Treating Many-Body Quantum Systems by Means of Classical Mechanics

Daniel A. Uken

Simulation of quantum dynamics for many-body systems is an open area of research. For interacting many-body quantum systems, the computer memory necessary to perform calculations has an astronomical value, so that approximated models are needed to reduce the required computational resources. A useful approximation that can often be made is that of quantum-classical dynamics, where the majority of the degrees are treated classically, while a few of them must be treated quantum mechanically. When energy is exchanged very quickly between the quantum subsystem and classical environment, the dynamics is nonadiabatic. Most theories for nonadiabatic dynamics are unsatisfactory, as they fail to properly describe the quantum backreaction of the subsystem on the environment. However, an approach based on the quantum-classical Liouville equation solves this problem. Even so, nonadiabatic dynamics is difficult to implement on a computer, and longer simulation times are often inaccessible due to statistical error. There is thus a need for improved algorithms for nonadiabatic dynamics. In this thesis, two algorithms that utilise the quantum-classical Liouville equation will be qualitatively and quantitatively compared. In addition, stochastic sampling schemes for nonadiabatic transitions will be studied, and a new sampling scheme is introduced [D. A. Uken et al., Phys. Rev. E. 88, 033301 (2013)] which proves to have a dramatic advantage over existing techniques, allowing far longer simulation times to be calculated reliably.

D. J. Thouless, H. S. W. Massey, R. W. Hellwarth

P. Enders

The "quantum equation" should "carry the quantum conditions in itself" (Second Commun., p. 511); 2. There should be a special mathematical method for solving the stationary Schrödinger equation, which accounts for the non-classical character of the quantization problem, ie, which is different from the classical eigenvalue methods for calculating the (eigen)modes of strings, resonators and so on (ibid., p. 513);3. The derivation should uniquely decide, that the energy rather than the frequency values are discretized (ibid., pp.511, 519), since only the former means true quantization, while the latter corresponds to the classical discretization mentioned in requirement 2;4. The use of the classical expressions for the kinetic and potential energies should be justified (Fourth Commun., p. 113).Schrödinger's requirements 1 and 2 mean, that -contrary to the very title of the papersquantization is actually not an eigenvalue problem.For in the latter the discretization is imposed not by the differential equation itself, but by the boundary conditions, and this is the classical discretization for standing waves in organ pipes etc.I will fulfill all four requirements by treating quantization as a selection problem.The number of stationary states of a quantum system is smaller than that of a classical system [6].I will describe, (i), the selection of quantum systems out of the set of all mechanical systems and, (ii), the selection of their stationary states out of the continuum of classical stationary states.Earlier arguing [19][10] is improved and extended in several essential points. Elements of an Eulerian representation of classical mechanics Euler's axiomsLeonhard Euler [20][21][22][23][24][25] was the first to apply the calculus to all areas of mathematics and mechanics of his time, and he developed novel methods, such as the calculus of variations and topology.Moreover, he worked out an axiomatic of mechanics, where only Newton's Law (axiom) 1 concerning the conservation of stationary states is retained as an axiom, while Newton's Laws (axioms) 2 and 3 concerning the change of stationary states are treated as problems to be solved (for a detailed account, see [19][10][45]).This allows for introducing alternative equations of motion without loosing the contact to CM.The existence of stationary states is postulated in the following axioms (as in Newton's axioms, rotatory motion is not considered). Axiom E0: Every body is either resting or moving.This means, that the subsequent axioms E1 and E2 are not independent of each other; they exclude each other and, at once, they are in harmony with each other [22].

S. Cruz Barrios, Maria Carolina Nemes

In the present work we have set up a scheme to treat field theoretical Lagrangians in the same bases of the well known non-relativistic many-body techniques. We show here that fermions and bosons can be treated quantum mechanically in a symmetric way and obtain results for the mean field approximation.

Luca Salasnich