Chapter-2-The-Schrodinger-Equation-量子力学英文教案课件.ppt

1、University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia Chapter 2 The Schrodinger EquationlThe Interpretation of the Wave FunctionlThe principle of the superposition statelAverage value of dynamics quantity and Differential OperatorslSchrodinger Equation lTime-independen

2、t Schrodinger EquationlThe Heisenberg Uncertainty Relation123456backUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The Interpretation of the wave functionWave functionThe interpretation of the wave functionThe property of wave functionbackUniversity of Electroni

3、c Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia )(expEtrpiA problem？A plane wave for a free particle),(tr If a particle moving in one dimension experiences a force represented by the potential V(x)：describe a quantum mechanical particleIt is de Broglie wave and also is wave function of

4、 a free particle.(1)How to describe the state by wave function？(2)How to describe wave particle duality by wave function？(3)What does the wave function mean?wave function backback1 1 University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The Interpretation of the wave f

5、unctionElectron PPOQQOThe probability density distribution|(r)|(r)|2 2 The probability distribution|(r)|(r)|2 2 x y z x y zUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The property of wave functionThe probability：d W(r,t)=C|(r,t)|d W(r,t)=C|(r,t)|2 2 d d，（1 1）

6、The probability and probability density The probability density：(r,t)=dW(r,t)/d(r,t)=dW(r,t)/d=C|(r,t)|=C|(r,t)|2 2W(tW(t)=)=V V dW dW=V V(r,t)d(r,t)d=C=CV V|(r,t)|(r,t)|2 2 d dUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia（2）CC|(r,t)|(r,t)|2 2 d d=1,=1,C=1/C=1/

7、|(r,t)|(r,t)|2 2 d d221221),(),(),(),(trtrtrCtrC （3）|(A)(A)-1/2-1/2(r,t)(r,t)|2 2 d d=1=1（4）University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia A plane wave be unity I Dirac function def.def.0000)(xxxxxx)0(1)()(0000 dxxxdxxxxx)()()(00 xfdxxxxf )(0021)(xxikedkxx k=pk=

8、px x/,dk=dp,dk=dpx x/,xxxpidpexxx)(0021)()()()()(000 xxxfxxxf )(|1)(xaax )()(xx 0 x0 x)(0 xx dxeppxpxpxppixxxxxx)(021)(，University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia II II A plane wave be unity EtipEtrpiperAetr )(),(321)()()()(zpiypixpippprpipzyxzyxeAeAeAzyxAer

9、 A plane wave t=0t=0)(),(),(22*22xxtppippppedxtxtxxxxx dxxxexxxxpptEEi)()(*dxxxexxxxpptppi)()(*2222 dxxxxxpp)()(*)(221xxppA 若取若取 A A1 12 2 2 2 =1=1，则，则 A A1 1=2=2 -1/2-1/2,于是于是xpipxxex 21)()(xxpp A plane wave be unity)(xxpp dxtxtxxxpp),(),(*)(xxpp dxeAxppixx21 dxeppxppixxxx)(21)()()()()(000 xxxfxxxf

10、 University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia three dimension EtipEtrpiperetr )(21),(2/3 drredtrtrpptEEipp)()(),(),(*)()()()()()(*ppppppppdrrzzyyxxpp 2/332121 AAAA)()(ppppetEEi where2/321)(rpiper University of Electronic Science and Technology of China 2005-3-

11、1 Prof.Zhang Xiaoxia The principle of the superposition state（1）The principle of the superposition state（2）The wave function in momentum space backUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The principle of the superposition statel=C=C1 11 1+C+C2 22 2 l|2 2=

12、|C=|C1 11 1+C+C2 22 2|2 2 l =(C =(C1 1*1 1*+C+C2 2*2 2*)(C)(C1 11 1+C+C2 22 2)l =|C =|C1 1 1 1|2 2+|C+|C2 22 2|2 2+C+C1 1*C C2 21 1*2 2+C+C1 1C C2 2*1 12 2*P1 12 2S1S2electron The electron from the upper slit The electron from the lower slitThe interference term University of Electronic Science and

