Momentum space symmetry

Mermin s "generalised crystallography" works primarily with reciprocal space notions centered around the density and its Fourier transform. Behind the density there is however a wave function which can be represented in position or momentum space. The wave functions needed for quasicrystals of different kinds have symmetry properties - so far to a large extent unknown. Mermin s reformulation of crystallography makes it attractive to attempt to characterise the symmetry of wave functions for such systems primarily in momentum space. 

In the next setion we review some key concepts in Mermin s approach. After that we summarise in section III some aspects of the theory of (ordinary) crystals, which would seem to lead on to corresponding results for quasicrystals. A very preliminary sketch of a study of the symmetry properties of momentum space wave functions for quasicrystals is then presented in section IV. 

The symmetry properties of the momentum space wave functions can be obtained either from their position space counterparts or more directly from the counterpart of the Hamiltonian in momentum space. 

For many problems in solid state physics, the computational efficiency of the computer programs is the result of using a planewave basis set and performing part of the calculation in momentum space through the use of Fast Fourier transforms. A planewave basis set is naturally applicable to systems with translational symmetry and this is the key of the success of... 

FIGURE 5.5 Symmetry enhancement from coordinate to momentum space C2v to D2h for water molecule. 

There are two Fermi seas for a given quark number with different volumes due to the exchange splitting in the energy spectrum. The appearance of the rotation symmetry breaking term, oc p U 4 in the energy spectrum (16) implies deformation of the Fermi sea so rotation symmetry is violated in the momentum space as well as the coordinate space, 0(3) —> 0(2). Accordingly the Fermi sea of majority quarks exhibits a prolate shape (F ), while that of minority quarks an oblate shape (F+) as seen Fig. 1 3. ... 

We compare results in the chiral limit (mo = 0) with those for finite current quark mass mo = 2.41 MeV and observe that the diquark gap is not sensitive to the presence of the current quark mass, which holds for all form-factors However, the choice of the form-factor influences the critical values of the phase transition as displayed in the quark matter phase diagram (/j,q — T plane) of Fig. 2, see also Fig. 1. A softer form-factor in momentum space gives lower critical values for Tc and at the borders of chiral symmetry restoration and diquark condensation. 

Hyperspherical harmonics are now explicitly considered as expansion basis sets for atomic and molecular orbitals. In this treatment the key role is played by a generalization of the famous Fock projection [5] for hydrogen atom in momentum space, leading to the connection between hydrogenic orbitals and four-dimensional harmonics, as we have seen in the previous section. It is well known that the hyperspherical harmonics are a basis for the irreducible representations of the rotational group on the four-dimensional hypersphere from this viewpoint hydrogenoid orbitals can be looked at as representations of the four-dimensional hyperspherical symmetry [14]. 

Since momentum densities are unfamiliar to many. Section II outlines the connection between the position and momentum space representations of wavefunctions and reduced-density matrices, and the connections among one-electron density matrices, densities, and other functions such as the reciprocal form factor. General properties of momentum densities, including symmetry, expansion methods, asymptotic behavior, and moments, are described in... 

In eq. (12), R(o) is the function operator that corresponds to the (2-D) configuration-space symmetry operator R(o). In eq. (13), /3 is the infinitesimal generator of rotations about z (eq. (8)) exp(i /3) is the operator [/do)] in accordance with the general prescription eq. (3.5.7). Notice that a positive sign inside the exponential in eq. (2) would also satisfy the commutation relations (CRs), but the sign was chosen to be negative in order that /3 could be identified with the angular momentum about z, eq. (6). 

Pair transfer interaction between the states of an electron system components can cause the gauge symmetry breaking realized in superconductivity. This circumstance forms the basis of the two-band model of superconductivity known already during a considerable time [1,2], The basic advantage of such approaches consists in the possibility to reach pairing by a repulsive interband interaction which operates in a considerable volume of the momentum space. An electronic energy scale is... 

The starting point for the study of the symmetry of the superconducting state is the pairing Hamiltonian. For applications to condensed matter systems it is convenient to write this Hamiltonian in momentum space... 

To evaluate the heating, a relativistic 1-D Fokker-Planck code was used. The configuration space is 1-D but the momentum space is 2-D, with axial symmetry. This code is coupled to a radiation-hydrodynamic simulation in order to include energy dissipation via ionization processes, hydrodynamic flow, the equation-of-state (EOS), and radiation transport. The loss of kinetic energy from hot electrons is treated through Coulomb and electromagnetic fields. 

How many values of k are there As many as the number of translations in the crystal or, alternatively, as many as there are microscopic unit cells in the macroscopic crystal. So let us say Avogadro s number, give or take a few. There is an energy level for each value of k (actually a degenerate pair of levels for each pair of positive and negative k values. There is an easily proved theorem that E(k) = E( — k). Most representations of E(k) do not give the redundant (- ), but plot ( k ) and label it as E(k)). Also the allowed values of k are equally spaced in the space of k, which is called reciprocal or momentum space. The relationship between k = 2x7 X and momentum derives from the de Broglie relationship X = hip. Remarkably, k is not only a symmetry label and a node counter, but it is also a wave vector, and so measures momentum. 

Hybrids constructed from hydrogenic eigenfunctions are examined in their momentum-space representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the momentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 

This equation was solved for the hydrogen atom by Vladimir Fock [2,3]. The solution in p space revealed the four-dimensional symmetry responsible for the degeneracy of states with the same n but different I quantum numbers in the hydrogen atom. This is a fine example where the momentum-space perspective led to fresh and deep insight. Fock s work spawned much further research on dynamical groups and spectrum-generating algebras. 

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