Free classical particles have well-defined momentum and position, while free quantum particles have well-defined momentum but a position fully delocalized over the sample volume. We develop a many-body formalism based on wave-packet operators that connects these two limits, the thermal energy being distributed between the state spatial extension and its thermal excitation. The corresponding mixed quantum-classical states, which render the Boltzmann operator diagonal, are the physically relevant states when the temperature is finite. The formulation of many-body Hamiltonians in terms of these thermally excited wave packets and the resulting effective scatterings is provided.
Free classical particles have well-defined momentum and position, while free quantum particles have well-defined momentum but a position fully delocalized over the sample volume. We develop a many-body formalism based on wave-packet operators that connects these two limits, the thermal energy being distributed between the state spatial extension and its thermal excitation. The corresponding mixed quantum-classical states, which render the Boltzmann operator diagonal, are the physically relevant states when the temperature is finite. The formulation of many-body Hamiltonians in terms of these thermally excited wave packets and the resulting effective scatterings is provided.
The classical and quantum representations of thermal equilibrium are strikingly different, even for free, non-interacting particles. While the first involves particles with well-defined positions and momenta, the second usually involves energy eigenstates that are delocalized over a confining volume. In this paper, we derive convex decompositions of the density operator for non-interacting, non-relativistic particles in thermal equilibrium that allow for a connection between these two descriptions. Associated with each element of the decomposition of the N-particle thermal state is an N-body wave function, described as a set of wave packets; the distribution of the average positions and momenta of the wave packets can be linked to the classical description of thermal equilibrium, while the different amplitudes in the wave function capture the statistics relevant for fermions or bosons.
We have developed a theoretical formalism to introduce temperature as a parameter into the framework of non-relativistic quantum mechanics using the laws of classical thermodynamics and the canonical ensemble scheme of statistical mechanics. A self-consistent Hamiltonian has then been constructed for a given quantum many-body system which includes the effect of temperature in the form of correction terms added to the corresponding zero-temperature Hamiltonian of the system. Investigating some quantum mechanical systems with exact zero-temperature solutions including the particle-in-a-box model, the free particle, and the harmonic oscillator within our finite-temperature approach up to the first order of self-consistency has led to temperature-dependent Hamiltonians describing these systems above absolute zero without encountering any physically unacceptable brand of behavior for their wave functions and energy spectra. Results firmly support the view that a quantum mechanical system at a finite temperature behaves as if it is in a zero-temperature excited state.
It is well known that physical particles are thermally dissipative at finite temperature. In this paper we reformulate both the equilibrium and nonequilibrium thermal field theories in terms of stable quasiparticles. We will redefine the thermal doublets, the double tilde conjugation rules and the thermal Bogoliubov transformations so that our theory can be consistent for most general situations. All operators, including the dissipative physical particle operators, are realized in a Fock space defined by the stable quasiparticles. The propagators of the physical particles are expressed in terms of the operators of such stable quasiparticles, which is a simple diagonal matrix with the diagonal elements being the temporal step functions, same as the propagators in the usual quantum field theory without thermal degrees of freedom. The proper self-energies are also expressed in terms of these stable quasiparticle propagators. This formalism inherits the definition of on-shell self-energy in the usual quantum field theory. With this definition, a self-consistent renormalization is formulated which leads to quantum Boltzmann equation and the entropy law. With the aid of a doublet vector algebra we have an extremely simple recipe for computing Feynman diagrams. We apply this recipe to several examples of equilibrium and nonequilibrium two-point functions, and to the kinetic equation for the particle numbers.
Bose-Einstein condensates are bosonic many-body systems in which the quantum mechanical ground state is macroscopically occupied. If the interparticle interaction between the single bosons is attractive, this ground state is metastable and separated from the collapse by a barrier. A sufficient thermal excitation of the condensate can enable the crossing of this barrier and induce its collapse. In this thesis, the corresponding decay mechanism of the thermally induced coherent collapse of Bose-Einstein condensates with additional long-range interaction is investigated for the first time. For the description of the condensate’s dynamics a variational approach to the quantum wave function is used, which allows for the precise determination of the system’s transition state that governs the reaction dynamics. In order to quantitatively calculate the decay rate and lifetime of the condensate, respectively, a novel transition state theory for quantum wave packets is developed. This is based on a variational approach as well as normal form expansions of the dynamical equations and the energy functional in the vicinity of the transition state. The dynamical equations obtained from the variational approach induce a symplectic structure onto the space of variational parameters. Furthermore, the normal form expansions serve as a tool to locally construct canonical coordinates on this symplectic manifold. For this purpose, the dynamical equations as well as the energy functional are expanded and diagonalized. The eigenvectors of the linearized equations of motion thereby define a symplectic basis of the system. Afterwards, the transformations of the higher-order terms are carried out in two steps via nonlinear near-identity transformations: In the first step, the nonresonant terms of the expansions are eliminated and in the second step the remaining resonant terms are transformed in such a way that they fulfill canonical equations. The combination of these methods finally allows for the application of the well-established classical transition state theory. The procedure is valid for a wide