0,x?0??0?x?a?0.05nm?U1,U??U2,a?x?2a??0,2a?x?
在每个区写出薛定鄂方程
d?dx222????2?????2E??0,x?00?x?ad?2?2
dx22(E?U1)??0,(E?U2)??0,E??0,2a?xd?dx222?2
a?x?2ad?dx22?2有题给E设k0??1eV?U2?2eV?U1?5eV
k2?2?(U2?E)?22?E?2,k1?2?(U1?E)?2,
得到解为
??Ae??Ce??Fe??Keik0xk1xk2xik0x?Be?De?Ge,?ik0x,x?00?x?aa?x?2a(此区域只有透射波(向右的))?k1x?k2x,,
2a?x用波函数连续及导数连续的条件把K和A联系起来, 透射几率为 在x=0处
A?B?C?D,ik0(A?B)?k1(C?D)
T?KA22
在x?a处
?A??(1?k1/ik0)/2?????B??(1?k1/ik0)/2(1?k1/ik0)/2??C???? (1?k1/ik0)/2??D?
Cek1a?De?k1a?Fek2a?Ge?k2a,k1(Cek1a?De?k1a)?k2(Fek2a?Ge?k2a)
/2??F???? /2??G? 在x?2a处
Fe?C??(1?k2/k1)e21/2????(k2?k1)a/2?D??(1?k2/k1)e(k?k)a(1?k2/k1)e(1?k2/k1)e?(k1?k2)a?(k2?k1)a2k2a?Ge?2k2a?Kei2k0a,k2(Fe2k2a?Ge?2k2a)?ik0Kei2k0a
(ik?k)2a?F??(1?ik0/k2)e02/2????0?G??0(1?ik0/k2)e(ik0?k2)2a??K????/2??0?
所以
?A??(1?k1/ik0)/2?????B??(1?k1/ik0)/2?(1?ik0/k2)e??0?(ik0?k2)2a(1?k1/ik0)/2??(1?k2/k1)e(k2?k1)a/2??(1?k1/ik0)/2??(1?k2/k1)e(k2?k1)a/2/20(1?ik0/k2)e(ik0?k2)2a(1?k2/k1)e(1?k2/k1)e?(k1?k2)a?(k2?k1)a/2??/2???K????/2??0?(1?k2/k1)e(1?k2/k1)e?(k1?k2)a?(k2?k1)a
?A??(1?k1/ik0)/2?????B??(1?k1/ik0)/2?(1?ik0/k2)e??0?(ik0?k2)2a(1?k1/ik0)/2??(1?k2/k1)e(k2?k1)a/2??(1?k1/ik0)/2??(1?k2/k1)e(k2?k1)a/2K/2???/2??/2?
?A??(1?k1/ik0)/2?????B??(1?k1/ik0)/2(1?k1/ik0)/2??(1?k2/k1)e(k2?k1)a(1?ik0/k2)e(ik0?k2)2aK/4????(1?k1/ik0)/2??(1?k2/k1)e(k2?k1)a(1?ik0/k2)e(ik0?k2)2aK/4?
(ik?k)2a(ik(k?k)a(k?k)a?A??(1?k1/ik0)(1?k2/k1)e21(1?ik0/k2)e02K/8?(1?k1/ik0)(1?k2/k1)e21(1?ik0/k2)e????(ik?k)2a(ik(k2?k1)a(k?k)a(1?ik0/k2)e02K/8?(1?k1/ik0)(1?k2/k1)e21(1?ik0/k2)e?B??(1?k1/ik0)(1?k2/k1)e
A?K8(1?ik0/k1)eik0ae?k2a?(1?k1/ik0)(1?k2/k1)e?k1a?(1?k1/ik0)(1?k2/k1)ek1a???
透射系数
T?KA?22?64(1?k0/k1)e2?2k2a22?2k2a1?(1?k1/ik0)(1?k2/k1)e?k1a?(1?k1/ik0)(1?k2/k1)ek1a??(1?k1/ik0)(1?k2/k1)???1/k)(1?k/k)?(1?k/k)?(1?k/k)e222222?2k1a64(1?k/k)e2?2(1?k2?(1?k/k)e22k1a??01211210?21代入
k2?E2?(U1?E)U2?E)0??2,k1??2,k2?2?(?2
E?1eV,U1?5eV,U2?2eV
k0/k1?1/2,k2/k1?1/2
2k2aT?64?4e15?9
?2k???92?5(4e1a?14e2k1a)???k2?(U21?E)E)1??2?2?a(U1??2a2?1U1?Ea13.6057EV
?1414a13.6057?5.29177?10?1113.6057/m?10/nm
k2?1U2?Ea13.6eV?5/nm
(a是波尔半径,
?22?a2?13.6057eV是氢原子基态能的值)
T?64?4e2k2a1?505??40?e
??9?5(9e?2k1a12ka??24?4e1)??
21?