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1、复旦量子力学讲义qmapter1 Still waters run deep.流静水深流静水深,人静心深人静心深 Where there is life,there is hope。有生命必有希望。有生命必有希望Chapter 1Foundation of Quantum Mechanics1.1 State vector,wave function and superposition of statesThis chapter evolves from an attempt of a brief review over the basic ideas and formulae in unde
2、rgraduate-level quantum mechanics.The details of this chapter can be found in the usual references of quantum mechanics1.1 State vector,wave function and superposition of states1.1 State vector,wave function and superposition of states1.1 State vector,wave function and superposition of states1.2 Sch
3、rdinger equation and its solutions1.2 Schrdinger equation and its solutions1.2 Schrdinger equation and its solutionsv1D Schrdinger equationvInfinite potential well1.2 Schrdinger equation and its solutionsvInfinite potential well1.2 Schrdinger equation and its solutionsvHarmonic oscillator1.2 Schrdin
4、ger equation and its solutionsvHarmonic oscillator1.2 Schrdinger equation and its solutionsvHarmonic oscillator1.2 Schrdinger equation and its solutionsvHarmonic oscillator1.2 Schrdinger equation and its solutionsvHarmonic oscillator1.2 Schrdinger equation and its solutionsv3D Schrodinger equationvC
5、entral potential1.2 Schrdinger equation and its solutionsvCentral potential1.2 Schrdinger equation and its solutionsvCoulomb potential1.2 Schrdinger equation and its solutionsvCoulomb potential1.3 OperatorsvAccording to the Born statistical interpretation,The probability of finding a particle at pos
6、ition r is just the square of its wave function1.3 Operators1.3 Operators1.3 Operators1.3 Operatorsvpi-ih/2ivCartesian rectangular coordinatesv1st convention:pure coordinate part pure momentum partv2nd convention:mixed part1.3 Operators1.3 OperatorsvCommutator1.3 OperatorsvCommutator1.3 OperatorsvCo
7、mmutator1.3 OperatorsvHermitian operator1.3 OperatorsvEigenequation1.3 OperatorsvO-representation1.3 OperatorsvO-representation1.4 Approximation methodvPerturbation independent of timevNon-degenerate1.4 Approximation methodvNon-degenerate1.4 Approximation methodvNon-degenerate1.4 Approximation metho
8、dvDegenerate1.4 Approximation methodvDegenerate1.4 Approximation methodvAdvantages of this choice are1.4 Approximation methodvDegeneracy may be removed1.4 Approximation methodvPerturbation depending on timevKey:How to calculate the transition amplitude1.4 Approximation methodvPerturbation depending
9、on time1.4 Approximation methodvPerturbation depending on time1.4 Approximation methodvVariational methodvKey:How to choose the trial wave function1.4 Approximation methodvVariational method1.5 WKB method(Wentzel-Kramers-Brillouin)vBasic idea:(Q.M.)(C.M)when h0vWKB Semi-Classical method:To find an e
10、xpansion of h and solve stationary Schrdinger equation1.5 WKB method(Wentzel-Kramers-Brillouin)1.5 WKB method(Wentzel-Kramers-Brillouin)1.5 WKB method(Wentzel-Kramers-Brillouin)vFor 1D case1.5 WKB method(Wentzel-Kramers-Brillouin)vFor 1D case1.5 WKB method(Wentzel-Kramers-Brillouin)vFor 1D case1.5 W
11、KB method(Wentzel-Kramers-Brillouin)vThree regions:vE U(x)1.5 WKB method(Wentzel-Kramers-Brillouin)vConservation of the probability1.5 WKB method(Wentzel-Kramers-Brillouin)vE=U(x)Turning points:The semi-classical approximation is not applicable1.5 WKB method(Wentzel-Kramers-Brillouin)vE=U(x)1.5 WKB
12、method(Wentzel-Kramers-Brillouin)vE=U(x)1.5 WKB method(Wentzel-Kramers-Brillouin)vE U(x)1.5 WKB method(Wentzel-Kramers-Brillouin)Example I:1.5 WKB method(Wentzel-Kramers-Brillouin)vE U(x)1.5 WKB method(Wentzel-Kramers-Brillouin)vE U(x)1.5 WKB method(Wentzel-Kramers-Brillouin)va1,b1 region1.5 WKB met