variety of trial wave functions, for arbitrary dimensions of the variational space, in arbitrary order of the expansions, and it is independent of the precise form of the Hamilton operator. In addition, a procedure to handle bifurcations in the transition state is developed using a uniform rate formula in connection with an appropriate normal form of the potential energy surface. In contrast to the usual treatment which breaks down at the point of the bifurcation, the new method allows to cross the bifurcation smoothly, because it is valid on both sides of the bifurcation as well as arbitrarily close to it. At the same time, one can calculate the flux over the whole configuration of saddles which emerge in the bifurcation. The applicability of the newly developed transition state theory for wave packets is demonstrated for a cubic potential well. The thermal decay rates of a point particle and a frozen Gaussian wave packet, respectively, are calculated within the classical and quantum normal form approaches as well as the variational one. It is shown that the variational approach is capable to perfectly reproduce both the classical and quantum normal form results of the point particle in different limits. In the limit of a narrow trial wave function it agrees perfectly with the results using the classical normal form. The results with a broader trial wave function go even beyond the classical approach, i.e. they coincide with those of the quantum normal form. In the field of Bose-Einstein condensates with long-range interaction the application of transition state theory for quantum wave packets is extended to the field of nonlinear Schrödinger systems. For condensates with attractive 1/r-interaction, the thermal decay rates are calculated for different numbers of coupled Gaussian wave functions as well as different normal form orders. Significant corrections of the decay rate are obtained with increasing order of the normal forms. In the low-temperature regime convergence is observed in eighth order of the expansions. Finally, the thermally induced coherent collapse of dipolar condensates is investigated. These exhibit the feature of a biconcave ground state density distribution for certain values of the physical parameters. It is shown that such condensates reveal the astonishing effect of a symmetry-breaking collapse dynamics. This effect is directly related to the occurrence of the structured ground state, and it depends on the geometry of the external trapping potential. In case of a “conventional” density distribution, the collapse possesses an s-wave symmetry, while it has a d-wave symmetry in case of a blood-cell shaped ground state. The thermally induced coherent collapse plays a major role close to the stability threshold where the energy barrier is small and the decay rates are high. In case of Cr atoms, dipolar condensate have been first realized with, the region of interest is on the order of a Bohr radius above the critical scattering length. There, the lifetime of the condensate can be reduced to a millisecond. For the elements Dy and Er which have also been condensed in the laboratory, the effects become even stronger, i.e. the parameter region of importance becomes larger, the energy barriers become smaller, and the decay rates are higher.
We present in this paper a systematic nonperturbative cluster-cumulant method for deriving thermal averages of operators in quantum many-body systems. The method combines the advantages of the cumulant expansion scheme of thermodynamic perturbation theory, the approach of thermofield dynamics as a finite-temperature field theory, and the time-dependent coupled-cluster theory extended to ``imaginary time.'' We have generalized the concepts of cumulants in a nonperturbative manner and have posited on the statistical operator an exponential-like ansatz containing connected, size-extensive operators in the exponent. These latter cumulantlike operators have been termed ``cluster cumulants'' by us. For a compact treatment, we have derived an alternative thermal field theory in which a time-ordered product is expanded in terms of ``thermal normal products'' of operators and thermal contractions---leading to a ``thermal Wick's theorem.'' The thermal normal products are the finite-temperature analogs of the ordinary normal products and have zero thermal averages. Operators in these products commute (anticommute) under permutations for bosons (fermions). This thermal representation is shown to be unitarily related to the traditional thermofield dynamics formulation, but has the advantage of using only the physical variables. The imaginary-time evolution of the statistical operator is treated by our recently formulated time-dependent cluster-cumulant theory. The partition function is evaluated as an exponential of a connected quantity. As an illustrative example, we have computed the partition function of an anharmonic oscillator with equally weighted cubic and quartic perturbation for a wide range of coupling, extending to the strongly nonperturbative regime. We study the behavior of free energy in the low-temperature limit and verify numerically the validity of the Kohn-Luttinger theorem [Phys. Rev. 118, 41 (1960)] for this system. We also show that our formalism is a natural nonperturbative analog of the thermodynamic perturbative theory by showing that a perturbative solution of the thermal-cluster-cumulant equations generates a variation of the Bloch--Balian--de Dominicis theory.
Since the systems described by quantum fields are by definition extended, it is natural to think that in some limit they may reasonably well be approximated as fluids. This means that the state of the system is parameterized by a few locally well-defined fields such as temperature or energy density, obeying a set of hyperbolic equations of motion. A concrete example is the extensive use of fluid models to describe high-energy collisions [BelLan56, CarDuo73, CoFrSc74, Bjo83, CarZac83]. Our earlier derivation of quantum kinetic theory suggests a way to put this insight on a formal basis. Within its range of validity, the Boltzmann equation will drive the one-particle distribution function towards local thermal equilibrium. On scales much larger than the local thermalization scale, we expect to see hydrodynamical behavior [BeCoPa02]. This is, after all, the usual way of deriving hydrodynamics from kinetic theory [Hua87]. Beware, notwithstanding, that even at the level of classical kinetic theory there are still open questions regarding the cross-over from the kinetic to the hydro regime [KarGor02, KarGor03]. If we understand hydrodynamics as stated in the first paragraph of this chapter, then a system defined in terms of a quantum field may not have a hydrodynamic limit. This has been shown in [Elz02] for the case of a free Fermi field.