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1、飞秒波包动力学 韩永昌 1 OUTLINE 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
2、面 5 最终得到各个电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？ 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BO近似 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 interaction ）。 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 核之间的相对动能 不妨
3、取 reduced Schrdinger equation 8 9 10 思考：dipole moment的物理意义 11 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 we have in the field an oscillating component which) We have dropped the other
4、, 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 then we obtain the two-state Eq. local Rabi frequency for the surfaces n and
5、 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 The laser-induced resonances appear as surface crossings. Thus we have reduced resonant laser-induced transitions to surface crossings which look very much like those arising from the failures of the Born-Oppenheimer approximatio
6、n. 14 Condon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 15 16 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 A wave-packet represents a quantum system that is localized in its position coordinate. In the context of dia
7、tomic 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 wavefunction 19 consider only a single electronic state described by a potential U(R) These equations indicate the trajectory of the wave-pa
8、cket20 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 wavefunction of a harmonic potential. 22 the lowest vibrational state is the ground sta
9、te, 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 potential of exactly the same shape, but displaced from the centre, it will simply os
10、cillate. 24 注意： 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 centre will exactly follow the classical trajectory. However, if the potential is harmonic with a different period, the packet will breathe while moving along the classical trajectory. The breathing means that the packet changes its
11、 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 harmonic well, it will become narrower than the local ground state when it is at the centre of the
12、well 25 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 only slight we may have an anharmonic potential, which, for example, may be of the form. If a Gaus
13、sian 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 spread about the potential surface. 29 A remarkable feature of the anharmonic well is that it ca
14、n 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 1. treat the quantum mechanical motion by discretizing the wave-packet and propagating it on a lattice of points 2. one finds
15、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 two 31 The lattice approach 以一维问题为例 32 The most straightforward approach would be to utilize a formal solution over a short time
16、 step so that 33 this method is numerically unstable and this has resulted in the development of other methods second-order finite differences Split-operator Fourier transform method The essence of this method is to split apart the two operator components and treat each operator separately 34 the heart of the split-operator methods 35 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
