The non-commutativity of the position and momentum operators is formulated as an effective potential in classical phase space and expanded as a series of successive many-body terms, with the pair term being dominant. A non-linear partial differential equation in temperature and space is given for this. The linear solution is obtained explicitly, which is valid at high and intermediate temperatures, or at low densities. An algorithm for solving the full non-linear problem is given. Symmetrization effects accounting for particle statistics are also written as a series of effective many-body potentials, of which the pair term is dominant at terrestrial densities. Casting these quantum functions as pair-wise additive, temperature-dependent, effective potentials enables the established techniques of classical statistical mechanics to be applied to quantum systems. The quantum Ornstein-Zernike equation is given as an example.
The non-commutativity of the position and momentum operators is formulated as an effective potential in classical phase space and expanded as a series of successive many-body terms, with the pair term being dominant. A non-linear partial differential equation in temperature and space is given for this. The linear solution is obtained explicitly, which is valid at high and intermediate temperatures, or at low densities. An algorithm for solving the full non-linear problem is given. Symmetrization effects accounting for particle statistics are also written as a series of effective many-body potentials, of which the pair term is dominant at terrestrial densities. Casting these quantum functions as pair-wise additive, temperature-dependent, effective potentials enables the established techniques of classical statistical mechanics to be applied to quantum systems. The quantum Ornstein-Zernike equation is given as an example.
The exchange of conserved variables between a subsystem and a reservoir or environment entangles their wave functions. Their subsequent collapse, together with the uniform weight of wave space, gives rise to quantum statistical mechanics formulated as the trace of the probability and observable operators. A dual representation in position and momentum space allows a permutation loop expansion for particle symmetrization, and a second expansion accounting for the non-commutativity of the position and momentum operators. The leading term gives classical statistical mechanics, and higher-order terms give successive quantum corrections that become increasingly important for low temperatures, high densities and small masses. The probability density in phase space is stable under the stochastic dissipative equations of motion. Trapped Bose-Einstein condensates may be modeled as a system of non-interacting harmonic oscillators. Most bosons in the condensate are in the ground state, in contrast to the -transition in 4He where the fraction in the ground state is negligible.
By means of a quantum mechanical phase space distribution function
introduced by von Roos, the Schroedinger equation for a non-relativistic
system of N identical particles with scalar interactions is transformed
into a quantum mechanical generalization of the Liouville equation,
thereby formulating the problem in terms of a generalized density in
phase space, a quantity of primary interest in most treatments of the
corresponding classical system (or plasma). This transformation permits
a parallel development of the theories of classical and quantum plasmas
and thus allows the quantum many-body problem to be discussed virtually
completely in classical terms. In particular, a kinetic theory of
quantum plasmas is obtained by deriving the quantum analogue of the BBGKY
hierarchy, and applying thereto approximation techniques similar to those
of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from
similar previous studies based on the Wigner distribution function is that
the proper exchange symmetry can be tractably introduced into the formalism. Attention is first focused on the Hartree and Hartree-Fock approximations,
in which case the quantum BBGKY system reduces to a simple quantum
generalization of the Vlasov equation. This equation is used to study
the response of spatially homogeneous systems to weak external forces,
and the associated problems of plasmon and spin-wave excitations. It is
also used to derive the quantum and exchange corrected equations of
inviscid hydrodynamical transport which are then applied to the problem
of sound propagation in the degenerate electron gas. The second part of the study is concerned with the theory of the
many-electron atom in the Hartree and Hartree-Fock approximations.
The relevant quantum Vlasov equations lead naturally to a statistical
theory of the atom which reduces to the Thomas-Fermi-Amaldi and Thomas Fermi
models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and
exchange corrections to these models are simultaneously generated. The
quantum hydrodynamical theory developed earlier is used to determine
the influence of these corrections on the boundary conditions of the
model, and a theory of the compressed atom is consequently obtained.
