【作者】 徐峰；

【导师】 郑雨军；

【作者基本信息】 山东大学 ， 原子与分子物理， 2014， 博士

【摘要】 传统的量子理论在处理高维度、非线性、强耦合体系时计算非常复杂困难,虽然近几年来计算机和计算方法都得到了很大的发展但是处理这样的体系依然是一个难题。用传统的量子方法处理复杂的体系即使得到了这些问题的数值结果,也没有办法用形象的物理图像来描述体系的动力学过程。量子相空间理论恰好能够解决这个问题,量子相空间理论是Wigner为了修正热力学系统的量子效应提出的,其核心是引入了量子相空间分布函数——Wigner函数。量子相空间分布函数可以代替波函数描述量子体系,它对体系的描述是准确和完备的,但是Wigner函数只能看做准概率分布函数,因为即使初始值处处为正,在演化的过程中它也会出现负值,但这并不影响它对物理量的计算。量子相空间理论的应用主要有二个方面的优点：第一,利用相空间分布函数可以避免量子力学中复杂的算符运算,可以作为一种有效的数学工具；第二,可以用来模拟量子动力学过程并给出量子过程的直观的物理图像,有利于研究量子和经典的对应性关系。量子相空间理论在物理学的许多领域已有广泛的运用,比如量子光学,统计物理,碰撞理论,以及非线性物理等。在量子光学中,基于Wigner函数引入密度算符定义了高次关联函数并讨论了量子光学相干现象；在统计物理中用Wigner函数研究了玻色爱因斯坦凝聚；在碰撞理论中,Wigner函数被用来研究粒子在无限深势阱和势能台阶中的运动情况,并且计算了氦原子与氢分子及氢原子和氢分子的反应几率等；在非线性物理中,量子相空间理论用来研究体系的量子混沌效应。因为在相空间理论中经典物理和量子物理具有相同的空间基础,它们的异同很容易反应在相空间的动力学过程中,因此量子相空间理论为研究量子力学和经典力学的对应关系提供了桥梁。几乎从量子力学诞生起,量子力学和经典力学的对应关系一直是人们关注但又晦涩的物理内容,并引起玻尔、爱因斯坦等世界顶级物理学家的关注。经典物理和量子物理对应性关系的基本表述是：在大量子数极限下,量子物理学将回到经典物理学。它还可表述为：当普朗克常量趋于零时,量子物理学将回到经典物理学。对应性原理是量子物理和经典物理的桥梁,它可以用来联系量子物理和经典物理,为人们用半经典物理探索量子体系提供理论基础。量子相空间理论为应用经典物理的理论框架研究量子物理提供了可能性；同时,量子相空间有利于处理复杂的量子多体问题,用经典语言描述体系的量子特性。量子理论的计算复杂度跟体系的维度成正指数关系,这样传统量子力学的计算量随维度的增加增长太快,对于高维体系精确的量子计算是非常困难的。基于量子相空间理论的经典分子反应动力学方法在复杂体系的模拟上有一定的优越性,它选取合适的初始轨迹分布来代替初始的Wigner函数,通过经典的哈密顿正则演化来得到系统的末态,这样的方法得到了许多可以和实验对比的结果。但是由于演化过程中的量子效应被忽略,对于量子效应比较明显的体系,经典的分子反应动力学方法是不适用的。经典分子反应动力学中的轨线演化是经典的,它没有体现体系的量子效应,为了把体系的量子效应直观的展现出来,人们发展了量子水动力学方法,它通过把体系的量子效应等效成量子力作用在轨线上得到了具有量子效应的轨线。经典分子反应动力学方法得不到系统演化过程中的量子效应,量子水动力学方法需要先求解体系的薛定谔方程才能反应出系统的量子效应。2010年Marten-s等发展的纠缠轨线分子动力学方法既不需要先求解薛定谔又能在轨线的演化过程中包含体系的量子效应,它把所有的轨迹看做是一个整体,轨线的演化不再是独立的而是纠缠在一起的,通过轨线之间的相互作用得到体系的量子效应。这一方法成功的处理了许多的问题,包括一维模型和二维模型,它计算了体系的反应几率和隧穿速率得到了和精确量子计算符合的非常好的结果,并且给出了量子隧穿效应经典的物理图像：轨线之间存在着相互作用,初始能量低于势垒的轨线,在演化的过程中可以从其它轨线中借取能量,使其能越过势垒发生反应。我们主要用纠缠轨线分子动力学方法来模拟量子过程,做了以下方面的工作：1.我们运用纠缠轨线分子动力学方法计算了水分子体系的第一激发态光解的反应截面,这是此方法第一次运用于真实的二维体系,得到了和精确量子计算符合的比较好的结果,证实了这种方法的适用性,并得到了量子力学中光解过程的经典物理图像。这也是纠缠轨线方法第一次运用于没有明显量子隧穿效应的体系,结果表明对于这样的体系,纠缠轨线方法依然可以给出体系部分的量子效应并得到很好的计算结果。2.我们运用纠缠轨线方法模拟了二个耦合粒子间的纠缠动力学行为,我们研究了经典极限下的纠缠动力学并把它和二个经典体系之间的经典关联演化对应起来。我们还研究了纠缠和混沌之间的关系,给出了混沌迅速提升纠缠的原因,我们也研究了单条轨线对于量子纠缠的贡献,我们发现当轨线位于Wigner函数中心区域时,它的贡献是很小的甚至是负值,这反应了非平衡态和量子熵的性质,当体系经过长时间演化后,我们发现所有轨线的贡献几乎相同。3.我们用纠缠轨线方法模拟了物质波的双缝干涉过程,我们定性的得到了物质波的干涉图样,但是因为现有纠缠轨线方法的缺陷使我们没有定量的得到和精确量子计算符合很好的结果。我们解释了为什么我们的结果没能定量的反应出量子干涉效应,因为我们采用的正定假设在此过程中是不能很好的描述体系的状态。在量子干涉过程中,Wigner函数的负值区域明显,它的作用不能忽略。4.我们发展了一种新的模拟Wigner函数的方法,这种方法可以模拟具有负值的Wigner函数,我们再利用刘维尔定理来重新定义纠缠轨线方程,利用刘维尔定理的缺陷和量子刘维尔方程相结合的方式精确的演化体系,得到用精确的具有负值的Wigner函数所表示的系统演化末态。本文的主要内容如下：第一章中我们综述了量子过程模拟的几种轨线方法：经典分子反应动力学,量子水动力学方法,纠缠轨线动力学方法。我们给出相应的量子轨线方程,并讨论各种方法的优缺点。第二章介绍了本文用到的纠缠轨线动力学方法的基础理论,介绍了几种常用的量子相空间分布函数：Wigner函数、Husimi分布函数,标准序和非标准序分布函数以及正则和反正则分布函数,我们讨论它们的基本特性。接下来,我们详细的推导出了纠缠轨线的演化方程,我们介绍选择初始态的过程,对比了初始等概率取点的初始轨迹和对它采用热平衡演化后的轨迹模拟初始函数的精度,简单提到了其它的取样方法。最后讨论纠缠轨线的演化过程中量子效应的非局域性,我们给出一个特别的例子来证明量子效应一直存在在轨线的演化过程中,即使长时间演化以后轨迹之间彼此离散的很远。在第三章中,我们用纠缠轨线分子动力学方法模拟了水分子在第一激发态的光解过程,并且我们计算它局部和总的光解反应截面,对比我们的结果和经典分子反应动力学方法与精确量子计算结果,结果表明纠缠轨线方法能够部分的反应出体系的量子效应,相对于经典分子反应动力学,我们的结果和精确量子计算结果符合的比较好。在这一过程,我们采用了一个被广泛验证的势能面,这是纠缠轨线方法第一次运用于复杂的量子体系,事实证明了纠缠轨线方法的适用性。我们对比拥有相同初始态的经典轨线和纠缠轨线,我们发现由于量子作用的存在,纠缠轨线在量子相空间表现出非常复杂的行为。事实证明纠缠轨线方法不仅能给出水分子第一激发态光解截面,并且能够通过纠缠轨线给出这一量子过程的经典物理图像。第四章中,我们用纠缠轨线方法模拟二个耦合粒子间的纠缠动力学过程,我们研究这一过程中的经典和量子的对应性关系,我们发现当h→0时,二个耦合粒子间量子纠缠动力学可以和二个经典体系的经典关联演化对应起来。量子纠缠和混沌的关系一直是人们关心的内容,很多文献证实了当二个粒子处于混沌动力学下它们之间的纠缠会被迅速的提升,我们从解析公式和经典物理图像出发,给出了这一现象的原因——混沌行为使粒子在空间的分布弥散导致偶合粒子间的纠缠增大。在我们的公式中,粒子间的纠缠可以被分解为单条轨线的贡献总和,我们讨论了单条轨线对纠缠的作用。