13、hod(Wentzel-Kramers-Brillouin)vE U(x)Asymptotic solutions1.5 WKB method(Wentzel-Kramers-Brillouin)1.5 WKB method(Wentzel-Kramers-Brillouin)1.5 WKB method(Wentzel-Kramers-Brillouin)vb2,a2 region1.5 WKB method(Wentzel-Kramers-Brillouin)This is the Bohr-Sommerfeld quantized condition1.5 WKB method(Went
14、zel-Kramers-Brillouin)vExample 2:Barrier penetration1.5 WKB method(Wentzel-Kramers-Brillouin)vBarrier penetration1.5 WKB method(Wentzel-Kramers-Brillouin)vBarrier penetration1.5 WKB method(Wentzel-Kramers-Brillouin)vBarrier penetration1.5 WKB method(Wentzel-Kramers-Brillouin)vBarrier penetration1.5
15、WKB method(Wentzel-Kramers-Brillouin)vConnection formulae(dU/dx0)1.5 WKB method(Wentzel-Kramers-Brillouin)vConnection formulae(dU/dx0)1.6 Density matrixvProblem:Can we get a new formula to calculate the expectation value like quantum statisticsvQ.M.=vQ.S.=tr(A)=tr(exp(-H)A)1.6 Density matrixvKey:Wha
16、t is density matrix 1.6 Density matrixvExample:Two level system1.6 Density matrixvExample:Two level system1.6 Density matrixvProperties of density matrixvHermitian matrix1.6 Density matrixvProperties of density matrix1.6 Density matrixvProperties of density matrix1.6 Density matrixvProperties of den
17、sity matrixvThe eigenvalue of density matrix are 0 or 11.6 Density matrixvProperties of density matrixvTensor Product1.6 Density matrixvProperties of density matrix1.6 Density matrixvProperties of density matrix1.6 Density matrixvProperties of density matrixvEvolution equation of density matrix1.6 D
18、ensity matrixvProperties of density matrixvVector p is a polarization vector of the state which points in direction1.6 Density matrixvProperties of density matrix1.7 Coherent StatesvConsider a forced linear Harmonic oscillator1.7 Coherent States1.7 Coherent StatesvThe last equation can be solved by
19、Greens functions1.7 Coherent States1.7 Coherent Statesvwhere ain is the solution of the corresponding homogeneous equation when tt2vSuppose f(t)0 when t1tin(forced)|inout,in particular,to find outin1.7 Coherent States1.7 Coherent States1.7 Coherent StatesvS|0 is the coherent states1.7 Coherent State
20、s1.7 Coherent States1.7 Coherent StatesvProperties of coherent statesvCoherent states is the eigenstate of operator a1.7 Coherent StatesvProperties of coherent statesvCoherent states is the eigenstate of operator a1.7 Coherent StatesvNormalization,but do not orthogonal1.7 Coherent StatesvNormalizati
21、on,but do not orthogonal1.7 Coherent StatesvOvercomplete set1.7 Coherent StatesvOvercomplete set1.7 Coherent StatesvOvercomplete set1.7 Coherent StatesvCoherent state is the state which satisfies the minimum uncertainty principle1.7 Coherent States1.7 Coherent States1.7 Coherent States1.7 Coherent S
22、tates1.8 Schrdinger picture,Heisenberg picture and interaction picturevSchrdinger picture(Lab coordinates)1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction picturevuns(x)does not depend on tvOs does not depend on tvs depends on
23、 t1.8 Schrdinger picture,Heisenberg picture and interaction picturevHeisenberg picture(co-moving coordinates)1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction pic
24、turevunH(x,t)depend on tvOH(t)depend on tvH does not depend on t1.8 Schrdinger picture,Heisenberg picture and interaction picturevDiscussion:1.8 Schrdinger picture,Heisenberg picture and interaction picturev=1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heis
25、enberg picture and interaction picturevFor energy representation1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction picturevInteractional picturevTo futher study pe
26、rturbation1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction picturevTo find the evolution operato
27、r1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction picture1.8 Schrdinger picture,Heisenberg picture and interaction picture