13、Technology of China 2005-3-1 Prof.Zhang Xiaoxia=C=C1 11 1+C+C2 22 2+.+C+.+Cn nn n +.+.=C=C1 11 1+C+C2 22 2The principle of the superposition stateUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The wave function in momentum space exp21)(2/3rpirp ）（rdtrrtpcp),()()

14、,(pdrtpctrp)(),(),(dxdydzrpitrexp),(212/3 ）（zyxdpdpdprpitpcexp),()2(12/3 The wave function in momentum space can be defined by the fourier transform The inverse fourier transform isUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia pdtpctpcpdtpcshow),(),(|),(|2pdrdr

15、trrdrtrpp)(),()(),(pdrrrdrdtrtrpp)()(),(),()(),(),(rrrdrdtrtr 1),(),(rdtrtrrdtrrtpcp),()(),(University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia Average value of dynamics quantity and Differential OperatorsAverage value of dynamics quantity （1）Average value of positio

16、n（2）Average value of momentumDifferential Operators （1）The Momentum Operator（2）The Kinetic Energy Operator（3）The Angular Momentum Operators（4）Hamilton OperatorbackUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia（1）Average value of position dxxxxx2|)(|drxxx2|)(|（2

17、2）Average value of momentumxxxxxxxdppcpppdxxipxpc22/1|)(|)/exp()()2(1)(backUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia Differential Operators（1）The Momentum Operatorxxxxxxxxxdppcppcdppcppp)()(|)(|2 xxxxpidppcpdxexx)()(21 xxxxpidxdppcpexx)()(21 xxxpidxdppcedxd

18、ixx)()(21 )(21)(xxxpidppcedxdixdxx University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia izkyjxiiprrxx dxdipx three dimension：one dimensionUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia One dimensiondxxFxFFdxxpxppdxxxxxxxxx)()()()(

19、)()(rdrrrdrFrFF)()()()(F is any OperatorrdrFrFFrdrprpprdrxrxxxxx)()()()()()(Three dimensionUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia（2）The Kinetic Energy Operator22:22()()ppTheKineticEnergy TOperator TmmTTr Tr dr（3 3）The Angular Momentum Operators prLprL ()

20、()()xzyyxzzyxLypzpiyzzyLzpxpizxxzLxpypixyyx rdrLrL)()(University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia 22If a particle moves in a static potential()()()2V rHTVHTV rV rm （4 4）Hamilton OperatorUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zh

21、ang Xiaoxia Schrodinger Equation（1 1）(r,t):describe a quantum mechanical particle（2 2）The Schrodinger equation for a free particle（3 3）A particle in potential V(r)（4 4）The Schrodinger equation for many particles（5）The probability current density backUniversity of Electronic Science and Technology of

22、 China 2005-3-1 Prof.Zhang Xiaoxia How to describe a quantum mechanical particle?The particles are subject to forces and Newtons Second Law can then be used to describe the motion of the particle in terms of a second-order differential equation.（1 1）Classical PhysicsClassical Physics0000,t tdrttrpmd

23、t，22d rFmdtNewtons laws：University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia（2 2）quantum mechanical 2The wave equation must be consistent with the classical correspondence principle.1The wave function must be the solution of a linear differential equation.返回返回Universi

24、ty of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The Schrdinger equation for a free particle）（1 EtiEit )(expEtrpiAThe wave function for a free particle:The time derivative of the wave packet is，2222)(xxEtzpypxpipxpiAexxzyx(1)(2)(1)(2)12222222222zyxpppzyx22222222yzpypz 222