我们发现这样的作用并不一定是积极的,有可能轨线对纠缠的作用是负值的,这表现出了轨线的量子特性,这也反应出系统处于非平衡态,当体系长时间演化后,所有的轨线的贡献几乎相同。在本章最后,我们讨论了量子隧穿效应在纠缠动力学中的影响,我们发现它可以降低纠缠。第五章中,我们模拟了物质波的双缝干涉过程,我们定性的给出了物质波的双缝干涉图样,它反应出了双缝干涉部分峰值位置。通过和精确量子计算对比,我们发现这样的结果只能是定性的反应出量子干涉效应,它没有能给出一个很好的干涉图样。我们用两个高斯波包在量子相空间中的干涉过程,来简单的分析现有的纠缠轨线方法在模拟这一过程中失败的原因,我们发现在此过程中Wigner函数的负值效应明显,现有的正定的模拟函数不能很好的反应出真实的Wigner函数的情况。在第六章中,我们采用新的模拟函数来模拟Wigner函数,这个模拟函数可以模拟出Wigner函数的负值部分。我们重新定义了我们的纠缠轨线演化方程,并提出了基于刘维尔定理缺陷和量子刘维尔方程的演化方法,这将使我们方法的结果更加精确。在本文的第七章,我们对前几章的工作进行了总结,并对下一步的工作进行了展望。更多还原

【Abstract】 For a many-body system, because of high dimension,nonlinear ity, strong correla-tion rigorous quantum dynamical simulation of molecular processes remains difficult in spite of concurrent advances in methodology and computer performance. Even if we get the numerical results of these problems, this is no way to use a vivid physical picture to describe its quantum dynamics process. Quantum phase space theory provides a powerful tool to solve this problem, it is introduced by Wign-er in1932to correct quantum effects in the thermodynamic system, its core is introduced quantum phase space distribution function which was named Wigner function. Wigner function can be used to replace wave function in quantum theroy to describe system, its description is accurate and complete. Wigner function only a quasi-probability function in phase space, because even for an initial conditions which positive everywhere, it can assume negative values in some regions of phase space with the evolution of quantum system, but it doesn’t affect its calculation of physical quantities. The application of quantum phase space theory has two major advantages:The first, because of using phase space distribution function, we can avoid complicated operator calculation in quantum mechanics, it can be used as a kind of effective mathematical tool; The other, it can be used to simulate quantum dynamics process and get an intuitive physical picture of quantum effect, it is very useful in the study of the correspondence principle between quantum and classical theory.Quantum phase space theory has been widely used in many fields of physics, such as quantum optics, statistics physics, collision theory and nonlinear physics. In quantum optics, density operator is introduced to define the high order correlation function and discusses quantum optical coherence phenomenon based on the Wigner function; In the statistical physics, it is used to study Bose-Einstein condensation; In the collision theory, Wigner function is used to study quantum dynamics in the infinite deep well and steps potential, and calculate the reaction probability of col- lision between helium molecules and hydrogen molecules or hydrogen atom collide with hydrogen molecules, etc.; In the nonlinear physics, quantum phase space the-ory is used to research quantum chaos phenomenon. Because classical physics and quantum physics has the same space base in phase space theory, so their similarities and differences in the dynamics process is easy to show in the phase space, quantum phase space theory provides a bridge for the research of corresponding relationship between quantum and classical mechanics. Almost since the birth of quantum mechanics, correspondence principle between quantum mechanics and classical me-chanics has been a concern but obscure physical