北京大学：《量子力学》课程教学资源（PPT课件讲稿）第四章 表象理论

量子力學 Chap 1- The Wave Function Chap 2- The Time-independent Schrodinger Equation Chap 3 Formalism in Hilbert Space Chap4-表象理論 2021/2/24

2021/2/24 1 量子力學 Chap 1 - The Wave Function Chap 2 - The Time-independent Schrödinger Equation Chap 3 - Formalism in Hilbert Space Chap 4 - 表象理論

Quantum Mechanics Schrodinger表象, Heisenberg表象和交互作用表象 1. Schrodinger表象 將力學量(不顯含t)的平均值及機率分佈隨時間的變化,全歸於 波函數隨時間的變化,而力學量(算符)本身是不隨時間變化的。 ∧ h202 (t) HY(t) where H=-+(x) (4.1) 2m ox iEt Y(x,)=(x)e h 2021/2/24 2

2021/2/24 2 Quantum Mechanics ► Schrödinger 表象, Heisenberg表象和交互作用表象 ( ) ˆ (t) H t t i  =     where 1. Schrödinger 表象 將力學量(不顯含t)的平均值及機率分佈隨時間的變化,全歸於 波函數隨時間的變化,而力學量(算符)本身是不隨時間變化的。 …(4.1)

Quantum Mechanics 2. Heisenberg表象 波函數是不能測量的,與實際觀測有關的是力學量的平均值 及測量結果的機率分佈,且隨時間變化,可用其它方式表達, Heisenberg表象是其中常用的一種表達方式 F()=i F() (4.2) where F(=U(t,OFU(,0)=expliHt/h]Fexp-iHt/h 2021/2/24

2021/2/24 3 Quantum Mechanics 2. Heisenberg表象 波函數是不能測量的,與實際觀測有關的是力學量的平均值 及測量結果的機率分佈,且隨時間變化,可用其它方式表達, Heisenberg表象是其中常用的一種表達方式。 / ] ˆ / ] exp[ ˆ F(t) =U (t,0)FU(t,0) = exp[iHt  F −iHt  + F t H  i F t dt d ˆ ( ), 1 ( )  = …(4.2) where

Quantum Mechanics From(4. 1) ih p(t)=hp(t) Let(t)=U/(t,0)y(0) (4.3) Where U(t, 0) is time-evolution operator and is unitary, i.e U(t,0)U(t,0)=U(1,0)(t,0)=1<>U(10)=U(t,0) Substitute(4.3)into(4.1)and get iU(10)(0)=HC(t,0)(0),(0isarbitrary 今iU(0)=(t、0)1c.U(0,0)=1(44) 2021/2/24

2021/2/24 4 Quantum Mechanics ( ) ˆ (t) H t t i  =    From (4.1)  Let (t) =U(t,0)(0) Where is time U(t,0) -evolution operator and is unitary, i.e., …(4.3) ( ,0) ( ,0) ( ,0) ( ,0) 1 ( ,0) ( ,0) 1 U t U t U t U t U t U t + + + − = =  = Substitute (4.3) into (4.1) and get ( ,0) (0) ˆ ( ,0)(0) =    U t HU t t i , is arbitrary (0) ( ,0) ˆ U(t,0) HU t t i =    i.c. U(0,0) =1 …(4.4)

Quantum Mechanics Solve(4. 4 )and get (1)H不顯含t U(t,0)=exp[-iHt/h (2)H顯含t U(.0)=ecl-() Assume h不顯含t and F(t=U(t,O)FU(, 0)=exp[iHt/h F exp[-iHt/h where F不顯含t 2021/2/24

2021/2/24 5 Quantum Mechanics Solve (4.4) and get / ] ˆ U(t,0) = exp[−iHt  ( ) ] ˆ ( ,0) exp[ 0 ' '  = − t H t dt i U t  (1) H 不顯含 t ˆ (2) H 顯含 t ˆ and / ] ˆ / ] exp[ ˆ F(t) =U (t,0)FU(t,0) = exp[iHt  F −iHt  + Assume H 不顯含 t ˆ where F 不顯含 t

