We present a software package DiracQ, for use in quantum many-body Physics. It is designed for helping with typical algebraic manipulations that arise in quantum Condensed Matter Physics and Nuclear Physics problems, and also in some subareas of Chemistry. DiracQ is invoked within a Mathematica session, and extends the symbolic capabilities of Mathematica by building in standard commutation and anticommutation rules for several objects relevant in many-body Physics. It enables the user to carry out computations such as evaluating the commutators of arbitrary combinations of spin, Bose and Fermi operators defined on a discrete lattice, or the position and momentum operators in the continuum. Some examples from popular systems, such as the Hubbard model, are provided to illustrate the capabilities of the package.
We present a software package DiracQ, for use in quantum many-body Physics. It is designed for helping with typical algebraic manipulations that arise in quantum Condensed Matter Physics and Nuclear Physics problems, and also in some subareas of Chemistry. DiracQ is invoked within a Mathematica session, and extends the symbolic capabilities of Mathematica by building in standard commutation and anticommutation rules for several objects relevant in many-body Physics. It enables the user to carry out computations such as evaluating the commutators of arbitrary combinations of spin, Bose and Fermi operators defined on a discrete lattice, or the position and momentum operators in the continuum. Some examples from popular systems, such as the Hubbard model, are provided to illustrate the capabilities of the package.
In several problems of quantum many body physics, one is required to handle complex expressions originating in the non-commutative nature of quantum operators. Their manipulation requires precise ordering and application of simplification rules. This can be cumbersome, tedious and error prone, and often a challenge to the most expert researcher. In this paper we present a software package DiracQ to facilitate such manipulations. The package DiracQ consists of functions based upon and extending considerably the symbolic capabilities of Mathematica. With DiracQ, one can proceed with large scale algebraic manipulations of expressions containing combinations of ordinary numbers or symbols (the c-numbers) and arbitrary sets of non-commuting variables (the q-numbers) with user defined properties. The DiracQ package is user extendable and comes encoded with the algebraic properties of several standard operators in popular usage. These include Fermionic and Bosonic creation and annihilation operators, spin operators, and canonical position and momentum operators. An example book is provided with some suggestive calculations of large-scale algebraic manipulations.
Computational many-body theories in quantum chemistry, condensed matter, and nuclear physics aim to provide sufficiently accurate description of and insights into the motion of many interacting particles. Due to their intrinsic complexity, the development of such theories generally involves very complex, tedious, and error-prone symbolic manipulations. Here a complete solution to automate the symbolics in many-body theory development is attempted. General data structures based on an existing computer algebra system are designed to specifically address the symbolic problems for which there is currently no satisfactory handling. Based on the data structures, algorithms are given to accomplish common symbolic manipulations and simplifications. Noncommutative algebraic systems, tensors with symmetry, and symbolic summations can all enjoy deep simplifications efficient enough for theories of very complex form. After the symbolic derivation, novel algorithms for automatic optimization of tensor contractions and their sums are devised, which can be used together with automatic code generation tools. In this way, the burden of symbolic tasks in theory development can be vastly reduced, with the potential to spare scientists more time and energy for the actual art and science of many-body theories.
Solving problems in quantum mechanics is an essential skill and research activity for scientists, engineers and others. Nowadays, the labour of scientific computation has been greatly eased by the advent of computer algebra packages. These do not merely perform number-crunching, but enable users to manipulate algebraic expressions and equations symbolically. For example, differentiation and integration can now be carried out algebraically by the computer. This book collects standard and advanced methods in quantum mechanics and implements them using REDUCE, a popular computer algebra package. Throughout, sample programs and their output have been displayed alongside explanatory text, making the book easy to follow. Selected problems have also been implemented using two other popular packages, MATHEMATICA and MAPLE, and in the object-oriented programming language C++. Besides standard quantum mechanical techniques, modern developments in quantum theory are also covered. These include Fermi and Bose Operators, coherent states, gauge theory and quantum groups. All the special functions relevant to quantum mechanics (Hermite, Chebyshev, Legendre and more) are implemented. A careful balance is struck between practical computation and the underlying mathematical concepts, making the book well-suited for use with quantum mechanics courses.
We present QCommute, a software tool implemented in C++ for symbolic computation of nested commutators between a Hamiltonian and local observables in quantum many-body spin-1/2 systems on one-, two-, and three-dimensional hypercubic lattices. The computation is performed algebraically directly in the thermodynamic limit, and the Hamiltonian parameters are kept symbolic. Importantly, this way the entire parameter space is covered in a single run. The implementation supports extensive parallelization to achieve high computational performance. QCommute can serve as a computational backend for Heisenberg-picture approaches to quantum dynamics in strongly correlated regimes, ranging from direct Taylor expansion in time to advanced techniques such as the recursion method.
Solving problems in quantum mechanics is an essential skill and research activity for physicists, mathematicians, engineers and others. Nowadays, the labor of scientific computation has been greatly eased by the advent of computer algebra packages, which do not merely perform number crunching, but also enable users to manipulate algebraic expressions and equations symbolically. For example, the manipulations of noncommutative operators, differentiation and integration can now be carried out algebraically by the computer algebra package. This book collects standard and advanced methods in quantum mechanics and implements them using SymbolicC++ and Maxima, two popular computer algebra packages. Throughout, the sample programs and their outputs are accompanied with explanatory text of the underlying mathematics and physics explained in detail. Selected problems have also been implemented using two other popular packages Mathematica and Maple while some problems are implemented in C++. Modern developments in quantum theory are covered extensively, beyond the standard quantum mechanical techniques. The new research topics added to this second edition are: entanglement, teleportation, Berry phase, Morse oscillator, Magnus expansion, wavelets, Pauli and Clifford groups, coupled Bose Fermi systems, super-Lie algebras, etc.
Few-body physics explores quantum systems of a small number of particles, bridging the gap between single-particle and many-body regimes. To provide an accessible tool for such studies, we present FewBodyToolkit.jl, a Julia package for quantum few-body simulations. The package supports general two- and three-body systems in various spatial dimensions with arbitrary pair-interactions, and allows to calculate bound and resonant states. The implementation is based on the well-established Gaussian expansion method and we illustrate the package’s capabilities through benchmarks and research examples. The package comes with documentation and examples, making it useful for research, teaching, benchmarking, and method development.