Theoretical Concepts and Numerical Methods

Paulsamy Muruganandam, Ramaswamy Radha

This book offers a detailed introduction to Bose-Einstein condensates (BECs) and nonlinear quantum systems, tailored for advanced undergraduates and researchers. It bridges foundational concepts with modern developments in quantum condensed matter. Starting with the history and experimental milestones of BECs, it explores theoretical models including mean-field and beyond-mean-field approaches for both scalar and vectorial condensates. Analytical techniques such as inverse scattering and soliton theory are covered alongside numerical methods for solving the Gross-Pitaevskii equation. Applications span optical solitons, quantum vortices, and exciton-polariton physics. Each topic is supported by examples, problem sets, and computational tools, equipping readers with both theoretical insight and practical skills to engage with cutting-edge research areas like quantum turbulence and supersolids. Part of IOP Series in Advances in Optics, Photonics and Optoelectronics. Key features • Introduces the subject at the very basic level. • Includes the analytical and theoretical methods. • Contains numerical methods. • Provides worked examples, end-of-chapter summaries, and homework problems.

Szilárd Szalay, Max J. Pfeffer, V. Murg +4 more

The treatment of high‐dimensional problems such as the Schrödinger equation can be approached by concepts of tensor product approximation. We present general techniques that can be used for the treatment of high‐dimensional optimization tasks and time‐dependent equations, and connect them to concepts already used in many‐body quantum physics. Based on achievements from the past decade, entanglement‐based methods—developed from different perspectives for different purposes in distinct communities already matured to provide a variety of tools—can be combined to attack highly challenging problems in quantum chemistry. The aim of the present paper is to give a pedagogical introduction to the theoretical background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. Among the various optimization tasks, we will discuss only those which are connected to a controlled manipulation of the entanglement which is in fact the key ingredient of the methods considered in the paper. The selected topics will be covered according to a series of lectures given on the topic “New wavefunction methods and entanglement optimizations in quantum chemistry” at the Workshop on Theoretical Chemistry, February 18–21, 2014, Mariapfarr, Austria. © 2015 Wiley Periodicals, Inc.

Malte Henkel

Lucjan Piela

Kristoffer Aronsen, Simon Sandell

In this report we examine the concept of PT -symmetric quantum mechanics and analyze two such systems. PT -symmetric quantum mechanics is a new branch of quantum mechanics, first proposed by Bender and Boettcher in 1998 and has been the subject of much research since then. Various important aspects of PT -symmetric quantum mechanics is explained, such as asymptotic behaviour of solutions of the Schrödinger equation, as well as the concept of the C operator that is unique to PT -symmetric quantum mechanics. We briefly discuss both the concept of broken and unbroken PT -symmetry, and the relation between PT -symmetric and Hermitian quantum mechanics through similarity transforms. We specifically direct our attention to two theoretical quantum mechanical systems described by Hamiltonians with the property of being PT -symmetric. For these systems we determine the energy levels, one by analytical means and the other by use of numerical methods. To determine the energy levels numerically we have used the builtin ODE-solver ode45 in MATLAB in conjunction with the equation solver fzero. Our results are compared to those of previous studies and our methods and results are discussed.