Quantum Mechanics F(t U*(t0)FU(,0)+U+(t,0)FU(,0) GU HFU+UFHU) GUTHUUTFU+UTFUUTHO) (-HF(t)+F()H) i F(),H Heisenberg equation; Heisenberg表象中力學量隨時間變化 F(1)=|F( 2021/2/24

2021/2/24 6 Quantum Mechanics F t H  i HF t F t H i U HU U FU U FUU HU i U HFU U FHU i U t dt d U t FU t U t F dt d F t dt d ˆ ( ), 1 ) ˆ ( ) ( ) ˆ ( 1 ) ˆ ˆ ( 1 ) ˆ ˆ ( 1 ( ) ( ,0) ( ,0) ( ,0) ( ,0)     = = − + = − + = − +  +      = + + + + + + + + Heisenberg equation: F t H  i F t dt d ˆ ( ), 1 ( )  = Heisenberg表象中力學量隨時間變化

Quantum Mechanics Ex: Free particle in Heisenberg表象 2 =0,p is conservative p(t)=p(o)=p iHi O),H=e h 21 2021/2/24

2021/2/24 7 Quantum Mechanics m p H 2 ˆ ˆ 2 = ․ Ex: Free particle in Heisenberg表象   0 ˆ  pˆ, H = , p ˆ is conservative , p ˆ(t) = p ˆ(0) = p ˆ   m p e m p e e m p e r i r t H i r t dt d iHt iHt iHt iHt ˆ ˆ 2 ˆ ˆ, 1 ˆ ˆ( ), 1 ˆ( ) ˆ ˆ ˆ 2 ˆ = =       = = − −      

Quantum Mechanics ● Heisenberg表象跟 Schrodinger表象是等價的 Ex:力學量守恆,不因採用表象不同而異 Schrodinger表象□少[F,H] Heisenberg表象□∵H(t)=U(t,0)HU(t0)=H iHt F(t),H(r Fe hh 0 2021/2/24

2021/2/24 8 Quantum Mechanics ․Heisenberg表象跟Schrödinger表象是等價的 Ex:力學量守恆,不因採用表象不同而異 Schrödinger表象 [F, H] = 0 Heisenberg表象     0 ˆ , ˆ ( ) , ˆ ( ), ˆ ( ,0) ˆ ( ) ( ,0) ˆ ˆ ˆ ˆ ˆ = =          = = = − − +      iHt iHt iHt iHt e F H e F t H t e Fe H H t U t HU t H

Quantum Mechanics 3交互作用表象 介於 Schrodinger表象和 Heisenberg表象之間的一種表象,在 散射理論和量子場論的微擾論處理,是常被採用的一種表達方 式 Assume the system Hamilton is H=h+v where v is interaction y: state vector in Schrodinger picture state vector in interaction picture (t)=exp ys(t) 2021/2/24

2021/2/24 9 Quantum Mechanics Assume the system Hamilton is H ˆ = H ˆ 0 +V where V is interaction I s : state vector in Schrödinger picture : state vector in interaction picture ( ) ˆ ( ) exp 0 t iH t t I  S       =   3.交互作用表象 介於 Schrödinger 表象和 Heisenberg表象之間的一種表象,在 散射理論和量子場論的微擾論處理,是常被採用的一種表達方 式

Quantum Mechanics i mna w,0-ma exp lin Ya exp Hs(t)+exp B+B厘(O iH ih p exp V()N() 2021/2/24

2021/2/24 10 Quantum Mechanics ( ) ( ) ( ) ( ) ˆ exp ˆ exp ˆ exp ( ) ˆ exp ( ) ˆ ˆ ˆ exp ( ) ˆ ( ) exp ˆ exp ˆ ( ) ˆ ( ) exp 0 0 0 0 0 0 0 0 0 0 V t t t iH t iH t V iH t V t iH t H H t iH t t t i iH t t iH iH t i t iH t t t i t i I I S S S S S I S =               −       =        =  − +       =                   +       =                   =               