In the mathematical framework of quantum mechanics, both the particle mass and the height of the energy barrier play important roles in determining the quantum tunneling probability. However, the quantitative effect of each factor can vary depending on the specific scenario.
The phenomenon of quantum tunneling occurs when a particle has a non-zero probability of crossing an energy barrier even though it does not possess enough energy to overcome the barrier classically. The probability of tunneling is governed by the wave function of the particle and the characteristics of the barrier.
The mass of the particle influences the de Broglie wavelength associated with its motion. According to the de Broglie wavelength, the larger the mass of a particle, the smaller its associated wavelength. As a result, particles with larger masses have shorter wavelengths, making it more difficult for them to tunnel through barriers. Therefore, for particles of different masses, the mass can have a quantitatively significant effect on the tunneling probability.
On the other hand, the height of the energy barrier is also a crucial factor. A higher energy barrier presents a greater challenge for particles to tunnel through. As the height of the barrier increases, the probability of tunneling decreases. Therefore, the height of the energy barrier can have a quantitatively significant effect on the tunneling probability.
In summary, both the particle mass and the height of the energy barrier contribute to the quantum tunneling probability, but their quantitative effects depend on the specific scenario and the relative magnitudes of these factors.