The higher the temperature, the more molecules with high energy, the more molecules in the gaseous state, and the greater the pressure in the gaseous state.

Liquid evaporation can be understood as a purely kinetic or thermodynamic process and does not require quantum theory such as the energy-time uncertainty principle. However, it can also be fully explained by quantum tunneling and calculated quantitatively. Gas-liquid equilibria can be modeled entirely with classical molecular dynamics methods, where each molecule has a definite position and a velocity.

At the beginning of the last century, when nuclear fission just came into sight, in order to describe the fission of the nucleus and calculate the binding energy of the nucleus, the earliest model was the nuclear droplet model. The decay of atomic nuclei, such as alpha decay, can also be understood in the liquid-gas transition. Nuclei with large atomic numbers can be regarded as droplets composed of alpha particles (helium 4 nuclei) (such as in the interaction boson model, IBM). Many alpha particles near the surface are bound by the nuclear force and repulsed by the Coulomb force. If there is also a complex dynamic process in the atomic nucleus, the energy of the alpha particle will also satisfy the Boltzmann distribution to a certain extent. There will always be a particular moment when an alpha particle gets enough energy to overcome the constraints of the nuclear force and be pushed away from the nucleus by the Coulomb force. This process, like the evaporation of water molecules, is deterministic, not probabilistic, but there is no problem with probabilistic descriptions in many cases. The same is valid for nuclear decay. If the internal dynamics are known, the decay occurrence is determined. For many identical nuclei, it is impossible to determine which nucleus will decay without knowing the internal dynamics, but we can describe it with probability. The probability of decay can manifest in the intensity of the radiation or the length of the half-life.

So in GAI, all quantum tunneling effects may have an actual dynamic process. The process itself is deterministic, but it cannot be measured, or there is no need to measure it (because we always use many nuclei). We can describe it in terms of probability.

When discussing dynamics, we used the concept of classical particle motion, but if the particle is not a classical particle, the principle is the same. The state of the next moment is calculable, although it may be too complicated.

The principle of thermodynamics tells us that as long as there is temperature or free energy, including the free energy of the internal degrees of freedom of the particle, the free energy will change dynamically in different degrees of freedom. If a particular change can cause its macroscopic state to change, the change will occur. The dynamic process is inherently complex in many degree-of-freedom systems, such as liquid, atomic nucleus, or even any single particle (consider the infinite number of order of the Feynman diagram in the Standard Model). Each state will obtain various particular states through ergodication. They happen naturally like evaporation, condensation, decay, and fusion. The probability description can be phenomenologically compared with experiments, so it cannot be proved wrong, but it is not necessarily the actual process.

In conclusion, quantum random events can come from dynamic (thermodynamic) processes rather than random quantum fluctuations that do not require a reason or energy lifetime uncertainty.