Considered in somewhat less detail are the effects of non-zero temperature,
net orbital angular momentum, relativity and correlations, as well as
time dependent processes. The final part deals with the problem of the degenerate electron
gas with a uniform neutralizing background. Going beyond the Hartree-Fock approximation, the pair functions for particles with
parallel and anti-parallel spin are obtained by neglecting three particle
correlations. From these functions, a quantum-mechanical collision
integral is derived which differs from that obtained by Silin
and Guernsey and conjectured by Wyld and Pines in that dynamical exchange
effects are included. Also obtained from the pair function
is an expression for the correlation which reduces in the high
density limit to the result of Gell-Mann and Brueckner. At intermediate
densities an additional term appears in the energy due to the screening
of the exchange interaction by the dielectric properties of the medium.
It is evaluated in the high density limit and found to be -0.151 r_s ln r_s
Ryd/electron in marked disagreement with the corresponding value obtained
by DuBois.
The seventh chapter is concerned with the formulation of quantum statistical mechanics from first principles. It is shown that interactions with the reservoir or environment cause the subsystem wave function to collapse into a mixture of pure entropy states, which obey the axioms of classical probability theory. A dual representation in position and momentum states allows a permutation loop expansion for particle symmetrisation, and a second expansion accounting for the non-commutativity of the position and momentum operators. The leading term gives classical statistical mechanics, and higher order terms gives successive quantum corrections that become increasingly important for low temperatures, high densities, and small masses. From the permutation loop expansion, explicit results for the quantum ideal gas and the quantum harmonic oscillator are derived. The role of entropy in the classical manifestation of the universe given the underlying quantum mechanical wave equation is discussed.
Quantum and classical results are often presented as being dependent upon separate postulates as if the two are distinct and unrelated, and there is little attempt to show how the quantum implies the classical. The transformation to classical phase space gives researchers access to a range of algorithms derived from classical statistical mechanics that promise results on much more favourable numerical terms. Quantum Statistical Mechanics in Classical Phase Space offers not just a new computational approach to condensed matter systems, but also a unique conceptual framework for understanding the quantum world and collective molecular behaviour. A formally exact transformation, this revolutionary approach goes beyond the quantum perturbation of classical condensed matter to applications that lie deep in the quantum regime. It offers scalable computational algorithms and tractable approximations tailored to specific systems. Concrete examples serve to validate the general approach and demonstrate new insights. For example, the computer simulations and analysis of the λ-transition in liquid helium provide a new molecular-level explanation of Bose–Einstein condensation and a quantitative theory for superfluid flow. The intriguing classical phase space formulation in this book offers students and researchers a range of new computational algorithms and analytic approaches. It offers not just an efficient computational approach to quantum condensed matter systems, but also an exciting perspective on how the classical world that we observe emerges from the quantum mechanics that govern the behaviour of atoms and molecules. The applications, examples, and physical insights foreshadow new discoveries in quantum condensed matter systems. Key features • A computationally efficient approach to quantum condensed-matter many-particle systems based on a formally exact transformation of quantum statistical mechanics to classical phase space. • Shows how the observed classical world emerges from the underlying quantum mechanics of atoms and molecules. • Describes successive quantum corrections to classical statistical mechanics. • Derives a number of expansions that account for the non-commutativity of quantum position and momentum operators when transformed to classical phase space. • Tests various computer simulation algorithms against benchmark results for model condensed matter systems.
The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.
Close to equilibrium, a normal Bose or Fermi fluid can be
described by an exact kinetic equation whose kernel is nonlocal in space and time. The general expression derived for the kernel is evaluated to second order in the interparticle potential. The result is a wavevector- and frequency-dependent generalization of the linear
Uehling-Uhlenbeck kernel with the Born approximation cross section. The theory is formulated in terms of second-quantized phase space operators whose equilibrium averages are the n-particle Wigner distribution functions. Convenient expressions for the commutators and anticommutators of the phase space operators are obtained. The two-particle equilibrium distribution function is analyzed in terms of momentum-dependent quantum generalizations of the
classical pair distribution function h(k) and direct correlation function c(k). The kinetic equation is presented as the equation of motion of a two -particle correlation function, the phase space density-density anticommutator, and is derived by a formal closure of the quantum BBGKY hierarchy. An alternative derivation using a projection operator is also given. It is shown that the method used for approximating the kernel by a second order expansion preserves all the sum rules to the same order, and that the second-order kernel satisfies the appropriate positivity and symmetry conditions.