[Physics] Where does particle borrow energy from to tunnel

Question

Where does particle borrow energy from to tunnel? It is implied that particle can borrow energy and leaped over to the other side wherever that is, the shorter the gap the more energy it borrows my question is where does a particle borrow its energy from and what criteria allows it to do the borrowing, last but not least how does the particle return the borrowed energy? Is it linked to quantum foam?

Answer 1

One should always remember that quantum mechanics predicts probabilities and not energy distributions . The energy a particle will have is an eigenvalue of the energy operator operating on the wavefunction, but the probability of finding a particle at (x,y,z) at time t is given by the complex conjugate square of the wavefunction which is the solution of the quantum mechanical equations with the boundary values imposed.

It is worth reading this link, and generally searching hyperphysics for such questions. "Hyperphysics tunneling" brings up this:

According to classical physics, a particle of energy E less than the height U0 of a barrier could not penetrate - the region inside the barrier is classically forbidden. But the wavefunction associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. The wavefunction must also be continuous on the far side of the barrier, so there is a finite probability that the particle will tunnel through the barrier.

So the energy level on which the particle resides is the same inside and outside the barrier. It is the probability of finding it outside that is smaller than the probability inside.