Statistical Quantum Mechanics

12 Key excerpts on "Statistical Quantum Mechanics"

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  eBook - PDF

  Commonly Asked Questions in Thermodynamics

  • Marc J. Assael, William A. Wakeham, Anthony R. H. Goodwin, Stefan Will, Michael Stamatoudis(Authors)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  59 2 Chapter What Is Statistical Mechanics? 2.1 INTRODUCTION Chapter 1 dealt with the definitions of many of the quantities required for the macroscopic description of the thermodynamic behavior of systems viewed as continua, including the definition of a system. However, we are familiar with the notion that all matter is made up of atomic or molecular entities, and it is the purpose of statistical mechanics to provide a microscopic descrip-tion of the behavior of a thermodynamic system in terms of the properties, interactions, and motions of the atoms or molecules that make up the system. Because macroscopic thermodynamic systems contain very large numbers of molecules, the task of statistical mechanics is not to describe exactly what hap-pens to every single molecule, but rather to derive results that pertain to the complete assembly of molecules that comprise the system in a probabilistic manner. The atoms and molecules that comprise the system are best described using quantum mechanics rather than classical mechanics so that is the basis for the development of the theory of statistical mechanics. The solution of Schrödinger’s equation of quantum mechanics is a wave that describes the probable state of the system that includes a description of the quantum states (eigenstates) and energy levels (energy eigenvalues) an individ-ual and the system can attain; in quantum theory the energy levels are discrete. It is very much easier to solve Schrödinger’s equation for a single molecule (or realistically for a single atom) than for a system of N molecules or atoms to obtain the quantum states or energy levels, so we begin with that problem. For a single, relatively simple molecule (such as nitrogen) the problem of solving Schrödinger’s equation is made tractable by separation of the modes of motion of the molecule (translation of the centre of mass, rotation, and vibration) so that each is handled independently.

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  Lectures on Quantum Mechanics

  A Primer for Mathematicians

  • Philip L. Bowers(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  408 An Introduction to Quantum Statistical Mechanics suggests strongly that there will be significant differences between the behavior of aggregates of bosons and fermions. In this lecture, we offer an introduction to the statistical mechanics of quantum aggregates. Our goal is to derive the distribution functions that govern the behavior of the three types of aggregates – ensembles of distinguishable quanta, of bosons, and of fermions – and to determine their most likely configura- tions. Statistical mechanics is a vast subject, and a single lecture can accom- plish merely an inadequate preamble to this topic. The End Notes to the present lecture will point to various sources for more in-depth study. 26.1 Statistical Mechanics To avoid technicalities that have no bearing on the final results, we assume in this lecture that the state space H for a one-quantum system is of an unspecified, possibly very large but nonetheless finite, number of dimensions. This avoids the need to work in the completions of the tensor, bosonic, and fermionic products since the finite products are complete automatically. Consider, then, a system of N quanta for which the possible one- quantum energies are discrete, say ε 1 < ε 2 < · · · . These are the eigen- values of the Hamiltonian H on the state space H for the one-quantum system. We assume that the degeneracy of ε i is d i , meaning, of course, that the ε i -eigenspace is d i -dimensional. Let n = (n 1 , n 2 , . . . ) denote a multi-quanta configuration in which there are n i quanta in an ε i energy state. 1 Notice that a single configuration n may be realized by possibly many different linearly independent state vectors in the state space. We assume that the system is in thermal equilibrium and isolated from the environment with a total constant energy E. For the configura- tion n, we have ∑ i n i = N and ∑ i n i ε i = E.

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  Physical Chemistry

  Concepts and Theory

  • Kenneth S Schmitz(Author)
  • 2016(Publication Date)
  • Elsevier

    (Publisher)

  Chapter 13

  Quantum Statistical Mechanics

  Abstract

  Classical statistical mechanics and quantum statistical mechanics are both based on the statistical probability description of Nature. Classical statistical mechanics applies to a statistically significant number of particles, the bridge between the properties of individual particles and the bulk properties of a system. Temperature and pressure have no meaning if there is only one gas particle in a balloon. Quantum statistical mechanics is the study of probability potentials of individual particles as well as bulk properties. In both areas of statistical mechanics the correct counting of the degeneracy of states cannot be understated. The states of a pair of electrons and a pair of dice are compared to illustrate problems associated with the degeneracy of states.

