The reason the probability of quantum tunneling never goes completely down to zero in quantum mechanics is rooted in the fundamental principles of the theory. It is not solely due to the mathematical framework, but rather a consequence of the nature of quantum particles and the wave-particle duality.
Quantum mechanics describes particles as both particles and waves simultaneously. When a particle encounters a potential barrier that it classically does not have enough energy to overcome, there is still a finite probability that it can "tunnel" through the barrier and appear on the other side. This phenomenon arises from the wave nature of particles and is a consequence of the Heisenberg uncertainty principle.
According to the uncertainty principle, there is always an inherent uncertainty in the position and momentum of a quantum particle. Even when a particle is classically trapped by a potential energy barrier, there is a small but non-zero probability that it can be found on the other side of the barrier due to this uncertainty. The wave function of the particle extends into the classically forbidden region, allowing for the possibility of tunneling.
Mathematically, quantum tunneling is described by solving the Schrödinger equation, which is the fundamental equation of quantum mechanics. The solutions of the Schrödinger equation yield wave functions that extend into the barrier region and exhibit exponential decay, which represents the tunneling probability. The finite nature of the barrier and the exponential decay of the wave function ensure that there is always a non-zero probability of tunneling, even though it may be exceedingly small for barriers of practical significance.
In summary, the non-zero probability of quantum tunneling is not solely a result of the mathematical framework of quantum mechanics, but rather a consequence of the inherent wave-particle duality of quantum particles and the uncertainty principle. The mathematical framework of quantum mechanics provides a rigorous description and prediction of this phenomenon.