The probability for quantum tunneling is not derived through statistical approaches or methods directly. Instead, it arises from the mathematical framework of quantum mechanics itself, which incorporates the probabilistic nature of quantum phenomena.
In quantum mechanics, particles are described by wavefunctions, which are mathematical functions that assign probabilities to different states or positions of a particle. When a particle encounters a potential barrier, such as an energy barrier or a classically forbidden region, there is a non-zero probability that the particle can tunnel through the barrier and appear on the other side.
The phenomenon of quantum tunneling can be understood by solving the Schrödinger equation, which is the fundamental equation of quantum mechanics. The solutions to this equation provide the wavefunctions that describe the behavior of quantum particles. When a particle encounters a potential barrier, the wavefunction describes a superposition of states where the particle can exist both inside and outside the barrier.
Through the mathematical formalism of quantum mechanics, the wavefunction can be used to calculate the transmission coefficient, which represents the probability that the particle tunnels through the barrier. The transmission coefficient depends on various factors, such as the height and width of the barrier, the energy of the particle, and the shape of the potential.
It's important to note that while the probabilities in quantum mechanics have a statistical interpretation, they are not derived from statistical methods in the classical sense. Instead, the probabilities arise from the fundamental principles and mathematical formalism of quantum mechanics, which involve complex numbers, wavefunctions, and the superposition of states.