Quantum phase

The quantum phase factor is the exponential of an imaginary quantity (i times the phase), which multiplies into a wave function. Historically, a natural extension of this was proposed in the fonn of a gauge transformation, which both multiplies into and admixes different components of a multicomponent wave function [103]. The resulting gauge theories have become an essential tool of quantum field theories and provide (as already noted in the discussion of the YM field) the modem rationale of basic forces between elementary particles [67-70]. It has already been noted that gauge theories have also made notable impact on molecular properties, especially under conditions that the electronic... 

Another way to study the quantum dynamics of a system is to consider quantum phase space representations, that can be compared directly with classical results. Although there is no unique way to define a phase space representation of quantum mechanics, the most popular are the Wigner (Wigner, 1932) and Husimi (Husimi, 1940) functions. The Wigner transform... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

C. N. Yang, Concept of off-diagonal long-range order and quantum phases of liquid helium and... 

Information on the ranks of the optimal matrices P and Q can be used to gain efficiency since then the factor R need not have full rank, and the number of parameters is reduced accordingly. In the problem of quantum phases discussed in the following sections, the ranks of both P and Q can be predicted, which allows such efficiencies to be deployed in the solution process. 

Beiyan Jin, Quantum Phases for Tivo-Body Spin-Irwariant Nearest Neighbor Interactions, doctoral dissertation. Queens University, Kingston, Ontario, 1998. 

For 7=1 the Hamiltonian reduces to the Ising model and for y = 0 to the XY model. For the pure homogeneous case, /,, +i = / and = B, the system exhibits a quantum phase transition at a dimensionless coupling constant... 

Dimensional scaling theory [109] provides a natural means to examine electron-electron correlation, quantum phase transitions [110], and entanglement. The primary effect of electron correlation in the D 00 limit is to open up the dihedral angles from their Hartree-Fock values [109] of exactly 90°. Angles in the correlated solution are determined by the balance between centrifugal effects, which always favor 90°, and interelectron repulsions, which always favor 180°. Since the electrons are localized at the D 00 limit, one might need to add the first harmonic correction in the 1/D expansion to obtain... 

S. Sachdev, Quantum Phase Transitions, Cambridge University lYess, Cambridge, 1999. 

This derived expression satisfies conditions a-d mentioned above and based on numerical computatiotf 6-2 seems to bound the exact result from above. It is similar but not identical to Wigner s original guess. The quantum phase space function which appears in Eq. 52 is that of the symmetrized thermal flux operator, instead of the quantum density. 

Note that quantum phase is not a physical observable (at least in current formulations). The observable is n / or / 2. If, however, we have two (or more)... 

As emphasized by Berry and others [58], the formal limit h —- 0 presents an essential singularity in the mathematical expressions involving the quantum phases exp(iSp/h) such that the analyticity of the relevant quantities is lost near h - 0. For this reason, we may expect a variety of different behaviors as this limit is approached, depending on the type of systems considered and, especially, on die type of observables. Motivated by this remark, we would like to point out the existence of a regime different from the semiclassical one, in which the quantization of the energy levels is not the dominant feature. 

Now, the eigenenergies of the Hamiltonian can be detected directly if the time dependence of the above average exhibits quantum beats. This will be the case if the spectrum is not too dense and the linewidths are smaller than the level spacings. From a Fourier transform of the autocorrelation function, we then obtain an expression of the form (2.26)-(2.27), which can be evaluated semiclassically in terms of periodic orbits and their quantum phases. 

However, if the initial state is a thermal state, such as the canonical den-sity matrix p - (1/Z)exp(- 3//), the autocorrelation is no longer given by a single quantum amplitude but becomes a sum of quantum amplitudes in which quantum phases are randomized. In the classical limit ft - 0, the leading expression becomes the purely classical autocorrelation function with the dynamics being ruled by the classical Liouvillian operator ci = Hci> ... 

The set of these relations describe the deformed relativistic quantum phase space. 

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