Quantum Dynamics Process Research In Phase Space

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 hâ†'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.