Quantum tunneling is a phenomenon in which a particle has a non-zero probability of crossing an energy barrier even when its energy is lower than the barrier height. While it is true that energy must be conserved overall, quantum mechanics allows for temporary violations of energy conservation at small timescales due to the uncertainty principle.
In the case of an electron escaping from a solid, it encounters a potential barrier created by the attractive forces between the electron and the solid's atoms or ions. Classically, the electron would need to possess energy higher than the barrier height in order to escape. However, in quantum mechanics, the electron's position and momentum are described by wavefunctions, which represent probabilities rather than definite values.
When the electron encounters the potential barrier, its wavefunction extends into the barrier region. According to quantum mechanics, the wavefunction does not abruptly drop to zero at the barrier, but instead it decays exponentially within the barrier region. This means that there is a non-zero probability of finding the electron beyond the barrier, even though its energy is lower than the barrier height.
The phenomenon of quantum tunneling allows the electron to pass through the barrier despite not having enough energy to overcome it classically. This occurs because the wavefunction of the electron extends into the barrier, allowing it to "tunnel" through the barrier and emerge on the other side. The probability of tunneling depends on various factors, such as the barrier height, the width of the barrier, and the mass of the particle.
It's important to note that while the electron can tunnel through the barrier, its total energy must still be conserved. In the process of tunneling, the electron's kinetic energy may decrease, but this is compensated by other changes in the system, such as the potential energy of the electron. The overall energy conservation is satisfied, even though at a given instant, the energy of the electron may be lower than the barrier height.
In summary, quantum tunneling allows for the non-zero probability of an electron to escape from a solid, even when its energy is lower than the potential barrier. This is due to the probabilistic nature of wavefunctions in quantum mechanics and the temporary violations of energy conservation allowed by the uncertainty principle.