How is quantum tunneling possible?

Question

How is quantum tunneling possible?

According to quantum mechanics, each particle is represented by a probability density function. This function must be continuous, and therefore when we look at a particle near a potential barrier, we deduce that there is a finite probability for finding the particle inside the barrier (and as a result, beyond the barrier). If the particle can be found inside the barrier, his energy will be negative. This state sounds impossible. Where does the extra energy come from?

Answer 1

The instantaneously computed value for the kinetic energy would appear to be negative, as you propose. However, the reason for quantum tunneling has to do with Heisenberg uncertainty relations. You may find a particle in a classically forbidden region, but you are much less likely to find it here. The length of time of any departure from this energy conservation condition is limited by the energy departure, so that \begin{equation} \Delta E \Delta t \ge \hbar/2. \end{equation} The larger the apparent energy violation, the more fleeting the event. The uncertainty principle essentially allows the system to momentarily have enough energy for the particle to be in the forbidden region, with the proviso that it may not do so for very long. (This weirdness is also responsible for the Casimir effect.)