Tunneling Time Of The Triangle Barrier

Quantum tunneling is a quantum mechanical phenomenon,and the question of how long it takes a particle to get across a barrier still remains controversial.This paper discusses this problem,and briefly introduces several conventionally defined tunneling times in the literature,namely phase time,dwell time and Larmor time.Starting from a simple model,we calculate the tunneling time of a pair of mirror-symmetric right angle triangle barriers.According to whether the left slope of the hypotenuse of a right angle triangle barrier is positive or negative,we name the triangular barrier as the positive or negative slope triangle barrier(PSTB or NSTB)for short.The transmission tunneling time for a pair of mirror-symmetric PSTB and NSTB is equal.The difference between phase time and dwell time is the main reason for the oscillation phenomenon,namely the self-interference delay time,which can be affected by the shape of the barrier.At the same time,we define the tunneling velocity,so that we can more intuitively observe whether the Hartman effect is present or not in the PSTB and NSTB like the rectangle barrier.By comparing different tunneling times,we find that when the energy of incident particles is low,the reflected Larmor time is equal to the dwell time;when the energy of the incident particle is high,the transmitted Larmor time is equal to the dwell time,and the reflected Larmor time is equal to the reflected phase time,accompanied by the obvious oscillation phenomenon caused by interference.Finally,we extend the above special shapes of right angle triangular barriers to a general triangle barrier,which have two acute angles between the two sides of the barrier with the horizontal axis.The conclusion of the right angle triangle barrier is further verified by the discussion of the general triangle barrier.Compared with the above asymmetric triangular barrier,for the symmetric triangular barrier,both the transmitted Larmor time and the reflected Larmor time are equal to the dwell time.When the energy of the incident particle is higher,the reflected phase time is also equal to the dwell time.