复旦量子力学讲义qmapter1.ppt

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1、Quantum Mechenics II,Ru-Keng Su2005.1.5,Chapter 1Foundation of Quantum Mechanics,1.1 State vector,wave function and superposition of states,This chapter evolves from an attempt of a brief review over the basic ideas and formulae in undergraduate-level quantum mechanics.The details of this chapter ca

2、n be found in the usual references of quantum mechanics,1.1 State vector,wave function and superposition of states,1.1 State vector,wave function and superposition of states,1.1 State vector,wave function and superposition of states,1.2 Schrdinger equation and its solutions,1.2 Schrdinger equation a

3、nd its solutions,1.2 Schrdinger equation and its solutions,1D Schrdinger equationInfinite potential well,1.2 Schrdinger equation and its solutions,Infinite potential well,1.2 Schrdinger equation and its solutions,Harmonic oscillator,1.2 Schrdinger equation and its solutions,Harmonic oscillator,1.2 S

4、chrdinger equation and its solutions,Harmonic oscillator,1.2 Schrdinger equation and its solutions,Harmonic oscillator,1.2 Schrdinger equation and its solutions,Harmonic oscillator,1.2 Schrdinger equation and its solutions,3D Schrodinger equationCentral potential,1.2 Schrdinger equation and its solu

5、tions,Central potential,1.2 Schrdinger equation and its solutions,Coulomb potential,1.2 Schrdinger equation and its solutions,Coulomb potential,1.3 Operators,According to the Born statistical interpretation,The probability of finding a particle at position r is just the square of its wave function,1

6、.3 Operators,1.3 Operators,1.3 Operators,1.3 Operators,pi-ih/2iCartesian rectangular coordinates1st convention:pure coordinate part pure momentum part2nd convention:mixed part,1.3 Operators,1.3 Operators,Commutator,1.3 Operators,Commutator,1.3 Operators,Commutator,1.3 Operators,Hermitian operator,1.

7、3 Operators,Eigenequation,1.3 Operators,O-representation,1.3 Operators,O-representation,1.4 Approximation method,Perturbation independent of timeNon-degenerate,1.4 Approximation method,Non-degenerate,1.4 Approximation method,Non-degenerate,1.4 Approximation method,Degenerate,1.4 Approximation method

8、,Degenerate,1.4 Approximation method,Advantages of this choice are,1.4 Approximation method,Degeneracy may be removed,1.4 Approximation method,Perturbation depending on timeKey:How to calculate the transition amplitude,1.4 Approximation method,Perturbation depending on time,1.4 Approximation method,

9、Perturbation depending on time,1.4 Approximation method,Variational methodKey:How to choose the trial wave function,1.4 Approximation method,Variational method,1.5 WKB method(Wentzel-Kramers-Brillouin),Basic idea:(Q.M.)(C.M)when h0WKB Semi-Classical method:To find an expansion of h and solve station

10、ary Schrdinger equation,1.5 WKB method(Wentzel-Kramers-Brillouin),1.5 WKB method(Wentzel-Kramers-Brillouin),1.5 WKB method(Wentzel-Kramers-Brillouin),For 1D case,1.5 WKB method(Wentzel-Kramers-Brillouin),For 1D case,1.5 WKB method(Wentzel-Kramers-Brillouin),For 1D case,1.5 WKB method(Wentzel-Kramers

11、-Brillouin),Three regions:E U(x),1.5 WKB method(Wentzel-Kramers-Brillouin),Conservation of the probability,1.5 WKB method(Wentzel-Kramers-Brillouin),E=U(x)Turning points:The semi-classical approximation is not applicable,1.5 WKB method(Wentzel-Kramers-Brillouin),E=U(x),1.5 WKB method(Wentzel-Kramers

12、-Brillouin),E=U(x),1.5 WKB method(Wentzel-Kramers-Brillouin),E U(x),1.5 WKB method(Wentzel-Kramers-Brillouin),Example I:,1.5 WKB method(Wentzel-Kramers-Brillouin),E U(x),1.5 WKB method(Wentzel-Kramers-Brillouin),E U(x),1.5 WKB method(Wentzel-Kramers-Brillouin),E U(x),1.5 WKB method(Wentzel-Kramers-B