  The distinguishing feature of quantum theory that has no classical counterpart is superposition of wave functions. The manner in which wave functions of particles with integer spin (Bose–Einstein particles) and half-integer spin (Fermi–Dirac particles) can be superimposed, and quantum states filled greatly affect the degeneracy of quantum states. Under conditions that the de Broglie wavelength (or thermal wavelength) exhibits substantial overlap, the differences in the statistical nature of Bose–Einstein and Fermi–Dirac particles are amplified.

  The grand partition function and weighting factors for Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac particles are compared. Bose–Einstein condensation and the lambda transition for helium-4 are examined, and overcounting of states is quantitatively examined with the Scatchard model for binding on a linear lattice.

  Keywords

  Bose–Einstein condensation; Bose–Einstein statistics; Bosons; Degeneracy of states; Dice as an analogue to quantum states and degeneracy; Exchange of particles; Fermi–Dirac statistics; Fermions; Lambda transition; Maxwell–Boltzmann statistics

  13.0. Introduction

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  Quantum Mechanics In Phase Space: An Overview With Selected Papers

  An Overview with Selected Papers

  • Thomas L Curtright, David B Fairlie, Cosmas K Zachos(Authors)
  • 2005(Publication Date)
  • World Scientific

    (Publisher)

  Yet there are serious difficulties in effecting such a reformulation. Classical statistical mechanics is a ‘ crypto-deter- ministic ’ theory, where each element of the probability distribution of the dynamical variables specifying a given system evolves with time according to deterministic laws of motion; the whole uncertainty is contained in the form of the initial distribu. tions. A theory based on such concepts could not give a satisfactory account of such non-deterministic effects as raclioa,ctive decay or spontaneous emission (cf. Whit- taker (2)). Classical statistical mechanics is, however, only a special case in the general theory of dynamical statistical (stochastic) processes. In the general case, there is the possibihty of ‘ dEusion’ of the probability ‘fluid’, so that the transformation with time of the probability distribution need not be deterministic in the classical sense. In this paper, we shall attempt to interpret quantum mechanics as a form of such a general statistical dynamics. I. QUA.NTUM KDTEMATICS 2. THE EXISTENCE OF PHASE-SPAUE DISTRIBUTIONS I N QUANTUM THEORY In the accepted statistical interpretation of quantum theory, the possible values of a dynamical variable s are the eigenvalues si of the corresponding operator (observable ) 7-2 168 100 J. E. MOYAL S in the Hilbert space of the state vectors. The probability of finding sc in a state @ is then equal to the square of the modulus I a$ l 2 of the projection a, of @ on the corre- sponding eigenvector 1C.i. A complete or irreducible representation for a given mechanical system is given by a set of commuting observables s such that their eigenvectors pi span the whole space, i.e. such that any ? , b = Ca,$i. Hence we obtain directly from II.

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  The Undivided Universe

  An Ontological Interpretation of Quantum Theory

  • David Bohm, Basil J. Hiley(Authors)
  • 2006(Publication Date)
  • Routledge

    (Publisher)

  4 ].) However, in this book it is not our main purpose to discuss quantum statistical mechanics, but rather to focus on the fact that an ontological interpretation of the basic quantum theory is possible. What we have done in this section is to show the general lines along which one can include quantum statistical mechanics within the framework of our approach.

  9.5 The stochastic explanation of quantum probabilities

  We have thus far been explaining quantum probabilities in terms of chaotic motions that are implied by the quantum laws themselves, with pure ensembles representing chaotic motions of the particles and mixed ensembles bringing in also chaotic variations in the quantum field. Whenever we have statistical distributions of this kind, however, it is always possible that these chaotic motions do not originate in the level under investigation, but rather that they arise from some deeper level. For example, in Brownian motion, small bodies which may contain many molecules undergo chaotic velocity fluctuations as a result of impacts originating at a finer molecular level. If we abstract these chaotic motions and consider them apart from their possible causes we have what is called a stochastic process which is treated in terms of a well-defined mathematical theory [5 ]. We can have two attitudes to such a stochastic process. The first is that it is a result of deeper causes that do not appear at the level under discussion. The second is that there is some intrinsic randomness in the basic motions themselves. In so far as we apply the ordinary mathematical treatment, we need not commit ourselves to either attitude. But of course, if we are thinking of possible models for the process then our attitude may make a difference, because the assumption of deeper causes implies that the stochastic treatment will break down at the finer level at which these causes are operating. (We shall discuss this possibility in chapter 14

