Basic notions of quantum mechanics

Quantum axioms

Let us start with a brief recapitulation of quantum mechanics on the “how to” level. According to the standard lore, the instantaneous state of any quantum system (that is, everything that can be known about it at a given moment of time) is given by its wave function (state vector) – a complex-valued vector in some abstract Hilbert space; the nature of this space is determined by the system. All the observables (i.e., physical quantities defined for the system and determined by its state – e.g., the position or momentum of a free particle, the energy of an oscillator) are described by Hermitian operators defined in the same Hilbert space. All three elements – the Hilbert space, the state vector, and the set of observables – are necessary to describe the outcome of any experiment one could perform with the system. Since humans cannot directly observe the behaviour of quantum objects, these outcomes are also called measurements, being the result of using some classical apparatus in order to translate the state of a quantum system into the state of the apparatus, which can then be read out by the experimentalist. The classical (i.e., non-quantum) nature of the apparatus is essential, as we shall see in a moment.

It is always risky to combine well-known and well-tested notions in order to describe something new, since the future usage of such combinations is unpredictable. After “quantum leaps” were appropriated by the public at large, nobody, except physicists and some chemists, seems to realize that they are exceedingly small, and that breathless descriptions of quantum leaps in policy, economy, engineering and human progress in general may actually provide an accurate, if sarcastic, picture of the reality. When the notion of the “marketplace of ideas” was embraced by academia, scientists failed to recognize that among other things this means spending 95% of your resources on marketing instead of research. Nevertheless, “quantum engineering” seems a justified and necessary name for the fast-expanding field, which, in spite of their close relations and common origins, is quite distinct from both “nanotechnology” and “quantum computing” in scope, approaches and purposes. Its subject covers the theory, design, fabrication and applications of solid-statebased structures, which can maintain quantum coherence in a controlled way. In a nutshell, it is about how to build devices out of solid-state qubits, and how they can be used.

The miniaturization of electronic devices to the point where quantum effects must be taken into account produced much of the momentum behind nanotechnology, together with the need to better understand and control matter on the molecular level coming from, e.g., molecular biology and biochemistry (see, e.g., Mansoori, 2005, Chapter 1).

Josephson effect

Superconductivity: A crash course

The transition from the theoretical description of hypothetical building blocks of a quantum coherent device to something which can be actually fabricated and controlled is made much easier by the existence of superconductivity. This phenomenon, roughly speaking, allows a macroscopic quantum coherent flow of electrons in a sufficiently cold piece of an appropriate material by establishing a specific long-range order among them. Due to this one can, for example, use macroscopically different states of a superconductor as quantum states of a qubit, with the obvious advantage over “microscopically quantum” systems (like actual atoms) from the point of view of control, measurement and, last but not least, fabrication of structures with the desired parameters and on the desired scale.

The phenomenon of superconductivity – the history of its discovery, experimental manifestations, theoretical explanation, open questions, relevance to other branches of physics, and technological applications – requires a thorough treatment, which can be found in any number of books. For our purposes Tinkham (2004) will provide more than sufficient background.

We will not wander into the field of exotic/high-temperature superconductors for two simple reasons: the quantum coherent behaviour of the kind we need to realize qubits or other quantum coherent devices has not yet been properly and/or routinely achieved in these systems; and the fabrication of such devices is not yet reliable enough or even feasible.

