Yes, according to quantum mechanics, there is a non-zero probability for an electron to tunnel through a potential barrier that is higher than its classical energy. This phenomenon is known as quantum tunneling.
In classical physics, if an object does not have enough energy to overcome a potential barrier, it would be completely reflected by the barrier. However, in quantum mechanics, particles such as electrons can exhibit wave-like behavior and can exist in a superposition of states.
When an electron encounters a potential barrier, there is a finite probability that it can tunnel through the barrier and appear on the other side, even though it doesn't have enough energy to classically overcome it. This probability is determined by the wave function of the electron and the characteristics of the barrier.
The wave function of the electron describes its probability distribution, and it extends both before and after the barrier. Although the probability of tunneling decreases exponentially with the height and width of the barrier, it is not zero.
In experimental setups, this phenomenon has been observed in various systems, such as electron tunneling through thin insulating barriers in electronic devices. It is an important concept in quantum mechanics and plays a crucial role in understanding the behavior of particles in the presence of potential barriers.