第三章能级跃迁光与原子共振相互作用一、二能级近似二、 相干迭加与统计混合三、微扰含时Schrodinger方程Hy=ina/atA = Ho(r) +(t)W = wo +ow,Wo = (E2 - Ei)/h
第三章 能级跃迁 光与原子共振相互作用 一、二能级近似 二、相干迭加与统计混合 三、微扰 含时Schrodinger方程 𝐻̰𝛹 = 𝑖ℏ𝜕 𝛹Τ𝜕 𝑡 𝐻̰ = 𝐻̰ 0(𝑟) + 𝑉̰(𝑡) 𝜔 = 𝜔0 + δω, 𝜔0 = (𝐸2 − 𝐸1 )Τℏ
Ho=inai/atΦ;(r,t) = Φi(r)exp(-iE;t/h) (i = 1,2)Ho(r)Φ;(r) = EiΦi(r)(i = 1,2)c (t);(r)exp(-iE; t/h)Y(r,t) =Ci(t)b1(r)e-iE;t/h + C2(t)2(r)e-iEzt/hHy=inay/atH = Ho(r) +(t)(ci(t)V11 + C2(t)V12e-inot)c(t) =Vj =(ilV(t)li)= / iV(t);d3ri(Gi(t)V21e-iot + C2(t)V22)C2(t) 方
𝐻̰ 0𝜓𝑖 = 𝑖ℏ𝜕 𝜓𝑖Τ𝜕 𝑡 𝜓𝑖 (𝑟,𝑡) = 𝜓𝑖 (𝑟)exp(−iE𝑖 𝑡Τℏ) 𝑖 = 1,2 𝐻̰ 0(𝑟)𝜓𝑖 (𝑟) = 𝐸𝑖𝜓𝑖 (𝑟) 𝑖 = 1,2 𝛹(𝑟,𝑡) = 𝑖 𝑐𝑖 (𝑡)𝜓𝑖 (𝑟)exp(−iE𝑖 𝑡Τℏ) 𝑐1(𝑡)𝜓1(𝑟)𝑒 −iE1𝑡Τℏ + 𝑐2(𝑡)𝜓2(𝑟)𝑒 −iE2𝑡Τℏ . 𝐻̰𝛹 = 𝑖ℏ𝜕 𝛹Τ𝜕 𝑡 𝐻̰ = 𝐻̰ 0(𝑟) + 𝑉̰(𝑡) 𝑐Ǘ1(𝑡) = − 𝑖 ℏ (𝑐1(𝑡)𝑉11 + 𝑐2(𝑡)𝑉12𝑒 −𝑖𝜔0𝑡 ) 𝑐Ǘ2(𝑡) = − 𝑖 ℏ (𝑐1(𝑡)𝑉21𝑒 −𝑖𝜔0𝑡 + 𝑐2(𝑡)𝑉22) 𝑉ij = ⟨𝑖|𝑉̰(𝑡)|𝑗⟩ = න 𝜓 𝑖 𝑉(𝑡)𝜓𝑗𝑑 3 𝑟
(t) = er · E(t)E(t) = (Eo, 0,0)coswt(t) = exEocoswtexEoiwto-iwt2exEti xb;d3r2Eoint +e-iwt)ij121(ci(t)V11 + C2(t)V12e-iwot)c(t)hEo12(ei(-)t + e-i(a+)t)c2(t)=12hEoμ12((e-i(-wo)t + ei(w+wo)t)ci(t)C2(t)2h
𝑉̰(𝑡) = er ⋅ 𝐸(𝑡) 𝐸(𝑡) = (𝐸0, 0,0)cos𝜔𝑡 𝑉̰(𝑡) = exE0cos𝜔𝑡 = exE0 2 (𝑒 𝑖𝜔𝑡 + 𝑒 −𝑖𝜔𝑡) 𝑉ij(𝑡) = exE0 2 (𝑒 𝑖𝜔𝑡 + 𝑒 −𝑖𝜔𝑡)න 𝜓𝑖 𝑥𝜓𝑗𝑑 3 𝑟 = − 𝐸0 2 (𝑒 𝑖𝜔𝑡 + 𝑒 −𝑖𝜔𝑡)𝜇ij 𝑐Ǘ1(𝑡) = − 𝑖 ℏ (𝑐1(𝑡)𝑉11 + 𝑐2(𝑡)𝑉12𝑒 −𝑖𝜔0𝑡 ) = 𝑖 𝐸0𝜇12 2ℏ 𝑒 𝑖(𝜔−𝜔0)𝑡 + 𝑒 −𝑖(𝜔+𝜔0)𝑡 𝑐2(𝑡) 𝑐Ǘ2(𝑡) = 𝑖 𝐸0𝜇12 2ℏ 𝑒 −𝑖(𝜔−𝜔0)𝑡 + 𝑒 𝑖(𝜔+𝜔0)𝑡 𝑐1(𝑡)
Ci(t) =r(ei(w-o)t + e-i(a+o)t)c2(t)c2(t) =r(e-i(w-wo)t +ei(w+wo)t)ci(t)The weak-field limit:ci(t) = 02r(e-i(α-wo)t + ei(w+wo)t)c2(t) =-iowt-1ei(w+o)t - 1~QR2-iowi(w + wo)sin0wt/2OR1c2(t)/220w/22Rt2C2(t)2
𝑐Ǘ1(𝑡) = 𝑖 2 𝛺𝑅 𝑒 𝑖(𝜔−𝜔0)𝑡 + 𝑒 −𝑖(𝜔+𝜔0)𝑡 𝑐2(𝑡) 𝑐Ǘ2(𝑡) = 𝑖 2 𝛺𝑅 𝑒 −𝑖(𝜔−𝜔0)𝑡 + 𝑒 𝑖(𝜔+𝜔0)𝑡 𝑐1(𝑡) The weak-field limit: 𝑐Ǘ1(𝑡) = 0 𝑐Ǘ2(𝑡) = 𝑖 2 𝛺𝑅 𝑒 −𝑖(𝜔−𝜔0)𝑡 + 𝑒 𝑖(𝜔+𝜔0)𝑡 𝑐2(𝑡) = 𝑖 2 𝛺𝑅 𝑒 −𝑖δω𝑡 − 1 −𝑖δω − 𝑒 𝑖(𝜔+𝜔0)𝑡 − 1 𝑖(𝜔 + 𝜔0) 𝑐2(𝑡) 2 = 𝛺𝑅 2 2 sinδω𝑡Τ2 δωΤ2 2 𝑐2(𝑡) 2 = 𝛺𝑅 2 2 𝑡 2
1u(w)dw02sin(w - o)t/2μ12dac2(t)/2u(w)260h(w - wo)/2Tcohzui2u(wo)t元2uB1230212
1 2 𝜀0𝐸0 2 = න 𝑢(𝜔)𝑑𝜔 𝑐2(𝑡) 2 = 𝜇 12 2 2𝜀0ℏ න 𝑢(𝜔) sin(𝜔 − 𝜔0) 𝑡Τ2 (𝜔 − 𝜔0 )Τ2 𝑑𝜔 = 𝜋 𝜀0ℏ2 𝜇12 2 𝑢(𝜔0)𝑡 𝐵12 𝜔 = 𝜋 3𝜀0ℏ2 𝜇12 2