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  Course of Theoretical Physics

  • L. D. Landau, E. M. Lifshitz(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  (5.4) has a twofold nature. It comprises both the averaging due to the probabilistic nature of the quantum description (even when as complete as possible) and the statistical averaging necessitated by the incompleteness of our information concerning the object considered. For a pure state only the first averaging remains, but in statistical cases both types of averaging are always present. It must be borne in mind, however, that these constituents cannot be separated; the whole averaging procedure is carried out as a single operation, and cannot be represented as the result of successive averagings, one purely quantum-mechanical and the other purely statistical.

  The statistical matrix in quantum statistics takes the place of the distribution function in classical statistics. The whole of the discussion in the previous sections concerning classical statistics and the, in practice, deterministic nature of its predictions applies entirely to quantum statistics also. The proof given in § 2 that the relative fluctuations of additive physical quantities tend to zero as the number of particles increases made no use of any specific properties of classical mechanics, and so remains entirely valid in the quantum case. We can therefore again assert that macroscopic quantities remain practically equal to their mean values.

  In classical statistics the distribution function (p , q ) gives directly the probability distribution of the various values of the coordinates and momenta of the particles of the body. In quantum statistics this is no longer true; the quantities

  wn

  give only the probabilities of finding the body in a particular quantum state, with no direct indication of the values of the coordinates and momenta of the particles.

  From the very nature of quantum mechanics, the statistics based on it can deal only with the determination of the probability distribution for the coordinates and momenta separately, not together, since the coordinates and momenta of a particle cannot simultaneously have definite values. The required probability distributions must reflect both the statistical uncertainty and the uncertainty inherent in the quantum-mechanical description. To find these distributions, we repeat the arguments given above. We first assume that the body is in a pure quantum state with the wave function (5.1)

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  eBook - ePub

  Quantum Mechanics

  An Introduction

  • Thomas Banks(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 12

  Quantum Statistical Mechanics

  12.1 Introduction

  The key difference between classical and quantum statistical mechanics of particles has to do with identity of particles. In both classical and quantum mechanics (QM), one can write a Hamiltonian for an N particle system, which has an S N permutation symmetry exchanging the particles. In both cases, we can consider states that are invariant under the symmetry, as well as states that are not. There is no fundamental principle, enunciated in our description of QM, which requires us to reject states that are not invariant under the symmetry group. Rather, the experimental facts of the world we live in require this rejection.

  The first indication of this was found by Gibbs, in his discussion of the entropy of mixing between two boxes of gas. If a partition is removed between two boxes containing different gases, then an elementary calculation shows that the entropy increases. There are more states available. However, if we treat the molecules of a single gas as distinguishable particles, which is to say particles whose states1 need not be symmetric, even though the Hamiltonian is, then exactly the same calculation shows that the entropy increases when we remove a partition between two identical boxes of gas. This does not agree with experimental observations on the thermodynamics of uniform gases.

  The solution, in both classical and quantum mechanics, is to view particles as localized excitations of fields. In this interpretation, fields are fundamental, particles are special states of such fields, and the identity of particle states under permutation is a consequence of the fact that permutation of the particles gives the same field configuration. The details of how this works in classical and quantum mechanics are quite different. In classical mechanics, particle-like behavior implies that the state is eternally localized in space, and only special nonlinear field equations have such soliton solutions. In QM, however, we have seen that eigenstates of the quantized field Hamiltonian, for linear field theories, have a particle interpretation. The field is an operator which adds or subtracts a particle to/from the system and the field configuration associated with a particle is interpreted as the matrix element of the field

  ϕs

  (x ) = ⟨0|ϕ (x )|s 〉 between the state |0〉 with no particles and a single particle state |s 〉. For linear field theories, this is equal to the single particle wave function of the particle state, and the spreading of wave packets is understandable in terms of the fact that free particle states are not localized eternally, because of the Heisenberg uncertainty relations.