25、2221(2)22pp or The spatial derivative of this wave packet University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia )2()2(222 pEtiSatisfy some requirements 22224ppipptiE）（2232it （）2a free particle2pE，0)2(2 pE(1)(2)(1)(2)backUniversity of Electronic Science and Technology o

26、f China 2005-3-1 Prof.Zhang Xiaoxia A particle in potential V(r)Schrdinger equation22(,)()(,)2(,)Operatorir tV rr ttHr tHHamilton :HrVpE )(22 )(22rVpE backUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The Schrodinger equation for many particlesi i(i=1,2,.,N),(r

27、(i=1,2,.,N),(r1 1,r,r2 2,.,r,.,rN N ;t);t)U Ui i(r(ri i),V(r),V(r1 1,r,r2 2,.,r,.,rN N)The Schrodinger equation for many particles：);,(),()(2);,(211212221trrrrrrVrUtrrrtiNNiNiiiiN NiNiiiirrrVrUH12122),()(2 University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The proba

28、bility current density2|),(|),(),(),(trtrtrtr probability current density 0),(dtrdtdThe Conservation of Probability University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia 222Vti222Vti2222 titi22 ）（tiThe complex conjugate of this expression isNow consider the one-dimensi

29、onal Schrodinger equationUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia dddtdi22 ）（consider the integration of this equation：0),(dtrdtd0 Jt diddtd2 ）（dJdtrdtd ),(),(),(trSdJdtrdtdSGauss Gauss THTHJ is the probability current density2 iJSdS University of Electron

30、ic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia 0),(dtrdtd0 Jtmass conservationcharge conservation0 eeJt )(2|),(|2iJJtr )(2|),(|2 ieJeJtreeeebackUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The time-independent Schrodinger equationlThe time-inde

31、pendent Schrodinger equationlHamiltonian operator and the eigenvalue equation for energylThe step in finding general solutions of stationary stateslThe property of stationary statesbackUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The time-independent Schroding

32、er equation),()(2),(22trrVtrti )()(),(tfrtr )(2)()()(22rVtftfdtdri E )()(2)()(22rErVtEftfdtdi separation of variables/)(iEtetf Etiertr )(),(That isseparation of variables)(2)(1)()(122rVrtfdtdtfi Dividing this equation by()()givesr f tUniversity of Electronic Science and Technology of China 2005-3-1

33、Prof.Zhang Xiaoxia Hamiltonian operator and the eigenvalue equation for energy（1 1）Hamilton Hamilton operator),()(2),(22trrVtrti The classical hamiltonianHHamiltonianHamiltonian operator，:)()(2)()(22rErVtEftfdtdi EVEti22 HVti222)(r exp/iEt/expiEt SchrodingerSchrodinger equation equationUniversity of

34、 Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia（2）The eigenvalue equation for energylThe eigenvalue equationlThe The eigenfunctionslThe The eigenvalueHE EV 22 backEUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The step in finding general

35、 solutions of stationary states)()(222rErV 1212eigenvalue ,eigenfunctions ,nnEEE，/exp)(),(tiErtrnnn 1|)(|2 drCnnSchrodingerSchrodinger equationeigenenergiesstationary states eigenfunctionsNormalizable C Cn nbackUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The

36、property of stationary states（2）The probability current density nnntr ),(2),(nnnnnitrJ （1）the probability density)/exp()/exp(tiEtiEnnnn )/exp()/exp(tiEtiEnnnn )()(rrnn )/exp()/exp()/exp()/exp(2tiEtiEtiEtiEinnnnnnnn )()()()(2rrrrinnnn )(rJn University of Electronic Science and Technology of China 200

37、5-3-1 Prof.Zhang Xiaoxia dtrFtrFnn),(),(（3）The average value dtiErFtiErnnnn)/exp()()/exp()(drFrnn)()(University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia The Heisenberg Uncertainty RelationFGik，222()()()4kFG 222)4xxxpixp ，（22)22xxxpxp （University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia Home worksP 123:3,4,