13、rillouin),a1,b1 region,1.5 WKB method(Wentzel-Kramers-Brillouin),E U(x),Asymptotic solutions,1.5 WKB method(Wentzel-Kramers-Brillouin),1.5 WKB method(Wentzel-Kramers-Brillouin),1.5 WKB method(Wentzel-Kramers-Brillouin),b2,a2 region,1.5 WKB method(Wentzel-Kramers-Brillouin),This is the Bohr-Sommerfel

14、d quantized condition,1.5 WKB method(Wentzel-Kramers-Brillouin),Example 2:Barrier penetration,1.5 WKB method(Wentzel-Kramers-Brillouin),Barrier penetration,1.5 WKB method(Wentzel-Kramers-Brillouin),Barrier penetration,1.5 WKB method(Wentzel-Kramers-Brillouin),Barrier penetration,1.5 WKB method(Wentz

15、el-Kramers-Brillouin),Barrier penetration,1.5 WKB method(Wentzel-Kramers-Brillouin),Connection formulae(dU/dx0),1.5 WKB method(Wentzel-Kramers-Brillouin),Connection formulae(dU/dx0),1.6 Density matrix,Problem:Can we get a new formula to calculate the expectation value like quantum statisticsQ.M.=Q.S

16、.=tr(A)=tr(exp(-H)A),1.6 Density matrix,Key:What is density matrix,1.6 Density matrix,Example:Two level system,1.6 Density matrix,Example:Two level system,1.6 Density matrix,Properties of density matrixHermitian matrix,1.6 Density matrix,Properties of density matrix,1.6 Density matrix,Properties of

17、density matrix,1.6 Density matrix,Properties of density matrixThe eigenvalue of density matrix are 0 or 1,1.6 Density matrix,Properties of density matrixTensor Product,1.6 Density matrix,Properties of density matrix,1.6 Density matrix,Properties of density matrix,1.6 Density matrix,Properties of den

18、sity matrixEvolution equation of density matrix,1.6 Density matrix,Properties of density matrixVector p is a polarization vector of the state which points in direction,1.6 Density matrix,Properties of density matrix,1.7 Coherent States,Consider a forced linear Harmonic oscillator,1.7 Coherent States

19、,1.7 Coherent States,The last equation can be solved by Greens functions,1.7 Coherent States,1.7 Coherent States,where ain is the solution of the corresponding homogeneous equation when tt2Suppose f(t)0 when t1tt2,1.7 Coherent States,1.7 Coherent States,ain aout via a unitary transformationsTo find

20、S:Noting,1.7 Coherent States,1.7 Coherent States,Our problem is:how to find the probability amplitude from|nin(forced)|inout,in particular,to find outin,1.7 Coherent States,1.7 Coherent States,1.7 Coherent States,S|0 is the coherent states,1.7 Coherent States,1.7 Coherent States,1.7 Coherent States,

21、Properties of coherent statesCoherent states is the eigenstate of operator a,1.7 Coherent States,Properties of coherent statesCoherent states is the eigenstate of operator a,1.7 Coherent States,Normalization,but do not orthogonal,1.7 Coherent States,Normalization,but do not orthogonal,1.7 Coherent S

22、tates,Overcomplete set,1.7 Coherent States,Overcomplete set,1.7 Coherent States,Overcomplete set,1.7 Coherent States,Coherent state is the state which satisfies the minimum uncertainty principle,1.7 Coherent States,1.7 Coherent States,1.7 Coherent States,1.7 Coherent States,1.8 Schrdinger picture,He

23、isenberg picture and interaction picture,Schrdinger picture(Lab coordinates),1.8 Schrdinger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,uns(x)does not depend on tOs does not depend on ts depends on t,1.8 Schrdinger picture,Heis

24、enberg picture and interaction picture,Heisenberg picture(co-moving coordinates),1.8 Schrdinger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,unH(x,t)depend on tOH

25、(t)depend on tH does not depend on t,1.8 Schrdinger picture,Heisenberg picture and interaction picture,Discussion:,1.8 Schrdinger picture,Heisenberg picture and interaction picture,=,1.8 Schrdinger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and inter

26、action picture,For energy representation,1.8 Schrdinger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,Interactional pictureTo futher study perturbation,1.8 Schrdin

27、ger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,To find the evolution operator,1.8 Schrdinger p

28、icture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,1.8 Schrdinger picture,Heisenberg picture and interaction picture,