坐标表象中的态矢量和算符

坐标表象中的态矢量和算符xx=ˆxp∂∂−=iˆ=nnnfFϕϕ||ˆ表象Fˆ∑=nnncϕψ||=ψϕ|nnc↔iccc21ψ{}mnAA=ˆ=mnnmAAϕϕ|ˆ|表象变换†)()(ˆˆˆˆUFUFAB=)()(|ˆ|ABU=ψψ=nmmnabU|)(|xxxψψψ==表象Aˆ=nnnaaaA||ˆ表象Bˆ=nnnbbbB||ˆ五、时间演化算符=)(|),(ˆ)(|00tttUtψψ),(ˆˆ),(ˆi00ttUHttUt=∂∂1),(ˆ00=ttU若不显含时间tHˆHttettUˆ)(i00),(ˆ−−=若)(ˆˆtHH=，但不同时刻的哈密顿量对易′′−=∫ttttHttU0d)(ˆiexp),(ˆ0若)(ˆˆtHH=，且不同时刻的哈密顿量不对易∫′′′+=tttttUtHttU0d),(ˆ)(ˆi11),(ˆ00∑∫∫∫∫∞=−+=12132100102010)(ˆ)(ˆ)(ˆdddd)i1(1),(ˆnttnttttttnntHtHtHttttttUn多次叠代0121tttttn≥≥≥≥≥−时序算符Tˆ)(ˆ)(ˆ)(ˆ)](ˆ)(ˆ)(ˆ[ˆ21kjintFtFtFtFtFtFT=∑∫∫∫∫∞=−+=12132100102010)(ˆ)(ˆ)(ˆdddd)i1(1),(ˆnttnttttttnntHtHtHttttttUn0121tttttn≥≥≥≥≥−′′=′−tttttt01)(θ)(ˆ)(ˆ)(ˆ)()()(dddd)i1(1),(ˆ2101211321100000nnttttttnttnntHtHtHttttttttttttU−−−⋅+=−∞=∫∫∫∫∑θθθkjittt≥≥≥按时间先后顺序从右到左排列若=1212212121)(ˆ)(ˆ)(ˆ)(ˆ)](ˆ)(ˆ[ˆtttFtFtttFtFtFtFT若若)(ˆ)(ˆ)(ˆ)()()(dddd)i1(1),(ˆ2101211321100000nnttttttnttnntHtHtHttttttttttttU−−−⋅+=−∞=∫∫∫∫∑θθθ对任意一组t1,t2,t3,…,tn)](ˆ)(ˆ)(ˆ[ˆ21ntHtHtHT)(ˆ)(ˆ)(ˆ)()()(211211nnntHtHtHtttttt−−−=−θθθ?=∑−−−=−αθθθ)(ˆ)(ˆ)(ˆ)()()(211211nnntHtHtHtttttt求和对n个ti的所有排列进行，共有n!项，但其中只有满足时序顺序要求的一项不为零。)(ˆ)(ˆ)()()(ˆ)(ˆ)()()](ˆ)(ˆ[ˆ121222121121tFtFtttttFtFtttttFtFT−−+−−=θθθθ两种表示是等价的！=1212212121)(ˆ)(ˆ)(ˆ)(ˆ)](ˆ)(ˆ[ˆtttFtFtttFtFtFtFT∫∫=tttttFtFTttI00)](ˆ)(ˆ[ˆdd2121∫∫−−=tttttFtFtttttt00)(ˆ)(ˆ)()(dd2121121θθ∫∫−−+tttttFtFtttttt00)(ˆ)(ˆ)()(dd1212212θθ21tt1t2t12tt∫∫=tttttFtFtt010)(ˆ)(ˆdd!22121])(ˆ)(ˆ)(ˆ[ˆddd)i1(!11),(ˆ121210000∑∫∫∫∞=+=nttnttnttntHtHtHTtttnttU)d)(ˆi1exp(ˆ),(ˆ00∫′′=ttttHTttU时间演化算符)(ˆ)(ˆ)(ˆ)()()(ddd)i1(1),(ˆ211211n2110000nnnttttttnntHtHtHtttttttttttU−−−+=−∞=∫∫∫∑θθθ01ttt≥≥一、运动方程经典物理0=t时刻txyxyrr′ooox′y′rθθ=′=′θθsincosryrx=′=′θθsincosryrx量子力学=)(||tψψ力学量通常与时间t无关Fˆ=fFfF||ˆ与时间t无关f|§4绘景picture=)(|),(ˆ)(|00tttUtψψ′′=∫ttttHTttU0d)(ˆi1expˆ),(ˆ0若一个么正变换显含时间量子力学的表示平行、等价的表示绘景(picture)表象(representation)=∂∂ψψ|ˆ|iHt)(ˆˆ)(ˆˆ†tUFtUF=′=′ψψ|)(ˆ|tU薛定谔方程显含时间)(ˆˆtFF′=′不一定是时间演化算符！！1.薛定谔绘景目前为止的讨论限于位形空间=)(||tψψFSˆ)(|tSψ可以有不同的表象(representation)=∂∂)(|ˆ)(|itHttSSSψψ运动方程随时间变化Fˆ通常不随时间变化薛定谔绘景态矢随时间的变化Schrödingerpicture=fFfF||ˆ与时间t无关f|2.海森伯绘景=)()|,(ˆ)(|00tttUtSSψψ′′=∫ttSttHTttU0d)(ˆi1expˆ),(ˆ0幺正算符=′)()|,(ˆ|0†tttUSψψ=)(|0tSψ=)()|,(ˆ)(|0†tttUtSHψψHeisenbergpicture用做幺正变换),(ˆ01ttU−),(ˆ),(ˆ0†01ttUttU=−),(ˆ0ttU=定义=)(|0tSψ)(|tHψ海森伯绘景下的态矢量=∂∂)(|ˆ)(|itHttSSSψψ物理量的算符随时间变化′′=∫ttSttHTttU0d)(ˆi1expˆ),(ˆ0),(ˆˆ),(ˆˆ00†ttUFttUFSH=)(ˆˆtFFHH=海森伯绘景0)(|=∂∂ttHψ态矢量不随时间变化==)(|)()|,(ˆ)(|00†ttttUtSSHψψψ实际上是不随时间变化的基矢随时间变化经典上—“坐标系”的反向“旋转”物理量算符的变换海森伯绘景下的运动方程),(ˆˆ),(ˆ),(ˆˆ)],(ˆ[ˆdd00†00†ttUtFttUttUFttUtFtSSH∂∂+∂∂=)],(ˆˆˆ),(ˆ),(ˆˆˆ),(ˆ[i1ˆdd00†00†ttUHFttUttUFHttUFtSSSSH+−=),(ˆˆ),(ˆi00ttUHttUtS=∂∂]ˆˆˆˆˆˆˆˆˆˆˆˆ[i1††††UFUUHUUHUUFUSSSS−=),(ˆˆ),(ˆˆ00†ttUFttUFSH=薛定谔绘景下，力学量与时间无关时间演化算符的方程而HttUttUtSˆ),(ˆ),(ˆi0†0†=∂∂−]ˆˆˆˆˆˆˆˆˆˆˆˆ[i1ˆdd††††UFUUHUUHUUFUFtSSSSH−=)ˆˆˆˆ(i1FHHFHHHH−=]ˆ,ˆ[ˆddiHFFtHHH=海森伯方程若不显含时间tHˆHttSHttHSSFFˆ)(iˆ)(i00eˆeˆ−−−=HttttUˆ)(i00e),(ˆ−−=HHHSHˆˆˆ==与薛定谔绘景中的薛定谔方程地位相当海森堡绘景中，基矢和力学量算符均显含时间，但态矢量不是时间的函数。显然FSˆ若显含时间t]ˆ,ˆ[)ˆ(iˆddiHFtFFtHHHH+∂∂=一般不使用海森伯绘景哈密顿算符与所选绘景无关HHHSHˆˆˆ==力学量的平均值与所选绘景无关),(ˆˆ),(ˆˆ00†ttUFttUFSH=)(ˆ)(tFtHHHψψ=)(ˆ)(ˆtFtFSSSSψψ=对守恒量Aˆ[]0ˆ,ˆ=HAS0ˆ=∂∂tASAAASHttSHttHSSˆeˆeˆˆ)(iˆ)(i00==−−−)(),(ˆˆ),(ˆ)(0†0tttUFttUtSHSψψ===)(|)()|,(ˆ)(|00†ttttUtSSHψψψFHˆ=3.