坐标动量算符的升降算符

坐标算符的升、降算符peAˆi†)(ˆξξ−=peAˆi)(ˆξξ=动量算符的升、降算符)(|xxxψψψ→=′=′xAxAxx|ˆ|)(|xxxxxxxxx′−=′=′δxeBˆi†)(ˆηη=xeBˆi)(ˆηη−=坐标表象中：xppexpxiπ21)(|==ψ)(i|ˆ|xxxxpxpxx−′∂∂=′=′δ动量表象？)(|xxxx′−=′δξξ+=xxA)(ˆ†ξξ-xxA=)(ˆηη+=)(ˆ†ppBηη-ppB=)(ˆ三、坐标表象的函数形式=ψϕ|ˆ|x=ψϕ|ˆ||xxx)()(xxψx=ϕ∫+∞∞−′′′=ψϕ||ˆ|dxxxxxxxxψ=ϕ|)(|ˆ|xxxxxxxxx′=′=′-δxψxxxxx′′−=′∫d)(δϕxxxψ=ϕ)(ˆ)(xxxψϕ=坐标表象中，坐标算符的函数形式。坐标表象)(xϕψ|坐标表象)(xψxˆ坐标表象xxx=ˆ动量算符的函数形式=ψϕ|ˆ|p=ψϕ|ˆ||pxx∫′′′=xxxpxxd||ˆ|ψϕ∫′′=′xxpxxd|ψ)(i|ˆ|xxxxpxpxx−′∂∂=′=′δ∫′′′=xxxpxψdxxxxx′∫′−′∂∂′=ψδ)](i[dxxxxxxxx′′∂∂′−−′−=∫′∞+∞−′d]i)[(|)(iψδψδxxψ∂∂−=i)(ixxψ∂∂−==ψϕ|ˆ|p)(i)(xxxψϕ∂∂−=)(ˆ)(xpxψϕ=xp∂∂−=iˆxxxxx′∫′−′∂∂−=ψδ)(di或)(ixxψ∂∂−=)(ixxxpxx′−∂∂−=′δ∫′′′=xxxpxxd||ˆ|ψϕ∫′′′=xxxpxψd)(i)(xxxψϕ∂∂−=ϕ|坐标表象)(xϕψ|坐标表象)(xψpˆ坐标表象x∂∂−i四、表象representation1.量子态的表象分立谱为例=nnnfFϕϕ||ˆ∑=nnncϕψ||=ψϕ|nnc↔iccc21ψ表象下Fˆ对任一态矢量ψ|=ψ|ˆ|Ag对力学量Aˆ=ψϕϕ|ˆ||Agnn∑=mmnmncAa↔nnnnnnAAAAAAAAAA212222111211ˆ=ψϕ|nnc=gann|ϕ=mnnmAAϕϕ|ˆ|∑=mmmnAψϕϕϕ||ˆ|2.表象变换同一个量子态(态矢量)，在不同的表象中，波函数的形式不同。ψ|表象Aˆ=nnnaaaA||ˆ表象Bˆ=nnnbbbB||ˆ=ψψ|)(nAna=ψψ|)(nBnb=jiAijaFaF|ˆ|)(=jiBijbFbF|ˆ|)(=ψψ|)(jBjb∑=iiijaabψ||∑=iAijiU)(ψ=ijjiabU|表象Aˆ表象Bˆ)()(|ˆ|ABU=ψψ=ijjiabU|1ˆˆˆˆ††==UUUU么正变换=jiBijbFbF|ˆ|)(∑=nmjnnmmibaaFaab,||ˆ||∑=nmjnAmnimUFU,*)(~†)()(ˆˆˆˆUFUFAB=)()(|ˆ|ABU=ψψ=jiijbaU|†1†ˆˆ−=UU=ψψ|)(iAia∑=jjjibbaψ||∑=jBjijU)(*)~(ψAˆBˆ表象Aˆ表象：Bˆ：Fˆ力学量五、时间演化算符=∂∂ψψ|ˆ|iHt⇒)(|)(|0ttψψ=)(|),(ˆ)(|00tttUtψψ1),(ˆ00=ttU),(ˆ0ttU时间演化算符012ttt若：),(ˆ),(ˆ),(ˆ011202ttUttUttU=1)(|)(=ttψψ),(ˆ|)(|)(0†0ttUttψψ=1)(|),(ˆ),(ˆ|)(000†0=tttUttUtψψ=)(|)(00ttψψ1),(ˆ),(ˆ00†=ttUttU幺正算符),(ˆ),(ˆ001ttUttU=−),(ˆ),(ˆ),(ˆ0010†ttUttUttU==−=∂∂ψψ|ˆ|iHt=)(|),(ˆ)(|00tttUtψψ坐标表象中rtrrttUrtr′′′=∫d)(||),(ˆ|)(|00ψψrtrtrtrUtr′′′=∫d),(),;,(ˆ),(00ψψ′=′rttUrtrtrU|),(ˆ|),;,(ˆ00若)(),(00rrtr−′=′δψ),;,(ˆ),(00trtrUtr=ψt0时刻位于处的粒子，演化到t时刻位于处的概率幅。0rr传播子，格林函数=∂∂ψψ|ˆ|iHt=)(|),(ˆ)(|00tttUtψψ=∂∂)(|),(ˆˆ)(|),(ˆi0000tttUHtttUtψψ),(ˆˆ),(ˆi00ttUHttUt=∂∂1),(ˆ00=ttU若不显含时间tHˆHttettUˆ)(i00),(ˆ−−=时间演化算符的方程)(|0tψ若是哈密顿量的本征态=)(|)(|ˆ00tEtHψψ==−−)(|)(|),(ˆ)(|0ˆ)(i000tetttUtHttψψψ=−−)(|0)(i0teEttψ定态=−−)(|)(|0ˆ)(i0tetHttψψ若不显含时间tHˆHttettUˆ)(i00),(ˆ−−=若)(ˆˆtHH=，但不同时刻的哈密顿量对易划分成许多小的时间段tt→0第i个时间段上,)(ˆ)(ˆitHtH=)(ˆ)(ˆ)(ˆ)(ˆ1221tHtHtHtH=小时间段上不随t变化)(ˆ)(ˆ)(ˆ)(ˆ2121tHtHtHtHeee+=)(ˆ)(i11),(ˆiiitHttiiettU−−−−=而),(ˆ),(ˆ),(ˆ011202ttUttUttU=)(ˆ)(i)(ˆ)(i101212tHtttHttee−−−−=)(ˆ)(i)(ˆ)(i101212tHtttHtte−−−−=于是可以写出′′−=∫ttttHttU0d)(ˆiexp),(ˆ0若不显含时间tHˆHttettUˆ)(i00),(ˆ−−=)(ˆ)(ˆ)(ˆ)(ˆ1221tHtHtHtH≠形式上∫′′′=′tttttttUtHttU00d),(ˆ)(ˆi1|),(ˆ00若)(ˆˆtHH=，但不同时刻的哈密顿量对易原方程被写成了一个积分方程′′−=∫ttttHttU0d)(ˆiexp),(ˆ0若)(ˆˆtHH=，且不同时刻的哈密顿量不对易)(ˆ)(ˆ)(ˆ)(ˆ2121tHtHtHtHeee+≠),(ˆˆ),(ˆi00ttUHttUt=∂∂∫′′′+=tttttUtHttU0d),(ˆ)(ˆi11),(ˆ00可逐次叠代求解∫∫++=tttttt,ttUtHtHttU0101202210dd)(ˆ)(ˆi11)(ˆi11),(ˆ∑∫∫∫∫∞=−+=12132100102010)(ˆ)(ˆ)(ˆdddd)i1(1),(ˆnttnttttttnntHtHtHttttttUn∫+=tttttUtHttU010110d),(ˆ)(ˆi11),(ˆ再一次叠代123033210ddd)(ˆ)(ˆi11)(ˆi11)(ˆi11),(ˆ}{][20100ttt,ttUtHtHtHttUtttttt∫∫∫+++=01ttt≥≥021tttt≥≥≥多次叠代0121tttttn≥≥≥≥≥−