What specific parameters of a potential barrier most significantly influence the probability of quantum tunneling, and how are they related?

The probability of quantum tunneling is most significantly influenced by the height and width of the potential barrier. The height of the barrier is the difference between the potential energy of the barrier and the total energy of the particle. The wider and taller the barrier is relative to the particle's energy, the lower the probability of tunneling. Specifically, the tunneling probability decreases exponentially with both the barrier width and the square root of the barrier height. This relationship is generally described by the following approximate equation: T ≈ exp(-2√(2m(V-E))W/ħ), where T is the tunneling probability, m is the mass of the particle, V is the potential energy of the barrier, E is the energy of the particle, W is the width of the barrier, and ħ is the reduced Planck constant. This equation shows that even a small increase in barrier width or barrier height can substantially reduce the tunneling probability. The mass of the particle also plays a role; heavier particles have a lower tunneling probability compared to lighter particles, assuming other parameters are the same. The energy of the particle relative to the barrier height is also a key factor; a particle with energy closer to the barrier height has a higher tunneling probability. Therefore, the barrier height, barrier width, and particle mass are the dominant parameters determining the likelihood of tunneling through a potential barrier.