相互作用绘景10ˆˆˆHHHSSS+=设)(ˆˆ11tHHSS=0†0ˆˆˆˆUFUFSI=000ˆˆˆHHHHS===)()|,(ˆ|0†0tttUSIψψ=)(|e0ˆitSHtψInteractionpicture介于海森堡绘景和薛定谔绘景之间的一种绘景不显含时间用做么正变换),(ˆ010ttU−00ˆ)(i00e),(ˆHttSttU−−=00ˆiˆieˆeHtSHtF−=已取t0=0])(|i[e)()|ˆ(e)(|i00ˆi0ˆi∂∂+−=∂∂tttHttSHtSHtIψψψ态矢的运动方程⋅⋅+−=)(|ˆˆ)ˆˆ(ˆ†000†0tUUHHUSSψ于是=∂∂)(|)(|i1tHttIIIψψ01†01ˆˆˆˆUHUHSI==)(|e)(|0ˆittSHtIψψ⋅=)(|ˆˆ)ˆ(ˆ†001†0tUUHUSSψ相互作用绘景下哈密顿量的含时部分形式上与薛定谔绘景下的运动方程一致()+−=)(|ˆˆ)(|ˆˆ†00†0tHUtHUSSSψψ=∂∂)(|)(|i1tHttIIIψψ0†0ˆˆˆˆUFUFSI=)ˆˆˆ(ddiˆddi0†0UFUtFtSI=]ˆ,ˆ[)ˆ(iˆddi0HFtFFtIII+∂∂=与海森伯方程形式相同力学量的演化0†000†00†00ˆˆˆiˆˆˆˆˆˆˆˆUtFUUHFUUFUHSSS∂∂++−=00ˆˆˆˆHFFHII+−=)ˆ(itFI∂∂+0ˆi0eˆHtSU−==∂∂)(|)(|i1tHttIIIψψ相互作用绘景亦称狄拉克绘景与薛定谔绘景中运动方程形式相同与海森伯方程形式相同薛定谔绘景求解含时微扰轮问题时，如果使用表象，其过程本质上就是取相互作用绘景0Hˆ=)(|e)(|0ˆittSHtIψψ)(ˆˆˆ10tHHHSS+=]ˆ,ˆ[)ˆ(iˆddi0HFtFFtIII+∂∂=01†01ˆˆˆˆUHUHSI=222ˆ212ˆˆxpHωµµ+=例题：谐振子求海森堡绘景下坐标算符与动量算符。解：]ˆ,ˆ[!1ˆ)(0ˆˆBAieBeiiAA∑∞=−=Baker-Hausdorff公式[]]ˆ,ˆ[,ˆ]ˆ,ˆ[)()1(BAABAnn=+BBAˆ]ˆ,ˆ[)0(=tHStHHSSFFˆiˆieˆeˆ−=tHtHHxxˆiˆieˆeˆ−=]ˆ,ˆ212ˆ[]ˆ,ˆ[222xxpxHµωµ+=µpˆi−=又：由题意：xxSˆˆ=ppSˆˆ=µpxHˆi]ˆ,ˆ[−=]ˆ,ˆ[!1ˆ)(0ˆˆBAieBeiiAA∑∞=−=tHStHHxxˆiˆieˆeˆ−=[]−=µµωpxxHHˆi,ˆ21]ˆ,ˆ[,ˆ22xˆ22ω=tHStHHxxˆiˆieˆeˆ−=xˆ=tpωµωˆ+2)(!2ˆtxω−+−3)(ˆ!31tpωµω++−=42!41!211cosxxx++−=53!51!31sinxxxx+−−+=32)(ˆ!31)(!2ˆˆˆˆtptxtpxxHωµωωωµωtptxωµωωsinˆcosˆ+=类似地，可得txtppHωµωωsinˆcosˆˆ−=利用运动方程求解：]ˆ,ˆ[ˆddiHFFtHH=]ˆ,ˆ[ˆddiHxxtHH=]ˆ,ˆ[ˆddiHpptHH=µpHxHˆi]ˆ,ˆ[=xHpHHˆi]ˆ,ˆ[2µω−=]ˆ,ˆ[ˆddiHxxtHH=]ˆ,ˆ[ˆddiHpptHH=µpHxHHˆi]ˆ,ˆ[=xHpHHˆi]ˆ,ˆ[2µω−=µpxtHHˆiˆddi=xptHHˆiˆddi2µω−=µpxtHHˆˆdd=xptHHˆˆdd2µω−=与经典正则方程形式相同tBtAxHωωsincosˆ+=tAtBpHωµωωµωsincosˆ−=0=t时xtxHˆ)0(ˆ==ptpHˆ)0(ˆ==tptxxHωµωωsinˆcosˆˆ+=txtppHωµωωsinˆcosˆˆ−=2220ˆ212ˆˆxpHωµµ+=例题：谐振子处于的第n个本征态0ˆHnnnEnHnω)21(ˆ)0(0+==t=0时