Calculation of the Quantum-Mechanical Tunneling in Bound Potentials

Table 1

Energy levels, splittings, and tunneling probability for the 𝑛={0,1} and 𝑛={2,3} pair of states of the symmetric rectangular double-well potential of Section 3.1. The reactant/product states overlap of the two-state description is given in the last column. The energy levels and level splittings are given in 𝐸_ℎ. Numbers in parentheses are the powers of 10.

𝐿 [𝑎0]	𝐸^QM0	Δ^QM	𝑇^QM	𝐸^QC0	Δ^QC	𝑇^QC	𝐸^2s0	Δ^2𝑠	𝑇^2𝑠	⟨𝜙𝑝∣𝜙𝑟⟩
For the 𝑛=0,1 pair of energy states
8	0.8800	1.919 (−9)	1.205 (−17)	0.8800	1.174 (−9)	4.39 (−18)	0.8800	1.913 (−9)	1.17 (−17)	3.554 (−9)
4	0.8800	4.192 (−5)	5.752 (−9)	0.8800	2.564 (−5)	2.095 (−9)	0.8800	4.210 (−5)	5.646 (−9)	4.408 (−5)
2	0.8770	6.196 (−3)	1.257 (−4)	0.8800	3.790 (−3)	4.577 (−5)	0.8769	6.224 (−3)	1.243 (−4)	4.035 (−3)
1	0.8422	0.0753	0.0185	0.8800	4.608 (−3)	6.765 (−3)	0.8425	7.564 (−2)	1.989 (−2)	3.398 (−2)
0.5	0.7433	0.2618	0.2105	0.8800	0.1607	8.225 (−2)	0.7483	0.2637	0.3064	0.0922
0.25	0.6098	0.4881	0.5900	0.8800	0.3000	0.2868	0.6289	0.4906	1.502	0.1476
For the 𝑛=2,3 pair of energy states
8	3.3053	1.099 (−4)	1.481 (−8)	3.3053	1.690 (−4)	6.450 (−9)	3.3053	0.974 (−4)	2.141 (−9)	4.712 (−4)
4	3.2993	1.227 (−2)	1.848 (−4)	3.3053	1.886 (−2)	8.031 (−5)	3.2999	1.087 (−2)	2.676 (−5)	2.790 (−2)
2	3.2477	0.1334	2.144 (−2)	3.3053	0.1992	8.962 (−3)	3.2472	0.1146	3.071 (−3)	0.1644
1	3.1337	0.4856	0.2267	3.3053	0.6474	0.0947	3.0860	0.3487	0.0315	0.3225
0.5	3.0068	1.0078	0.6166	3.3053	1.1672	0.3077	2.8262	0.5329	0.0877	0.3905
0.25	2.9086	1.4864	0.8529	3.3053	1.5672	0.5547	2.5604	0.6101	0.1401	0.3961