  ϕs

  (x ) cannot be interpreted as a classical observable of the system. By contrast, coherent states, which are superpositions of states with different particle numbers, DO behave classically in the fully quantized theory.2

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  Equilibrium Statistical Mechanics

  • E. Atlee Jackson(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  3

  Statistical Mechanics

  1. THE BASIC ASSUMPTION OF STATISTICAL MECHANICS

  As we have already noted in the introduction, the object of statistical mechanics is to predict the macroscopic properties of systems, using certain assumptions concerning the microscopic composition of the system (types of atoms, forces between atoms, and so on), together with certain statistical assumpttons. It was argued that we need use only statistical information about the system because the macroscopic properties (pressure, temperature, internal energy, and so on) are due to the average behavior of all the atoms of the system. To illustrate this point, we shall now consider how “pressure” results from the microscopic interaction between the atoms of the system and a wall (or pressure gauge).

  In a box containing a gas, the atoms are moving in all directions. When they strike the wall of the box, their velocity is changed so that they move back toward the interior of the box. In order to alter the atom’s velocity, the wall must, of course, exert a force on the atom. An equal and opposite force is then exerted on the wall (action-reaction). Now the number of atoms that strike one square centimeter of the surface in one second is roughly a million billion billion, i.e., 1024 (for air at STP)! Obviously, a pressure gauge cannot differentiate between such frequent impacts. Therefore, it does not record the force that results from the reflection of each individual atom, but only the average force due to a vast number of impacts. It is this average force, divided by the area, which we call the “pressure.” This example illustrates the general fact that all macroscopic properties of a system are averages of some microscopic behavior

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  Modern Physics

  • Kenneth S. Krane(Author)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Thus p(2) = 4∕40 = 0.10, and we expect that 10% of our measurements would find E = 2. Similarly, p(1) = 0.30 and p(0) = 0.60. The statistical analysis of a complex system gives us a way of describing the state of the system, its average properties, and its evolution in time. Physics is often concerned with exactly these details, so the statistical analysis is very useful indeed. It can be applied to systems in which the number of particles is very small, such as a nucleus with the order of 10 2 particles, or very large, such as a star like the Sun with 10 57 particles. Our next task is to determine whether there are differences between the statistical behaviors of classical and quantum systems. 10.2 CLASSICAL AND QUANTUM STATISTICS To illustrate the differences between classical and quantum statistics, we first consider another example similar to the one of the previous section: the ∗ The second law is often expressed in terms of the entropy S by saying that an isolated system must evolve such that ΔS ≥ 0. The Austrian physicist Ludwig Boltzmann (1844–1906) developed the relationship between the entropy of a system and the multiplicity of its macrostate: S = k ln W, where k is the Boltzmann constant. The increase in the entropy of a system as it changes from one macrostate to another is thus equivalent to an increase in the multiplicity. 10.2 Classical and Quantum Statistics 307 6 5 4 3 2 1 0 Macrostate A B C D E F G H I J 5 30 20 10 10 60 30 20 20 5 Multiplicity Energy units FIGURE 10.3 The macrostates of a system in which five identical particles share 6 units of energy. distribution of a total of six energy units to a collection of five identical but distinguishable oscillator-like particles, each of which can absorb energy in equal increments of 1 unit. Instead of a tabulation similar to Table 10.1, the energy distribution is illustrated in Figure 10.3.

