QMI_09.dvi

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1 V 0 > 0 V ) = 0 < a) V 0 a a ) 0 a<) 6.1) a 0 V ) a V 0 6.1: < a 1 a a 2a< 3 1 incident wave reflected wave transmitted wave E E >0 Ct) Ct) = e iωt ω = Ē h 6.2) ω

2 m 1 k d 2 u) = Eu) 6.3) d2 2mE k = N 6.4) ψ in, t) = NCt) e ik = Ne ik ωt) < a ) 6.5) 1 R ψ ref, t) = RCt) e ik = Re ik+ωt) < a ) 6.6) ) u) k 3 ψ trans, t) = TCt) e ik = Te ik ωt) a<) 6.7) T m d 2 d 2 u) = E V 0 ) u) 6.8) E >V 0 E<V E>V 0 K = 2mE V 0 ) 6.9) ψ barr, t) = Ct)Ae ik + Be ik ) = Ae ik ωt) + Be ik+ωt) a a ) 6.10)

3 ), 6.6), 6.7) [ ] ψ 1, t) = e iωt u 1 ) = e iωt Ne ik + Re ik < a ) [ ] ψ 2, t) = e iωt u 2 ) = e iωt Ae ik + Be ik a a ) 6.11) [ ] ψ 3, t) = e iωt u 3 ) = e iωt Te ik a<) N N = a Ne ika + Re ika = Ae ika + Be ika ik Ne ika Re ika ) = ik Ae ika Be ika ) 6.12) = a Ae ika + Be ika = Te ika ik Ae ika Be ika ) = ik T e ika 6.13) 6.12) K ) k 1 K k Ae ika + ) Ae ika + 1 K ) Be ika = 2Ne ika k 1+ K ) 6.14) Be ika = 2Re ika k 6.13) 2 3 A = k ) Te ik K)a, B = 1 1 k ) Te ik+k)a 6.15) K 2 K 6.15) 6.14) A B R T N T N = 4kK e 2ik K)a k + K) 2 k K) 2 e 4iKa, R N = k2 K 2 )1 e 4iKa )e 2ika k + K) 2 k K) 2 e 4iKa 6.16) E<V 0 κ = 2mV 0 E) 6.17)

4 66 6 E >V 0 ik κ ψ 1 ψ 3 [ ] ψ 1, t) = e iωt u 1 ) = e iωt Ne ik + Re ik < a ) [ ] ψ 2, t) = e iωt u 2 ) = e iωt Ae κ + Be κ a a ) 6.18) [ ] ψ 3, t) = e iωt u 3 ) = e iωt Te ik a<) = a = a Ne ika + Re ika = Ae κa + Be κa ik Ne ika Re ika ) = κ Ae κa Be κa ) 6.19) 6.19) 6.20) 1+ κ ) ik 1 κ ik Ae κa + Be κa = Te ika κ Ae κa Be κa ) = ik T e ika 6.20) Ae κa + ) Ae κa + 1 κ ) Be κa = 2Ne ika ik 1+ κ ) 6.21) Be κa = 2Re ika ik A = ik ) Te ika κa, B = 1 1 ik ) Te ika+κa 6.22) κ 2 κ 6.22) 6.21) A B R T N T N = 4ikκ e 2ika+2κa k iκ) 2 k + iκ) 2 e 4κa, R N = k2 + κ 2 )1 e 4κa )e 2ika k iκ) 2 k + iκ) 2 e 4κa 6.23) ) flu ψ, t) j, t) j, t) = 1 ψ ψ ) i 2m ψ ψ 6.24) 1 ψ 1 j = 1 [ e iωt N e ik + R e ik) e iωt ik) Ne ik Re ik) i 2m e iωt ik) N e ik R e ik) e iωt Ne ik + Re ik)] = k N 2 R 2) = v N 2 R 2) m

