Quantum tunneling is a consequence of the probabilistic nature of the wave function. In classical mechanics, we can use energy to understand how an object will move. For example, a skateboarder in a half pipe will have kinetic energy, and potential energy from gravity. The skateboarder starts with a fixed amount of energy which is that from gravity, measured at the top of the curve. The curved walls of the half pipe then, are a potential barrier for the skateboarder because momentum will take him to the other edge, and no further (under ideal conditions). This is because the energy level of the system is lower than or equal to the energy of the potential barrier. Therefore, motion is only defined within the potential valley that is the half pipe. Of course, if the energy level were to be increased, say by pushing, the skateboarder can be carried past the energy barrier and motion will then be defined outside of it.

Quantum systems behave differently from this. If you had a scattering photon with energy level E_1, and it was moving towards a potential barrier with energy level E_2 which is higher than E_1, it turns out that the quantum wave function near the barrier will actually extend through it, with exponential falloff. Of course, the quantum wave function is a probability distribution curve of various observable quantities, like position. That is to say that a quantum particle in motion may have its position defined on the other side of a potential barrier, despite the fact that its energy is too low to overcome it. The probabilistic nature of the wave function means sometimes a particle will reflect, and sometimes it will transmit. This is quantum tunneling.

Tunneling is when the wavefunction of a quantum system is nonzero in a region of space that would be classically forbidden.

In other words, it's when there's a possibility to find a particle in a region of space that would be impossible in classical mechanics.

A particle's location in quantum mechanics isn't precisely defined, it's more of a bell curve. It's most likely to be in the center, but there's a non-zero probability that it could be anywhere in the universe.

This means that if you put an "impassable" barrier somewhere close enough to the center of a particle's bell curve, you end up with a significant chance that the particle could be on either side of the barrier.

This behavior is confined to the quantum scale because in order for it to happen to a macroscopic object, all of its gajillions of particles would have to tunnel at exactly the same time, making the probability functionally zero. Furthermore, any macroscopic barrier would also be far too thick to be within the distance tunneling can realistically be expected.

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