Using Mathematica for Quantum Mechanics

This book revisits many of the problems encountered in introductory quantum mechanics, focusing on computer implementations for finding and visualizing analytical and numerical solutions. It subsequently uses these implementations as building blocks to solve more complex problems, such as coherent laser-driven dynamics in the Rubidium hyperfine structure or the Rashba interaction of an electron moving in 2D. The simulations are highlighted using the programming language Mathematica. No prior knowledge of Mathematica is needed; alternatives, such as Matlab, Python, or Maple, can also be used.

Demonstrates how complex problems in quantum mechanics can be solved using computational tools

Includes practical exercises after each chapter, promoting an active understanding of the subject

Bridges the gap between simple analytic calculations and large-scale computations for molecular structures, crystalline solids, and lattice models

Written for students with no prior background in the programming language Mathematica

Auteur

PD Dr. Roman Schmied studied physics at the École Polytechnique Fédérale de Lausanne and the University of Texas at Austin. He wrote his diploma thesis on helium nanodroplet spectroscopy at Princeton University with Kevin Lehmann and Giacinto Scoles, and later obtained his Ph.D. from the same group, working on the superfluidity of helium nanodroplets and on the spectroscopy of molecules solvated within these droplets. He carried out his postdoctoral work at the Max Planck Institute of Quantum Optics in Garching, Germany, where he first began working with quantum simulators and quantum simulations. After a short stay at the NIST ion storage group in Boulder, USA, he took on his current position at the University of Basel, where he was habilitated in 2017. Since 2016 he has also been working at the University s Human Optics Lab, where he is currently using digital technology for child health, particularly eye health.

Contenu

1 Wolfram language overview 1.1 introduction1.1.1 exercises1.2 variables and assignments1.2.1 immediate and delayed assignments1.2.2 exercises1.3 four kinds of bracketing1.4 prefix and postfix1.4.1 exercises1.5 programming constructs1.5.1 procedural programming1.5.2 exercises1.5.3 functional programming1.5.4 exercises1.6 function definitions1.6.1 immediate function definitions1.6.2 delayed function definitions1.6.3 functions that remember their results1.6.4 functions with conditions on their arguments1.6.5 functions with optional arguments1.7 rules and replacements1.7.1 immediate and delayed rules1.7.2 repeated rule replacement1.8 many ways to define the factorial function1.8.1 exercises1.9 vectors, matrices, tensors1.9.1 vectors1.9.2 matrices1.9.3 sparse vectors and matrices1.9.4 matrix diagonalization1.9.5 tensor operations1.9.6 exercises1.10 complex numbers1.11 units2 quantum mechanics 2.1 basis sets and representations 2.1.1 incomplete basis sets 2.1.2 exercises 2.2 time-independent Schrödinger equation 2.2.1 diagonalization 2.2.2 exercises 2.3 time-dependent Schrödinger equation 2.3.1 time-independent basis 2.3.2 time-dependent basis: interaction picture 2.3.3 special case: I (t), (tt)l = 0 (t, tt) H H2.3.4 special case: time-independent Hamiltonian 2.3.5 exercises 2.4 basis construction 2.4.1 description of a single degree of freedom 2.4.2 description of coupled degrees of freedom 2.4.3 reduced density matrices 2.4.4 exercises 3 spin systems 3.1 quantum-mechanical spin and angular momentum operators 3.1.1 exercises 3.2 spin-1/2 electron in a dc magnetic field 3.2.1 time-independent Schrödinger equation 3.2.2 exercises 3.3 coupled spin systems: 87Rb hyperfine structure 3.3.1 eigenstate analysis 3.3.2 magic magnetic field 3.3.3 coupling to an oscillating magnetic field 3.3.4 exercises 3.4 coupled spin systems: Ising model in a transverse field 3.4.1 basis set 3.4.2 asymptotic ground states 3.4.3 Hamiltonian diagonalization 3.4.4 analysis of the ground state 3.4.5 exercises 4 real-space systems 4.1 one particle in one dimension 4.1.1 computational basis functions 4.1.2 example: square well with bottom step 4.1.3 the Wigner quasi-probability distribution 4.1.4 1D dynamics in the square well 4.1.5 1D dynamics in a time-dependent potential 4.2 non-linear Schrödinger equation 4.2.1 ground state of the non-linear Schrödinger equation 4.3 several particles in one dimension: interactions 4.3.1 two identical particles in one dimension with contact interaction 4.3.2 two particles in one dimension with arbitrary interaction 4.4 one particle in several dimensions 4.4.1 exercises 5 combining space and spin 5.1 one particle in 1D with spin 5.1.1 separable Hamiltonian 5.1.2 non-separable Hamiltonian 5.1.3 exercises5.2 one particle in 2D with spin: Rashba coupling5.2.1 exercises 5.3 phase-space dynamics in the JaynesCummings model exercises