According to quantum mechanics, the probability for quantum tunneling to completely eliminate a true bound state is indeed zero. In quantum mechanics, bound states refer to states in which a particle is confined within a potential energy well.
Quantum tunneling occurs when a particle encounters a potential barrier that classically would be impossible to surmount due to insufficient energy. However, in quantum mechanics, there is a non-zero probability for the particle to "tunnel" through the barrier and appear on the other side, even if its energy is lower than the height of the barrier.
For a true bound state, the particle is already localized within the potential well. Quantum tunneling, in this case, refers to the possibility of the particle escaping from the bound state by tunneling through the potential barriers surrounding the well. However, the probability of tunneling completely eliminating the bound state is zero.
This is because a true bound state is characterized by a well-defined energy level and wave function within the potential well. The wave function describes the behavior and probability distribution of the particle. While tunneling allows for some probability of finding the particle outside the well, the majority of the wave function remains localized within the well due to the confinement provided by the potential. As a result, the bound state persists, albeit with a small leakage of probability outside the well.
In summary, the probability for quantum tunneling to completely eliminate a true bound state is negligible but not zero. The bound state retains a high probability density within the potential well, despite the possibility of tunneling outside the well.