5 v = p/m =k/m 3 ψ v j in = v N 2, j ref = v R 2, j trans = v T ) j I T = trans j in = T 2 j N, I R = ref j in = R 2 N. 6.26) 6.16) 6.23) E V 0 E<V 0 I T = I R = 4EV 0 E) 4EV 0 E)+V0 2 sinh2 κb 4EE V 0 ) 4EE V 0 )+V0 2 sin2 Kb V0 2 sinh2 κb V 0 E)+V0 2 sinh2 κb V0 2 sin2 Kb 4EE V 0 )+V0 2 sin2 Kb E<V 0 ) E V 0 ) E<V 0 ) E V 0 ) 6.27) 6.28) b =2a sinh κb sin Kb sinh κb = e+κb e κb 2mV 0 E) κ =, 6.29) 2 sin Kb = e+ikb e ikb 2mE V 0 ) K =, 6.30) 2i I T + I R = )

6 ) E/V E = V ) 2mV 0 b 2 / 2 4, 16, 36 E<V 0 0 tunnel effect b =2a E V 0 sinh 2 κb κ 2 b 2 = 2mV 0 E)b 2 2 I T 4EV 0 E) 4EV 0 E)+V0 2 2mV 0 E)b mV 0 b2 2 2mV 0 b 2 / 2 =4E = V 0 I T = I T mV 0 b 2 h E / V 0 6.2: E>V 0 I T =1 I T =1 0 sin Kb =0 Kb = nπ n =1, 2, ) 6.32) b =2a λ =2π/K π

7 E nπ) 2 = 1+ V 0 2mV 0 b 2 / 2 n =1, 2, ) 6.33) - Ramsauer-Townsend effect 6.3 E/V 0 =0.5 < a E = V 0 E/V 0 =1.5, 2.0 >a < a E >V a <<a 6.3: V 0 2a 2mV 0 2a) 2 / =64 E/V 0 =0.5, 1.0, 1.5, 2.0

8 V 0 > 0 V ) = 0 < a) V 0 a a ) 6.34) 0 a<) < a 1 a a 2a < 3 1 V ) a 0 a V 0 6.4: E mE + V 0 ) K = 6.35) V ) 6.27) V 0 I T = I R = 4EE + V 0 ) 4EE + V 0 )+V 2 0 sin2 Kb V 2 0 sin2 Kb 4EE + V 0 )+V 2 0 sin2 Kb 6.36) 6.37) 6.5 sin Kb =0 Kb = nπ n =1, 2, ) 6.38) 6.35) 2mV 0 b 2 ) 1+ EV0 = nπ) ) 2mV 0 b 2 / =4 4 <π 2 n =1 2mV 0 b 2 / =16 π 2 < 16 < 2π) 2

9 I T mV 0 b 2 h E / V 0 6.5: n =2 2mV 0 b 2 / =36 π 2 < 36 < 2π) 2 n =2 2mV 0 b 2 / =64 2π) 2 < 64 < 3π) 2 n =3 E/V 0 2mV 0 b 2 / a) E 0 V 0 <E<0 V 0 a V 0 a b) E<0 E 0 a) V 0 b) 0 0 V 0 6.6: E E 6.7 a <<a)

10 : V 0 2a 2mV 0 2a) 2 / =64 E/V 0 =2.0, 1.0, 0.5, 0.2

11 Ct) =e iet/ : 2mV 0 2a) 2 / =4 E = V 0

12

QMI_10.dvi

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4.6 (E i = ε, ε + ) T Z F Z = e βε + e β(ε+ ) = e βε (1 + e β ) F = kt log Z = kt log[e βε (1 + e β )] = ε kt ln(1 + e β ) (4.18) F (T ) S = T = k = k

4.6 (E i = ε, ε + ) T Z F Z = e βε + e β(ε+ ) = e βε (1 + e β ) F = kt log Z = kt log[e βε (1 + e β )] = ε kt ln(1 + e β ) (4.18) F (T ) S = T = k = k 4.6 (E i = ε, ε + ) T Z F Z = e ε + e (ε+ ) = e ε ( + e ) F = kt log Z = kt loge ε ( + e ) = ε kt ln( + e ) (4.8) F (T ) S = T = k = k ln( + e ) + kt e + e kt 2 + e ln( + e ) + kt (4.20) /kt T 0 = /k (4.20)

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