In this chapter we discuss a theory for the freezing of an isotropic liquid into a crystalline solid state with long-range order. The transformation is a first-order phase transition with finite latent heat absorbed in the process. We focus on a first-principles orderparameter theory of freezing that originated from the pioneering work of Ramakrishnan and Yussouff (1979). The theory approaches the problem from the liquid side and views the crystal as a liquid with grossly inhomogeneous density characterized by a lower symmetry of the corresponding lattice. This is in contrast to description of the crystal in terms of phonons. The crucial quantity characterizing the physical state of the system in this non-phonon-based model is the average one-particle density function n(x). The thermodynamic description of either phase involves a corresponding extremum principle for a relevant potential. The latter, obtained as a functional of the one-particle density n(x) and the stable thermodynamic state of the system, is identified by the corresponding density required for invoking the extremum principle. This approach, which is generally referred to as the density-functional theory (DFT) of freezing (Haymet, 1987; Baus, 1987, 1990; Singh, 1991; Löwen, 1994; Ashcroft 1996), has been improved over the years and successfully applied for the study of liquid-to-crystal transitions in various simple liquids, the solid–liquid interface, two-dimensional systems, metastable glassy states, etc. For applications of density-functional methods in statistical mechanics there exist general reviews (Evans, 1979; Henderson, 1992).

The fluctuating-hydrodynamics approach discussed earlier takes into account only the transport properties at the level of completely uncorrelated motion of the fluid particles. The corresponding dissipative processes are expressed in terms of bare transport coefficients of the fluid. The strongly correlated motion of the fluid particles which occurs at high density is not take into consideration here. This is reflected through the Markov approximation of the transport coefficients and the short correlation of the corresponding noise representing the fast degrees of freedom in the system. The Markovian equations for the collective modes involving frequency-independent transport coefficients constitute a model for the dynamics of fluids with exponential relaxation of the fluctuations. The corresponding equations of motion for the collective modes are linear. However, exceptions occur in certain situations in which the description of the dynamics cannot be reduced to a set of linearly coupled fluctuating equations with frequency-independent transport coefficients. In this chapter we will consider the nonlinear dynamics of the hydrodynamic modes for studying the strongly correlated motion of the particles in a dense fluid.

Nonlinear Langevin equations

We present in this section the formulation of a set of nonlinear stochastic equations for the dynamics of the many-particle system. We first discuss the physical motivation for extension of the fluctuating-hydrodynamics approach to include nonlinear coupling of the slow modes.

We have discussed the construction of the nonlinear Langevin equations for the slow modes in a number of different systems in the previous chapter. Next, we analyze how the nonlinear coupling of the hydrodynamic modes in these equations of motion affects the liquid dynamics. In particular, we focus here on the case of a compressible liquid in the supercooled region. In this book we will primarily follow an approach in which the effects of the nonlinearities are systematically obtained using graphical methods of quantum field theory. Such diagrammatic methods have conveniently been used for studying the slow dynamics near the critical point (Kawasaki, 1970; Kadanoff and Swift, 1968) or turbulence (Kraichnan 1959a, 1961a; Edwards, 1964). The present approach, which is now standard, was first described by Martin, Siggia, and Rose (1973). The Martin–Siggia–Rose (MSR) field theory, as this technique is named in the literature, is in fact a general scheme applied to compute the statistical dynamics of classical systems.

The field-theoretic method presented here is an alternative to the so-called memoryfunction approach. The latter in fact involves studying the dynamics in terms of non-Markovian linearized Langevin equations (see, for example, eqn. (6.1.1) which are obtained in a formally exact manner with the use of so called Mori–Zwanzig projection operators. This projection-operator scheme is described in Appendix A7.4). The generalized transport coefficients or the so-called memory functions in this case are frequency-dependent and can be expressed in terms of Green–Kubo forms of integrals of time correlation functions.

In Chapter 4 we introduced the Kauzmann temperature TK as a possible limiting temperature for the existence of the supercooled liquid phase. The original hypothesis due to Kauzmann proposes eventual crystallization in the supercooled liquid at very low temperatures as a possible way out of the paradoxical situation in which the entropy of the disordered state becomes less than that of the crystal. Another possible explanation of the Kauzmann paradox could be that the simple extrapolation of the high-temperature result to very low temperature is not correct and the entropy difference between supercooled liquid and crystal remains finite down to very low temperature (Donev et al., 2006; Langer, 2006a, 2006b, 2007), finally going to zero only near T = 0. Either of these resolutions, however, leaves us with no understanding of the dramatic slowing down and associated phenomenology of the supercooled region above Tg. The difference of the entropy of the supercooled liquid from that of the solid having only vibrational motion around a frozen structure represents the entropy due to large-scale motion and is identified with the configurational entropy Sc of the liquid. The rapid disappearance of the configurational entropy of the disordered liquid or the so-called “entropy crisis” poses an important question that is essential for our understanding of the physics of the glass-transition phenomena and the divergence of the relaxation time at Tg. Apart from having a characteristic large viscosity, the supercooled liquid shows a discontinuity in specific heat cp at Tg due to freezing of the translational degrees of freedom in the liquid.