content, it attracts the attention of the world’s top physicist like Bohr and Einstein. Correspondence principle’s basic statement is:under the big quantum number limit, quantum physics will return to classical physics. It also can be expressed as:when the Planck constant goes to zero, quantum physics will go back to classical physics. Correspondence principle is the bridge of quantum physics and classical physics, it can be used to contact quantum physics and classical physics, it provides a theoretical basis for people using semiclassical theory to explore quantum system. Quantum phase space the-ory provides the possibility of using the theoretical framework of classical physics to study quantum system; At the same time, quantum phase space is conducive to dealing with quantum complex many-body system, it allows us to describe the quantum properties of system with classics language.It is well known that computational complexity of quantum theory is expo-nential increase with the dimension of system, it is too fast so strictly quantum theory is very difficult for high-dimensional system. Classical molecular dynamics method based on quantum phase space theory does a lot of good works in simula-tion of complex system, it chooses appropriate initial trajectories replace the initial Wigner function and evolution with Hamiltonian canonical equation, then get the final state of system, this method got many good results which can compare with experimental ones. But due to quantum effects in the evolution are ignored, for a system with significantly quantum effects, it is not applicable. In the classical molecular dynamics method, trajectories’evolution is an classic Hamiltonian dy-namics, it doesn’t reflect the quantum effects of system, in order to show quantum effects of system intuitively, people developed quantum hydrodynamics method, it get trajectories with quantum effects through using the quantum effects of system as an equivalent force in the evolution of trajectories.Classical molecular dynamics method can’t get quantum effects in the evolu-tion of system, quantum hydrodynamics methods need to solve the evolution of system with the schrodinger equation. Entangled trajectories molecular dynamics trajectory method introduced by Martens in2001doesn’t need solving schrodinger equation and can get quantum effect in the evolution of system, it uses all trajecto-ries as a whole, the evolution of trajectories is no longer independent but entangled with each other, quantum effects of system is obtained by the interaction between trajectories. This method is successful to deal with a lot of problems, including one-dimensional and two-dimensional model, it is used to calculate the reaction probability and quantum tunneling rates of the system and get good results com-pared with accurately quantum calculation, it gives an perfect physical picture of quantum tunneling effect——there is interaction between entangled trajectories, so trajectory with initial energy lower than the potential barrier can borrow energy form other trajectories in the evolution, at the end, the trajectory across the barrier.We mainly used entangled trajectories molecular dynamics method to simulate quantum dynamics process, we did some works as follow:1. We used entangled trajectories molecular dynamics method to