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Centro-symmetric many-body Hamiltonians investigated with number-conserving operator equivalences and rotational symmetries

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Published Sep 10, 2026• 0 citations
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Abstract

The one- and two-body components of a centro symmetric Hamiltonian for a free ion taken as an example are cast into useful forms utilising equivalences based on the rotational properties of double tensor operators. Second-quantised number-conserving pairs may be expressed in terms of spin and orbital operators following the work of Stevens. The final result is expressed in a useful diagonal form in spin and orbital space by equating two-body matrix elements. (C) 1999 Elsevier Science B.V. All rights reserved.

DOI
10.82444/warw.33528583

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The one- and two-body components of a centro symmetric Hamiltonian for a free ion taken as an example are cast into useful forms utilising equivalences based on the rotational properties of double tensor operators. Second-quantised number-conserving pairs may be expressed in terms of spin and orbital operators following the work of Stevens. The final result is expressed in a useful diagonal form in spin and orbital space by equating two-body matrix elements. (C) 1999 Elsevier Science B.V. All rights reserved.
General remarks As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics – the Schrödinger equation – lies beyond the potentialities of modern mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. Symmetry concepts come about in physics in two ways. First, since any physical process occurs in a real space, we have to make use of one or another coordinate system. The isotropy and homogeneity of space make physically meaningful only those mathematical relationships that remain unchanged under rotations of the axes of the coordinate system, and that impose fairly rigorous constraints on the possible physical laws. Second, every physical object and process features a symmetry which should be taken into account by the physical theory. A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modern theory of complex atoms and ions is based.
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Matrix elements of two-body operators between many-particle permutational symmetry-adapted functions in hyperspherical coordinates are constructed. The matrix elements are evaluated using the appropriate hyperspherical coefficients of fractional parentage, the Raynal-Revai coefficients, and the hyperspherical recoupling coefficients. We use the power expansion of the two-body operators and obtain an analytic expression for the matrix element of each term. These expressions are studied numerically. The results allow precise evaluations of two-body matrix elements for few-body calculations in nuclear, atomic, and molecular physics.
For a system of NNN spinless particles in a state of definite angular momentum LLL with definite zzz component MMM, the rotational symmetry properties of the two-particle density matrix are obtained in a completely explicit fashion. Also studied is the 2-matrix associated with an equally weighted (unpolarized) ensemble of the various MMM states for a given LLL.
Abstract The representation of an N‐electron Schrödinger Hamiltonian on an orthonormal, spin‐free (freon) orbital product space is exactly modeled by a second‐degree polynomial in the infinitesimal generators of the unitary group. Symmetry adaptation of the orbital product space with respect to the symmetric group yields Gel'fand states which provide base for irreducible represntations of the unitary group. These exist in closed form as, in consequence, does the representation of the model Hamiltonian in this same basis. We retain as physical only those states characterized by tableaux with no more than two columns for which the spin labelling is one‐half the difference in the lengths of the two columns. The unitary group formulation is equivalent to standard, number‐conserving, second‐quantized, many‐body theory.