讲义5-0-波包动力学

1、飞秒波包动力学韩永昌1OUTLINE 导论 波包动力学基本理论 波包物理过程Rep. Prog. Phys. 58, 365(1995)2导论 波包的概念：limited to simple textbook examples, which help the undergraduate students to understand how Fourier transforms；本征态的叠加 波包的应用：超短激光脉冲与原子分子相互作用化学反应碰撞原子光学和半导体物理3波包动力学基本理论4 外场中分子的哈密顿量 引入BO近似（绝热表象） 相应的随时间变化的分子波函数为势能面势能面5 最终得到各个电

2、子态之间核波函数的耦合方程（矩阵形式？）思考：没有外场和有外场时，以上方程形式如何变化，体现什么物理意义？思考：没有外场和有外场时，以上方程形式如何变化，体现什么物理意义？没有外场时，退化成一个单电子态上的核波函数，没有外场时，退化成一个单电子态上的核波函数，处于某一电子态上的波函数不随其他波函数变化处于某一电子态上的波函数不随其他波函数变化BO近似近似有外场时，多电子态上的核波函数发生耦合有外场时，多电子态上的核波函数发生耦合6题外话（无外场情况下） BO近似中，我们一方面忽略了电子态之间的耦合，另一方面没有考虑相对论效应（自旋的影响，比如spin-orbital, spin-spin in

3、teraction）。 这些项，都会导致电子态的能量移动以及电子态之间的耦合（coupling）7考虑双原子分子的情况考虑双原子分子的情况 核之间的相对动能 不妨取 reduced Schrdinger equation8910思考：dipole moment的物理意义11激光与分子的相互作用激光与分子的相互作用 假设激光只耦合两个电子态，不妨命名对应的势能面为1(lower)和2(higher) 激光（电场）的描述：经典方法同时忽略电场的空间变化（为什么？）1.分子空间电场的波长2.hard，a.坐标系变换；b.分子质心运动12 在共振条件下(the resonance means that

4、 we have in the field an oscillating component which) We have dropped the other, rapidly oscillating interaction term, as it usually averages to zero and does not contribute to the real interaction process: rotating-wave approximation (RWA)local detuning总电场电场包络函数13 if we rewrite the state vector as

5、then we obtain the two-state Eq.local Rabi frequency for the surfaces nand m. It contains the field envelope (pulse shape)思考以上得到的两态方程，尤其关注右下对角项思考以上得到的两态方程，尤其关注右下对角项The laser-induced resonances appear as surface crossings.Thus we have reduced resonant laser-induced transitions to surface crossings wh

6、ich look very much like those arising from the failures of the Born-Oppenheimer approximation.14Condon近似近似 波包：我们可以将以上每一电子态上的波函数写成相应电子态上振动态的叠加1516 the Franck-Condon factors 注意：这里把偶极距看成了常数，是近似处理，否则需要带着一起积分（对R的积分） 波包方法的优势：短脉冲伴随大能量展宽，很多振动态、电子态、甚至连续态involved；强场下微扰理论不适用。17波包的一些性质波包的一些性质 A wave-packet repr

7、esents a quantum system that is localized in its position coordinate. In the context of diatomic molecules, the wavepacket represents the fact that there is some uncertainty in the separation of the two atoms.18 A typical wave-packet is the Gaussian wavefunction19 consider only a single electronic s

8、tate described by a potential U(R)These equations indicate the trajectory of the wave-packet20 In free space, U = 0, the wave-packet not only moves with speed hk/m, but also spreads.21 The reason for experimental interest in Gaussian wave-packets is that they are naturally found as the ground-state

9、wavefunction of a harmonic potential.22the lowest vibrational state is the ground state, which is of the form of the width of the Gaussian packet is related to the oscillator frequency (谐振子势的频率)23谐振子势在量子力学里有零点能，即最低振动能级不为零，因此动量和坐标即使在最低能量处也存在不确定度，高斯波包就体现了这一特点 if the packet finds itself in another pote

10、ntial of exactly the same shape, but displaced from the centre, it will simply oscillate.24注意：注意： 我们上面所说的高斯波包是个定态波函数我们上面所说的高斯波包是个定态波函数It will not change shape and its centrewill exactly follow the classical trajectory. However, if the potential is harmonic with a different period, the packet will br

11、eathe while moving along the classical trajectory. The breathing means that the packet changes its width as it moves about the well, but, unlike the case of dispersion, its original shape will return. If the packet is wider at the turning point than the natural ground-state wavefunction of the new h