calculate the partial and total cross section of photodissociation of H2O in its first ab-sorption band, this is the first time this method is applied to a realistic two-dimensional system, and our results are in reasonable agreement with the results of exact quantum mechanical, it is confirmed that this method is applicability for generally realistic system, we get vivid physical pictures of photodissociation in quantum theory.it is also the first time entangled trajec-tories method is applied to system which has no obvious quantum tunneling effect, the results show that for such system, entangled trajectories molecular dynamics method still can give part of quantum effects in the system and obtain a very good calculation results.2. We used entangled trajectories molecular dynamics method to simulate the entanglement dynamics between two coupled particles, we force on the en-tanglement dynamics under classical limit and correspond it to the classical dynamics describing classical correlations between two classical subsystem. We also studied the relationship between entanglement and chaos, we give a good reason why chaos can rapidly increase entanglement in the system, the contribution of single trajectory to quantum entanglement is investigated, we found that when the trajectory is located in center area of Wigner function, the contribution of it is small or even negative, this reflects the nonequilibrium property and the nature of the quantum entropy, when the system become equilibrium, we found that the contribution of all trajectories almost the same.3. The entanglement trajectory method is used to study the double-slit inter-ference phenomena of matter waves, we got an qualitative wave interference pattern, we explain why we failed to quantitatively reflect the results of quan-tum interference effect. We use a positive simulation function in our calcula-tion, but through analyze the process of Gaussian wave packet interference in quantum phase space, we know that in double-slit interference, the negative part of Wigner function has a great effect.4. We used a new simulation function to get the negative part of Wigner function, through Liouville theorem and its drawback we get an correctly results of the evolution of quantum Liouville equation, get accurately final Wigner function with negative values.The main content of this thesis is as follows:In the first chapter, we introduced three quantum trajectory methods for the quantum dynamics process simulation:Classical molecular dynamics method, quan- tum hydrodynamics methods, entangled trajectories dynamics method. We present the corresponding quantum trajectories equation and discussed the advantages and disadvantages of these methods. We introduced the theoretical basis of entangled trajectories dynamics method and present some common quantum phase space dis-tribution functions in the chapter two:Wigner function,Husimi distribution func-tion,standard (or anti-standard) and nomal (or anti-nomal) ordering distribution function, we give their basic properties. Then, we give simple derivation of our tra-jectories evolutions of entangled trajectories,and discuss the properties of entangled trajectories.In the third chapter, we use entangled trajectories dynamics method to calculate the partial and total cross section of photodissociation