of a system in terms of a wave function Ψ is by itself a description in terms of

a probability density: Ψ— Ψ(r 1 , . . . , r n , t) is the probability density that the

particles 1, . . . , n are at positions r 1 , . . . , r n at time t. Even if the initial state

is precisely de¬ned by a sharp wave function, the wave function evolves under

the quantum-dynamical equations to yield a probability distribution rather

than a precise trajectory. From the wave function evolution expectation

values (i.e., average properties over a probability distribution) of physical

observables can be obtained by the laws of quantum mechanics, but the

wave function cannot be interpreted as the (unmeasurable) property of a

single particle.

Such a description ¬ts in well with equations for the evolution of prob-

ability distributions in classical systems, but it is not compatible with de-

scriptions in terms of classical trajectories. This fundamental di¬erence in

interpretation lies at the basis of the di¬culties we encounter if we attempt to

use a hybrid quantum/classical description of a complex system. If we insist

on a trajectory description, the quantum-dynamical description should be

reformulated by some kind of contraction and sampling to yield trajectories

that have the same statistical properties as prescribed by the quantum evo-

lution. It is for the same reason of incompatibility of quantum descriptions

and trajectories that quantum corrections to classical trajectories cannot be

unequivocally de¬ned, while quantum corrections to equilibrium probability

distributions can be systematically derived.

1.4 Further reading

While Part I treats most of the theoretical models behind simulation and

Part II provides a fair amount of background knowledge, the interested

reader may feel the need to consult standard texts on further background

material, or consult books on aspects of simulation and modeling that are

not treated in this book. The following literature may be helpful.

1 S. Gasiorowicz, Quantum Physics (2003) is a readable, over 30 years

old but updated, textbook on quantum physics with a discussion of

the limits of classical physics.

2 L. I. Schi¬, Quantum Mechanics (1968). A compact classic textbook,

slightly above the level of Gasiorowicz.

3 E. Merzbacher, Quantum Mechanics (1998) is another classic text-

book with a complete coverage of the main topics.

4 L. D. Landau and E.M. Lifshitz, Quantum Mechanics (Non-relativis-

1.4 Further reading 15

tic Theory) (1981). This is one volume in the excellent series “Course

of Theoretical Physics.” Its level is advanced and sophisticated.

5 P. A. M. Dirac, The Principles of Quantum Mechanics (1958). By

one of the founders of quantum mechanics: advisable reading only

for the dedicated student.

6 F. S. Levin, An Introduction to Quantum Theory (2002) introduces

principles and methods of basic quantum physics at great length. It

has a part on “complex systems” that does not go far beyond two-

electron atoms.

7 A. Szabo and N. S. Ostlund, Modern Quantum Chemistry (1982)

is a rather complete textbook on quantum chemistry, entirely de-

voted to the solution of the time-independent Schr¨dinger equation

o

for molecules.

8 R. McWeeny, Methods of Molecular Quantum Mechanics (1992) is

the classical text on quantum chemistry.

9 R. G. Parr and W. Yang, Density Functional Theory (1989). An

early, and one of the few books on the still-developing area of density-

functional theory.

10 F. Jensen, Introduction to Computational Chemistry (2006). First

published in 1999, this is a modern comprehensive survey of methods

in computational chemistry including a range of ab initio and semi-

empirical quantum chemistry methods, but also molecular mechanics

and dynamics.

11 H. Goldstein, Classical Mechanics (1980) is the classical text and

reference book on mechanics. The revised third edition (Goldstein

et al., 2002) has an additional chapter on chaos, as well as other

extensions, at the expense of details that were present in the ¬rst

two editions.

12 L. D. Landau and E. M. Lifshitz, Mechanics (1982). Not as complete

as Goldstein, but superb in its development of the theory.

13 L. D. Landau and E. M. Lifshitz, Statistical Physics (1996). Basic

text for statistical mechanics.

14 K. Huang, Statistical Mechanics (2nd edn, 1987). Statistical mechan-

ics textbook from a physical point of view, written before the age of

computer simulation.

15 T. L. Hill, Statistical Mechanics (1956). A classic and complete, but

now somewhat outdated, statistical mechanics textbook with due at-

tention to chemical applications. Written before the age of computer

simulation.

16 Introduction

16 D. A. McQuarrie, Statistical Mechanics (1973) is a high-quality text-

book, covering both physical and chemical applications.

17 M. Toda, R. Kubo and N. Saito, Statistical Physics. I. Equilibrium

Statistical Mechanics (1983) and R. Kubo, M. Toda and N. Hashit-

sume Statistical Physics. II. Nonequilibrium Statistical Mechanics

(1985) emphasize physical principles and applications. These texts

were originally published in Japanese in 1978. Volume II in par-

ticular is a good reference for linear response theory, both quantum-

mechanical and classical, to which Kubo has contributed signi¬cantly.

It describes the connection between correlation functions and macro-

scopic relaxation. Not recommended for chemists.

18 D. Chandler, Introduction to Modern Statistical Mechanics (1987). A

basic statistical mechanics textbook emphasizing ¬‚uids, phase tran-

sitions and reactions, written in the age of computer simulations.

19 B. Widom, Statistical Mechanics, A Concise Introduction for Chem-

ists (2002) is what it says: an introduction for chemists. It is well-

written, but does not reach the level to treat the wonderful inventions

in computer simulations, such as particle insertion methods, for which

the author is famous.

20 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids

(1987). A practical guide to molecular dynamics simulations with

emphasis on the methods of solution rather than the basic underlying

theory.

21 D. Frenkel and B. Smit, Understanding Molecular Simulation (2002).

A modern, instructive, and readable book on the principles and prac-

tice of Monte Carlo and molecular dynamics simulations.

22 D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations

in Statistical Physics (2005). This book provides a detailed guide to

Monte Carlo methods with applications in many ¬elds, from quantum

systems to polymers.

23 N. G. van Kampen, Stochastic Processes in Physics andChemistry

(1981) gives a very precise and critical account of the use of stochastic

and Fokker“Planck type equations in (mostly) physics and (a bit of)

chemistry.

24 H. Risken, The Fokker“Planck equation (1989) treats the evolution

of probability densities.

25 C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chem-

istry and the Natural Sciences (1990) is a reference book for modern

developments in stochastic dynamics. It treats the relations between

stochastic equations and Fokker“Planck equations.

1.4 Further reading 17

26 M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (1986)

is the already classic introduction to mesoscopic treatment of poly-

mers.

27 L. D. Landau and E. M. Lifshitz, Fluid Mechanics (1987) is an excel-

lent account of the physics behind the equations of ¬‚uid dynamics.

28 T. Pang, Computational Physics (2006). First published in 1997,

this is a modern and versatile treatise on methods in computational

physics, covering a wide range of applications. The emphasis is on the

computational aspects of the methods of solution, not on the physics

behind the models.

29 F. J. Vesely, Computational Physics, An Introduction (2nd ed., 2001)

is an easily digestable treatment of computational problems in phys-

ics, with emphasis on mathematical and computational methodology

rather than on the physics behind the equations.

30 M. Griebel, S. Knapek, G. Zumbusch and A. Caglar, Numerische

Simulation in der Molek¨ldynamik (2003) gives many advanced de-

u

tails on methods and algorithms for dynamic simulation with par-

ticles. The emphasis is on computational methods including paral-

lelization techniques; programs in C are included. Sorry for some

readers: the text is in German.

31 D. Rapaport, The Art of Molecular Dynamics Simulation (2004) is

the second, reworked edition of a detailed, and readable, account of