12、armonic well, it will become narrower than the local ground state when it is at the centre of the well25 the reverse case：This happens because the initial wave-packet is narrower than the ground state wave-packet of the harmonic well.26 Real potentials may not be harmonic and if the distortion is on

13、ly slight we may have an anharmonic potential, which, for example, may be of the form. If a Gaussian wave-packet is placed in such a potential well, then ?27 it will initially oscillate in the well as it would in a harmonic potential.28 However, after a time it breaks up into pieces so that it is sp

14、read about the potential surface.29 A remarkable feature of the anharmonic well is that it can show revivals where the wave-packet reforms. Furthermore, fractional revivals take place where the wave-packet reforms as two (or even more) separate pieces.30波包演化的数值求解波包演化的数值求解1. treat the quantum mechani

15、cal motion by discretizing the wave-packet and propagating it on a lattice of points 2. one finds the eigenstates of the potential (which may also be found on a lattice) and expands the wave-packet in that basis, using time-dependent coefficients 3. Hybrid of the above two31 The lattice approach 以一维

16、问题为例32 The most straightforward approach would be to utilize a formal solution over a short time step so that33this method is numerically unstable and this has resulted in thedevelopment of other methodssecond-order finite differencesSplit-operator Fourier transform method The essence of this method

17、 is to split apart the two operator components and treat each operator separately34the heart of the split-operator methods35 动量算符利用傅里叶变换，与波函数发生作用36 It is straightforward if there is only one potential surface. However, if there are two coupled surfaces we must form the exponential of a 2 x 2 matrix.

18、 diagonalize to form the exponential37Ifthen38 For a time-independent problem it may be convenient to prepare the matrix Uv in advance of the wave-packet propagation as UV depends only on the potentials and coupling at the discretized time and space points. The method may also be used for time depen

19、dent potentials and couplings, but then no advance preparation is possible which adds to the numerical overheads.39 regarded as a form of second-order expansion of the exponential operator40Crank-Nicholson methodrecall the previously mentioned unstable one: the first-order expansionHomework RWA （11页

20、）具体怎么来的？ 18页公式怎么推出来的？ wave packet revival？按照以上所说的例子，验证一下 33页，为什么不稳定？ 37、38页，推导过程是什么？ 40页CN的具体推导过程是什么，该方法如何保证稳定性？41波包相关的物理过程42波包的产生波包的产生1. Instantaneous excitation One method to create a well defined quantum mechanical wave-packet is to apply a short laser pulse to a molecule in its equilibrium state

21、.43思考：要在激发态形思考：要在激发态形成波包，需要什么条成波包，需要什么条件？件？1. 共振条件2. 脉冲足够短 思考：脉冲不够短会思考：脉冲不够短会怎样？怎样？1. 波包在作用过程中演化2. 脉冲长，带宽小，不易共振（不完全共振） The use of ultrashort pulses reduced the excitation process to a simple dumping of the ground-state wave-packet on the excited-state potential surface, without any alterations in th

22、e spatial distribution. This is sometimes called Franck-Condon excitation.442. Rabi oscillations The pulse area determines the population distribution between the levels for ultrashort pulses, so only pulses with areas equal to odd multiples of can deplete the ground state completely If the pulse ar

23、ea is larger than we observe Rabi oscillations in the time evolution of the excited-state populations4546虚线为虚线为Rosen-Zener方方法（法（The local excitation approach ）得到的结果得到的结果实线为实线为wave packet方方法得到的结果法得到的结果思考：为什么随着作思考：为什么随着作用时间不同而不同？用时间不同而不同？波包组份的移动或者dispersion导致部分波包逃出了共振区域Processes induced by chirped pul

24、ses A time-dependent change in the laser frequency during the pulse is called chirping or frequency-sweeping The chirp in a pulse adds a new, externally controllable degree of freedom to the laser-molecule system It is usually easier to induce a chirp into a femtosecond pulse, than to obtain a speci

25、fic pulse shape 思考chirp脉冲的用处？47 利用以上知识学习 Phys. Rev. Lett. 91, 023002 (2003) 思考？Chirped pulse?或者其他？48Pump-probe schemes The concept of the pump-probe schemepump：preparation of the wave packetprobe：observe the motions4950Wave-packet interference Wave-packet dynamics can exhibit quantum mechanical interference of the matter wave 思考：坐标表象和动量表象的波函数图形？5152Two energy levels that me coupled by a laser at two points (xa and xb). The initial wave-packet PsiI (created by a short pulse at the pasition x0) can form PsiIV by two routes: by passing through PsiII, or