of H20in its first absorption band. We used the analytic potential energy surface which has been widely used, we compare our results with classical molecular dynamics method and exact quantum calculation, the results show that our results can reflect part quantum effects of the system, compared with the classical molecular dynamics method, our results are good agreement with exact quantum calculation. We compared with trajectory with same initial state under classical dynamics and quantum dynamics,we found that entangled trajectories showed a very complex behavior in the phase space because of the existence of the quantum effects.In the fourth chapter, we use entangled trajectories dynamics method to simulate entanglement dynamics of two coupled particles, we found correspondence principle between classical dynamics and quantum ones in this process, we found that when h→0, entanglement dynamics between two coupled particles does have a clas-sical analog in classical dynamics describing classical correlations between classical subensembles. The relationship between quantum entanglement and chaos has at-tracted lots of people’s attention, some literatures confirmed that when two particles under an chaotic dynamics, entanglement between them will rapidly increase, we gave the causes of this phenomenon through analytic equation and classical physics pictures-the chaotic behavior of particle make his distribution function diffuse in phase space. In our formula, entanglement between the particles can be considered is the sum of contribution of single trajectory, we discussed the contribution of s-ingle trajectory to entanglement. We found that it is not necessarily be positive, sometime is negative, it shows quantum properties of the trajectory and also re-flects that system state in a non-equilibrium state, after enough time evolution, the contribution of all trajectories is almost the same. Entangled trajectories method began to be used in the simulation of quantum tunneling effect, in the end of this chapter, we discuss the quantum tunneling effect in the entanglement dynamics, we find that it can reduce entanglement between particles.In the fifth chapter, we simulated double-slit interference process of matter waves, our gave a qualitatively interference pattern, it shows part of peak posi-tion in the double-slit interference. Compared with exact quantum calculation, we found that our result only can reflect the quantum interference effect in qualita-tively, it can’t give a good interference pattern. We use two gaussian wave packet interference in the phase space as an example to give a reason why our method is failure to simulate quantum interference process, we found that in this process negative part of the Wigner function is obviously, our positive simulation func-tion can’t reflect the real Wigner function well. In the sixth chapter, we adopt a new simulation function to simulate the wigner function, this simulation function can simulate the negative part of Wigner function. We redefined our entangled trajectories evolution equations, and got a new evolutionary method based on the drawback of Liouville theorem and quantum Liouville equation, it will make results of our method is more accurate.In the last chapter, we summarize our works and prospect the future works.更多还原