Quantum tunneling is mostly discussed in the Euclidean path integral formalism using instantons. On the other hand, it is difficult to understand quantum tunneling based on the real-time path integral due to its oscillatory nature, which causes the notorious sign problem. We show that recent development of the Lefschetz thimble method enables us to investigate this issue numerically. In particular, we find that quantum tunneling occurs due to complex trajectories, which are actually observable experimentally by using the so-called weak measurement.
A semiclassical method is applied to the problem of quantum tunneling in the presence of internal oscillator degrees of freedom. Comparison with coupled channel calculations verifies the accuracy of the semi-classical method. Providing the coupling has a physically reasonable form factor, the coupling produces an enhancement in the tunneling rate. The enhancement becomes larger when the tunneling degree of freedom couples with a harmonic oscillator with lower frequency. The semiclassical method is found to be quite useful in situations where the coupled channel method becomes numerically difficult.
We have applied the variational
R-matrix method to calculate the reflection and tunneling probabilities of particles tunneling through one-dimensional potential barriers for five different types of potential profiles -- truncated linear step, truncated exponential step, truncated parabolic, bell-shaped, and Eckart. Our variational results for the transmission and reflection coefficients are compared with exact analytical results and results obtained from other numerical methods. We find that our results are in good agreement with them. We conclude that the variational
R-matrix method is a simple, non-iterative, and effective method to solve one-dimensional quantum tunneling problems.