In the previous chapters we have discussed the transition of the liquid from a disordered fluid state to an ordered crystalline state through a first-order phase transition at the melting or freezing point Tm. In the present chapter we consider the behavior of the liquid supercooled below Tm and the associated phenomenon of the liquid–glass transition.

The liquid–glass transition

Almost all liquids can, under suitable conditions, be supercooled below the freezing point Tm while avoiding crystallization. The undercooled liquid continues to remain in the disordered state and is characterized by very rapidly increasing viscosity with decreasing temperature. The characteristic relaxation time τ of the liquid grows with increasing supercooling. Eventually, at low enough temperature, the supercooled liquid becomes so viscous that it can hold shear stress and behaves like a solid. At this stage the supercooled liquid is said to have transformed into a glass. The latter is an amorphous solid without long-range order. It is in fact in a nonequilibrium state on the time scale of the experiment. The relaxation time τ required for the supercooled liquid to equilibrate is longer than the typical time scale τexp of an experiment. Apart from the viscosity, other dynamic quantities such as the diffusion coefficient, dielectric response function, and conductivity change strongly with increasing supercooling. In contrast, thermodynamic properties such as the specific heat, enthalpy, compressibility, and static structure factor do not show any strong change with supercooling.

If the liquid is cooled beyond the corresponding freezing point Tm at which the liquid and crystalline phases coexist in equilibrium, a thermodynamic driving force builds up towards forming the crystal. In this chapter we will discuss how the liquid transforms into a crystal, focusing on how the changes in the liquid are initiated and on the nature of the crystalline region that is formed. This process is referred to as nucleation. The thermodynamic force favoring the formation of the crystal seed in the supercooled liquid competes with the process of forming an interface between the solid and the liquid. The cost of the interfacial free energy therefore presents a barrier to the formation of the new phase. Only when the driving force is made large enough by moving deep into the supercooled state does crystallization occur on laboratory time scales. Thus pure water can be cooled to -20 °C or below without freezing. Our focus here will be mainly on the process of crystallization of solid from the melt. The condensation of vapor into liquid is a very thoroughly studied process that has been discussed in various reviews (Stanley, 1971; Evans, 1979; ten Wolde et al., 1998). For condensation from a low-density gas or crystallization from dilute solution, it is easier to identify the nucleating bubbles since they differ widely in composition from the surrounding phase.

This book is aimed at teaching the important concepts of the various theories of statistical physics of dense liquids, freezing, and the liquid–glass transition. Both thermodynamic and time-dependent phenomena relating to transport properties are discussed. The standard tools of statistical physics of the dense liquid state and the associated technicalities needed to learn them are included in the presentation. Details of some of the calculations have been included, whenever needed, in the appendices at the ends of chapters. I hope this will make the book more accessible to beginners in this very active field of research. The book is expected to be useful for graduate students and researchers working in the area of soft-condensed-matter physics, chemical physics, and the material sciences as well as for chemical engineers.

We now give a brief description of what is in the book. The first chapter reviews the basics of statistical mechanics necessary for studying the physics of the liquid state. Key concepts of equilibrium and nonequilibrium statistical mechanics are presented. The topics covered here have been chosen keeping in mind the theories and concepts covered in the subsequent chapters of the book. Following this introductory chapter, we focus on the physics of liquids near freezing. In Chapter 2, we demonstrate how the disordered liquid state as well as the crystalline state of matter with long-range order can be understood in a unified manner using thermodynamic extremum principles.