scholarly journals Effective Hamiltonians and Wavefunctions for Electrons in Deformed Crystals

1983 ◽  
Vol 36 (3) ◽  
pp. 321 ◽  
Author(s):  
RA Brown
Keyword(s):  
Hilbert Space ◽  
Probability Density ◽  
Physical Interpretation ◽  
Wannier Function ◽  
Deformation Potential ◽  
Effective Hamiltonian ◽  
Wannier Functions ◽  
Modulating Functions ◽  
Strain Dependent ◽  
Effective Hamiltonians

An effective Hamiltonian for electrons in in homogeneously deformed crystals is derived by expanding the wavefunction in terms of Wannier functions of the homogeneously deformed crystal. The physical interpretation of the modulating functions which determine the amplitude of each Wannier function in the expansion, and which are governed by the effective Hamiltonian, is investigated. This leads to strain-dependent expressions for the probability density and current, averaged over the fluctuations within each unit cell. The operators which represent, in the Hilbert space of the . modulating functions, similarly averaged physical observables are introduced and explicit straindependent expressions for the velocity and momentum operators are obtained. Applications of the theory are foreshadowed and its relationship to previous deformation-potential theories is examined.

Functional Quantum Theory of Free Relativistic Fermi Fields

1970 ◽  
Vol 25 (5) ◽  
pp. 575-586
Author(s):  
H. Stumpf
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Physical Interpretation ◽  
Functional Equations ◽  
Dirac Field ◽  
Functional Hilbert Space ◽  
Free Fermi ◽  
Special Case ◽  
Relativistic Fermi

Functional quantum theory of free Fermi fields is treated for the special case of a free Dirac field. All other cases run on the same pattern. Starting with the Schwinger functionals of the free Dirac field, functional equations and corresponding many particle functionals can be derived. To establish a functional quantum theory, a physical interpretation of the functionals is required. It is provided by a mapping of the physical Hilbert space into an appropriate functional Hilbert space, which is introduced here. Mathematical details, especially the problems connected with anticommuting functional sources are treated in the appendices.

Von Neumann lattices of Wannier functions for Bloch electrons in a magnetic field

1986 ◽  
Vol 403 (1824) ◽  
pp. 135-166 ◽  
Keyword(s):  
Magnetic Field ◽  
Phase Space ◽  
Periodic Function ◽  
Degrees Of Freedom ◽  
Two Dimensions ◽  
Effective Hamiltonian ◽  
Hall Conductance ◽  
Von Neumann ◽  
Wannier Functions ◽  
Bloch Electrons

The problem of Bloch electrons in a magnetic field in two dimensions can be reduced to a one-dimensional problem with a Hamiltonian Ĥ that is a periodic function of x ^ and p ^ . Wannier functions can be defined for the sub-bands of the spectrum of this effective Hamiltonian. When the Chern class (quantized Hall conductance integer) of the sub-band is zero, the Weyl-Wigner formalism can be used to represent these Wannier functions by a von Neumann lattice. It is shown how this von Neumann lattice of Wannier functions can be defined for irrational as well as rational magnetic fields. An important benefit from using the Weyl-Wigner formalism is that symmetries of the periodic potential are reflected by symmetries of the effective Hamiltonian in phase space. It is shown how the Wannier functions can be defined so that their Wigner functions have the point symmetries of the effective Hamiltonian. An example of how these results can prove useful is given: if we take matrix elements of the Hamiltonian between the Wannier states of a sub-band, we derive a new effective Hamiltonian describing this sub-band, which is again a periodic function of coordinate and momentum operators. Since, by projecting onto a sub-band, we have also reduced the number of degrees of freedom, this operation is a renormalization group transformation. It is shown that the symmetry of the new effective Hamil­tonian in phase space is the same as that of the original one. This preservation of symmetry helps to explain some unusual properties of the spectrum when the Hamiltonian has fourfold symmetry.

Spin and Charge Correlations in the Extended Hubbard-Holstein Model

1999 ◽  
Vol 13 (09n10) ◽  
pp. 1183-1188 ◽  
Author(s):  
M. Acquarone ◽  
M. Cuoco ◽  
C. Noce
Keyword(s):  
Effective Hamiltonian ◽  
Phonon Coupling ◽  
Range Interaction ◽  
Electronic Interactions ◽  
Gaussian Shape ◽  
Wannier Functions ◽  
Static Spin ◽  
Electron Phonon Coupling ◽  
Charge Correlations ◽  
Long Range Interaction

From the complete extended Hubbard-Holstein Hamiltonian we obtain a polaronic effective Hamiltonian by successive application of generalized displacement and squeezing transformations, with wavevector dependent characteristic parameters which make possible a phonon-induced long range interaction between charges, whose sign can change with the distance. As an application we consider the four-site chain, in which the bare electronic interactions are evaluated by model Wannier functions of gaussian shape. The ground states for different ranges of interactions are obtained by exact diagonalization of the effective Hamiltonian, followed by simultaneous and independent optimization of the displacement and squeezing parameters. We consider the behaviour of the static spin and charge correlation functions for the half-filled case, upon varying the shape of the Wannier functions for given electron-phonon coupling. Special attention is devoted to the influence of the phonon coupling on the presence of competing or coexisting charge- and spin-ordered states.

Hilbert Space of Probability Density Functions Based on Aitchison Geometry

2006 ◽  
Vol 22 (4) ◽  
pp. 1175-1182 ◽  
Author(s):  
J. J. Egozcue ◽  
J. L. Díaz–Barrero ◽  
V. Pawlowsky–Glahn
Keyword(s):  
Hilbert Space ◽  
Probability Density ◽  
Probability Density Functions ◽  
Density Functions ◽  
Aitchison Geometry

scholarly journals Realization of the noncommutative Seiberg–Witten gauge theory by fields in phase space

2015 ◽  
Vol 30 (22) ◽  
pp. 1550135 ◽  
Author(s):  
R. G. G. Amorim ◽  
F. C. Khanna ◽  
A. P. C. Malbouisson ◽  
J. M. C. Malbouisson ◽  
A. E. Santana
Keyword(s):  
Hilbert Space ◽  
Gauge Theory ◽  
Phase Space ◽  
Gauge Symmetry ◽  
Probability Density ◽  
Symmetry Analysis ◽  
Wave Functions ◽  
Wigner Functions ◽  
Poincaré Symmetry

Representations of the Poincaré symmetry are studied by using a Hilbert space with a phase space content. The states are described by wave functions (quasi-amplitudes of probability) associated with Wigner functions (quasi-probability density). The gauge symmetry analysis provides a realization of the Seiberg–Witten gauge theory for noncommutative fields.

scholarly journals Notizen: Zur Deutung der unterschiedlichen Strahlen- Empfindlichkeit organischer Moleküle / On the Radiation Sensitivity of Organic Molecules

1973 ◽  
Vol 28 (10) ◽  
pp. 1729-1731 ◽  
Author(s):  
R. Spehr ◽  
H. Schnabl
Keyword(s):  
Carbon Atom ◽  
Probability Density ◽  
Radiation Damage ◽  
Model Calculation ◽  
Organic Molecules ◽  
Radiation Sensitivity ◽  
Benzene Molecule ◽  
Wannier Functions ◽  
Rough Model ◽  
Original Distribution

If an electron initially was localized at one carbon atom of a benzene molecule, its probability density will quickly dissipate over the whole ring. A rough model calculation using Wannier functions yields T1/2 ≳ 10-16 sec for the decay of the original distribution. A similar spreading will happen to a defect electron produced locally at the bond between two atoms. This effect of “healing” should be taken into consideration when the mechanism of radiation damage in organic molecules is discussed.

The Wannier Functions of the Lattice Electron in a Uniform Magnetic Field

1966 ◽  
Vol 21 (10) ◽  
pp. 1577-1579 ◽  
Author(s):  
A. Jannussis
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Wave Vector ◽  
Free Electron ◽  
Vector Potential ◽  
Uniform Magnetic Field ◽  
Wannier Function ◽  
Wannier Functions ◽  
Uniform External Magnetic Field

In this paper a generalization of the WANNIER functions is performed for the case of the lattice electron moving in a uniform magnetic field.In the case of the free electron moving in a uniform external magnetic field, the WANNIER function may be obtained directly by the Schrauben-function, if the wave vector k is substituted by the vector potential — (1/c) A (ai).

scholarly journals Effective Hamiltonians for Complexes of Unstable Particles

2013 ◽  
Vol 20 (03) ◽  
pp. 1340008 ◽  
Author(s):  
Krzysztof Urbanowski
Keyword(s):  
Evolution Equation ◽  
State Space ◽  
Total System ◽  
Effective Hamiltonian ◽  
Minimal Energy ◽  
Unstable Particles ◽  
The One ◽  
Unstable States ◽  
Instantaneous Energy ◽  
Effective Hamiltonians

Effective Hamiltonians governing the time evolution in a subspace of unstable states can be found using more or less accurate approximations. A convenient tool for deriving them is the evolution equation for a subspace of state space sometime called the Królikowski–Rzewuski (KR) equation. KR equation results from the Schrödinger equation for the total system under considerations. We will discuss properties of approximate effective Hamiltonians derived using KR equation for n-particle, two-particle and for one-particle subspaces. In a general case these effective Hamiltonians depend on time t. We show that at times much longer than times at which the exponential decay take place the real part of the exact effective Hamiltonian for the one-particle subsystem (that is the instantaneous energy) tends to the minimal energy of the total system when t → ∞ whereas the imaginary part of this effective Hamiltonian tends to zero as t → ∞.

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