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  Statistical Mechanics

  • Paul D. Beale(Author)
  • 1996(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  CHAPTER 5 FORMULATION OF QUANTUM STATISTICS

  THE scope of the ensemble theory developed in Chapters 2 Chapter 3 Chapter 4 is extremely general, though the applications considered so far were confined either to classical systems or to quantum-mechanical systems composed of distinguishable entities. When it comes to quantum-mechanical systems composed of indistinguishable entities, as most physical systems are, considerations of the preceding chapters have to be applied with care. One finds that in this case it is advisable to rewrite ensemble theory in a language that is more natural to a quantum-mechanical treatment, namely the language of the operators and the wave functions. Insofar as statistics are concerned, this rewriting of the theory may not seem to introduce any new physical ideas as such; nonetheless, it provides us with a tool which is highly suited for studying typical quantum systems. And once we set out to study these systems in detail, we encounter a stream of new, and altogether different, physical concepts. In particular, we find that the behavior of even a non-interacting system, such as the ideal gas, departs considerably from the pattern set by the classical treatment. In the presence of interactions, the pattern becomes even more complicated. Of course, in the limit of high temperatures and low densities, the behavior of all physical systems tends asymptotically to what we expect on classical grounds. In the process of demonstrating this point, we automatically obtain a criterion which tells us whether a given physical system may or may not be treated classically. At the same time, we obtain rigorous evidence in support of the procedure, employed in the previous chapters, for computing the number, Γ, of microstates (corresponding to a given macrostate) of a given system from the volume, ω, of the relevant region of its phase space, viz. Γ ω/h f , where f

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  Statistical Mechanics

  International Series of Monographs in Natural Philosophy

  • R.K. Pathria, D. ter Haar(Authors)
  • 2017(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 11 STATISTICAL MECHANICS OF INTERACTING SYSTEMS: THE METHOD OF QUANTIZED FIELDS IN THIS chapter we present yet another method of dealing with a system composed of mutually interacting particles. This method is based on the concept of a quantized field which is characterized by the field operators y(r), and their hermitian conjugates ^ ( r ) , that satisfy a set of well-defined commutation rules. In terms of these operators, one defines a number operator N and a Hamiltonian operator H which provide an adequate representa-tion for a system composed of any finite number of particles and possessing any finite amount of energy. In view of the formal similarity with the Schrödinger formulation, the formulation in terms of a quantized field is generally referred to as the second quantization of the system. For convenience of calculation, the field operators ip(r) and ^ ( r ) may be expressed as superpositions of a set of single-particle wave functions {w a (r)}, with coefficients a a and al; the latter turn out to be the annihilation and creation operators which again satisfy a set of well-defined commutation rules. The operators N and H find a convenient expression in terms of the operators a a and a, and the final formulation is well suited for a treatment based on the operator algebra; as a result, many calculations, which would otherwise be tedious, can be carried out in a more or less straightforward manner. We propose to demonstrate this by calculating the low-temperature properties of imperfect gases to the second order in a, where a is the scattering length of the interparticle potential. 11.1. The formalism of second quantization To represent a system of particles by a quantized field, we invoke the field operators y(r) andy^r), which are defined for all values of the position coordinate r and which oper-ate upon a Hilbert space; a vector in this space corresponds to a particular state of the quan-tized field.

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  Statistical Physics

  Volume 5

  • L D Landau, E.M. Lifshitz(Authors)
  • 2013(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  It must be emphasised that the averaging over various Ψ states, which we have used in order to illustrate the transition from a complete to an incomplete quantum-mechanical description, has only a very formal significance. In particular, it would be quite incorrect to suppose that the description by means of the density matrix signifies that the subsystem can be in various Ψ states with various probabilities and that the averaging is over these probabilities. Such a treatment would be in conflict with the basic principles of quantum mechanics.

  The states of a quantum-mechanical system that are described by wave functions are sometimes called pure states , as distrinct from mixed states , which are described by a density matrix. Care should, however, be taken not to misunderstand the latter term in the way indicated above.

  The averaging by means of the statistical matrix according to (5.4) has a twofold nature. It comprises both the averaging due to the probabilistic nature of the quantum description (even when as complete as possible) and the statistical averaging necessitated by the incompleteness of our information concerning the object considered. For a pure state only the first averaging remains, but in statistical cases both types of averaging are always present. It must be borne in mind, however, that these constituents cannot be separated; the whole averaging procedure is carried out as a single operation, and cannot be represented as the result of successive averagings, one purely quantum-mechanical and the other purely statistical.

  The statistical matrix in quantum statistics takes the place of the distribution function in classical statistics. The whole of the discussion in the previous sections concerning classical statistics and the, in practice, deterministic nature of its predictions applies entirely to quantum statistics also. The proof given in § 2

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