1、Quantum Mechenics II,Ru-Keng Su 2005.1.5,Chapter 1 Foundation 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 chapte
2、r can 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 equ
3、ation and its solutions,1.2 Schrdinger equation and its solutions,1D Schrdinger equation Infinite 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 oscillat
4、or,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,Harmonic oscillator,1.2 Schrdinger equation and its solutions,3D Schrodinger equation Central potential,1.2 Schrdinger equation and
5、 its solutions,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
6、function,1.3 Operators,1.3 Operators,1.3 Operators,1.3 Operators,pi-ih/2i Cartesian rectangular coordinates 1st convention: pure coordinate partpure momentum part 2nd convention: mixed part,1.3 Operators,1.3 Operators,Commutator,1.3 Operators,Commutator,1.3 Operators,Commutator,1.3 Operators,Hermiti
7、an operator,1.3 Operators,Eigenequation,1.3 Operators,O - representation,1.3 Operators,O - representation,1.4 Approximation method,Perturbation independent of time Non -degenerate,1.4 Approximation method,Non -degenerate,1.4 Approximation method,Non -degenerate,1.4 Approximation method,Degenerate,1.
8、4 Approximation method,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 time Key: How to calculate the transition amplitude,1.4 Approximation method,Perturbation depending on time,1
9、.4 Approximation method,Perturbation depending on time,1.4 Approximation method,Variational method Key: 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 h0 WKB Semi- Classical method: To find an
10、expansion of h and solve stationary 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 1
11、D case,1.5 WKB method (Wentzel-Kramers-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-Br
12、illouin),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-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-Kr
13、amers-Brillouin),E U(x),1.5 WKB method (Wentzel-Kramers-Brillouin),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
14、 method (Wentzel-Kramers-Brillouin),This is the Bohr-Sommerfeld 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 (
15、Wentzel-Kramers-Brillouin),Barrier penetration,1.5 WKB method (Wentzel-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 f
16、ormula to calculate the expectation value like quantum statistics Q.M. = Q.S. = 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 matrix Hermitian m
17、atrix,1.6 Density matrix,Properties of density matrix,1.6 Density matrix,Properties of density matrix,1.6 Density matrix,Properties of density matrix The eigenvalue of density matrix are 0 or 1,1.6 Density matrix,Properties of density matrix Tensor Product,1.6 Density matrix,Properties of density ma
18、trix,1.6 Density matrix,Properties of density matrix,1.6 Density matrix,Properties of density matrix Evolution equation of density matrix,1.6 Density matrix,Properties of density matrix Vector p is a polarization vector of the state which points in direction,1.6 Density matrix,Properties of density
19、matrix,1.7 Coherent States,Consider a forced linear Harmonic oscillator,1.7 Coherent States,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 tt2 Suppose f(t)0 when
20、t1tt2,1.7 Coherent States,1.7 Coherent States,ain aout via a unitary transformations To find 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 Coher
21、ent States,S|0 is the coherent states,1.7 Coherent States,1.7 Coherent States,1.7 Coherent States,Properties of coherent states Coherent states is the eigenstate of operator a,1.7 Coherent States,Properties of coherent states Coherent states is the eigenstate of operator a,1.7 Coherent States,Normal
22、ization, but do not orthogonal,1.7 Coherent States,Normalization, but do not orthogonal,1.7 Coherent States,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
23、 Coherent States,1.7 Coherent States,1.7 Coherent States,1.7 Coherent States,1.8 Schrdinger picture, Heisenberg picture and interaction picture,Schrdinger picture (Lab coordinates),1.8 Schrdinger picture, Heisenberg picture and interaction picture,1.8 Schrdinger picture, Heisenberg picture and inter
24、action picture,uns(x) does not depend on t Os does not depend on t s depends on t,1.8 Schrdinger picture, Heisenberg picture and interaction picture,Heisenberg picture (co-moving coordinates),1.8 Schrdinger picture, Heisenberg picture and interaction picture,1.8 Schrdinger picture, Heisenberg pictur
25、e and interaction picture,1.8 Schrdinger picture, Heisenberg picture and interaction picture,unH(x,t) depend on t OH(t) depend on t H does not depend on t,1.8 Schrdinger picture, Heisenberg picture and interaction picture,Discussion:,1.8 Schrdinger picture, Heisenberg picture and interaction picture
26、,=,1.8 Schrdinger picture, Heisenberg picture and interaction picture,1.8 Schrdinger picture, Heisenberg picture and interaction picture,For energy representation,1.8 Schrdinger picture, Heisenberg picture and interaction picture,1.8 Schrdinger picture, Heisenberg picture and interaction picture,1.8
27、 Schrdinger picture, Heisenberg picture and interaction picture,Interactional picture To futher study perturbation,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 inte
28、raction picture,1.8 Schrdinger picture, Heisenberg picture and interaction picture,To find the evolution operator,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,1.8 Schrdinger picture, Heisenberg picture and interaction picture,
