When an electron and a proton of the same kinetic energy encounter a potential barrier of the same height and width, which one of them will tunnel through the barrier more easily? Why?

Quantum tunneling is a phenomenon where a particle can tunnel through a potential barrier even if its energy is lower than the barrier's height. This happens because particles in quantum mechanics are described by wave functions, which can extend beyond the barrier, and there is a small but non-zero probability of the particle being found on the other side of the barrier.

The tunneling probability depends on several factors, such as the height and width of the potential barrier, as well as the mass and energy of the particle. The formula for tunneling probability is given by: \(P = e^{-2KL}\), where \(P\) is the tunneling probability, \(K\) is a constant that depends on the particle's energy, and \(L\) is the width of the potential barrier. The constant \(K\) is given by: \(K = \frac{\sqrt{2m(V-E)}}{\hbar}\), where \(m\) is the mass of the particle, \(V\) is the height of the potential barrier, \(E\) is the kinetic energy of the particle, and \(\hbar\) is the reduced Planck constant.

Given that the electron and proton have the same kinetic energy and encounter potential barriers of the same height and width, we can compare their tunneling probabilities by analyzing the constant \(K\). The larger the value of \(K\), the smaller the tunneling probability. From the formula of the constant \(K\), we can observe that \(K\) is inversely proportional to the square root of the mass of the particle. The mass of a proton is approximately 1836 times heavier than the mass of an electron. Therefore, the value of \(K\) for a proton would be smaller than that for an electron: \(K_p < K_e\) Since \(K_p < K_e\), according to the formula for tunneling probability, the tunneling probability for a proton will be larger than that for an electron: \(P_p > P_e\)

Based on our analysis, when an electron and a proton with the same kinetic energy encounter potential barriers of the same height and width, the proton will tunnel through the barrier more easily than the electron due to the proton's larger mass.