Publication Date:
2016-03-09
Description:
In the discussion of temporary behaviors of quantum tunneling, people usually like to focus their attention on rectangular barrier with steep edges, or to deal with smooth barrier with semi-classical or even numerical calculations. Very few discussions on analytic solutions of tunneling through smooth barrier appear in the literature. In this paper, we provide two such examples, a semi-infinite long barrier V ( x ) = A 2 [ 1 + tanh ( x / a ) ] and a finite barrier V ( x ) = A sech 2 ( x / a ). To each barrier, we calculate the associated phase time and dwell time after obtaining the analytic solution. The results show that, different from rectangular barrier, phase time or dwell time does increase with the length parameter a controlling the effective extension of the barrier. More interestingly, for the finite barrier, phase time or dwell time exhibits a peak in k -space. A detailed analysis shows that this interesting behavior can be attributed to the strange tunneling probability T s ( k ), i.e., T s ( k ) displays a unit step function-like profile Θ( k − k 0 ), especially when a is large, say, a ≫ 1/ κ , 1/ k . And k 0 ≡ m A / ħ is exactly where the peak appears in phase or dwell time k -spectrum. Thus only those particles with k in a very narrow interval around k 0 are capable to dwell in the central region of the barrier sufficiently long.
Print ISSN:
0022-2488
Electronic ISSN:
1089-7658
Topics:
Mathematics
,
Physics
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