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Physical Review A

Matrix elements between vector hyperspherical states

Nir Barnea, Victor B. Mandelzweig

Published Dec 1, 1991• 12 citations
Publisher

Abstract

Three-body symmetrized vector hyperspherical harmonic functions, realizing irreducible representations of the permutation group of three particles, are constructed and expressed in terms of Wigner D functions. Matrix elements of arbitrary two-body potentials between these hyperspherical states, including their velocity-dependent parts, are calculated analytically and expressed through the sum of products of the Wigner 3-j symbols and explicitly written functions of the radial variable.

DOI
10.1103/physreva.44.7053

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When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation $T$ of a real algebraic Lie group $G$. This requires defining an inner product on the Hilbert space $\\mathbb{H}$ that carries the representation $T$. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of $T$, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet ($\\mathfrak{F}_{\\mathbb{H}_D}$, $\\mathbb{H}_D$, $\\mathfrak{F}^{\\mathfrak{H}_D}$) of Bargmann spaces. Coherent state wave functions -- the elements of $\\mathfrak{F}_{\\mathbb{H}_D}$ and of $\\mathfrak{F}^{\\mathbb{H}_D}$ -- are used to define the inner product on $\\mathbb{H}_D$ in a way that simplifies the calculation of the overlaps. This inner product and the group action $\\Gamma$ of $G$ on $\\mathfrak{F}^{\\mathbb{H}_D}$ are used to formulate expressions for the matrix elements of $T$ with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for $\\Gamma$ relies on matrix factorizations in the complexification of $G$ even though the representation $T$ does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and $\\Gamma$ action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in $\\mathbb{H}$. The scalar and vector coherent state methods will both be applied to Sp($n$) and Sp($n,\\mathbb{R}$).

•Nov 1, 2010•0 citations
Three-body symmetrized vector hyperspherical harmonic functions, realizing irreducible representations of the permutation group of three particles, are constructed and expressed in terms of Wigner D functions. Matrix elements of arbitrary two-body potentials between these hyperspherical states, including their velocity-dependent parts, are calculated analytically and expressed through the sum of products of the Wigner 3-j symbols and explicitly written functions of the radial variable.
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.
Bases for SU(3) irreps are constructed on a space of three-particle tensor products of two-dimensional harmonic oscillator wave functions. The Weyl group is represented as the symmetric group of permutations of the particle coordinates of these spaces. Wigner functions for SU(3) are expressed as products of SU(2) Wigner functions and matrix elements of Weyl transformations. The constructions make explicit use of dual reductive pairs which are shown to be particularly relevant to problems in optics and quantum interferometry.
The introduction of a set of intrinsic coordinates to give an explicit construction of the intrinsic states of vector coherent state theory has greatly simplified earlier attempts to generalize this theory to include the construction of vector coherent state realizations of operators other than the group generators. The group U(3)⊇ U(2)×U(1) is used as a prototype. The construction of irreducible tensor operators with specific shift properties is illustrated with a number of examples. These show how the Wigner calculus for a higher symmetry group can be expressed solely in terms of the recoupling coefficients of the core subgroup and the simple K‐matrix elements of vector coherent state theory.
We investigate the adiabatic eigenfunctions and eigenvalues (and therefore the effective potentials) associated with the hyperspherical harmonic expansion of the wave function for three particles on a line subject to \ensuremath{\delta}-function interactions. Because of the special separability of the potential matrix elements, we obtain the effective potentials as the solutions of a transcendental equation and we also obtain the eigenfunctions in closed form. We demonstrate that the long-range (large hyper-radius \ensuremath{\rho}) behavior of the effective potentials is qualitatively different from the long-range behavior of the potential matrix elements.
When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation TTT of a real algebraic Lie group GGG. This requires defining an inner product on the Hilbert space mathbbH\\mathbb{H}mathbbH that carries the representation TTT. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of TTT, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (mathfrakFmathbbHD\\mathfrak{F}_{\\mathbb{H}_D}mathfrakFmathbbHD​​, mathbbHD\\mathbb{H}_DmathbbHD​, mathfrakFmathfrakHD\\mathfrak{F}^{\\mathfrak{H}_D}mathfrakFmathfrakHD​) of Bargmann spaces. Coherent state wave functions -- the elements of mathfrakFmathbbHD\\mathfrak{F}_{\\mathbb{H}_D}mathfrakFmathbbHD​​ and of mathfrakFmathbbHD\\mathfrak{F}^{\\mathbb{H}_D}mathfrakFmathbbHD​ -- are used to define the inner product on mathbbHD\\mathbb{H}_DmathbbHD​ in a way that simplifies the calculation of the overlaps. This inner product and the group action Gamma\\GammaGamma of GGG on mathfrakFmathbbHD\\mathfrak{F}^{\\mathbb{H}_D}mathfrakFmathbbHD​ are used to formulate expressions for the matrix elements of TTT with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for Gamma\\GammaGamma relies on matrix factorizations in the complexification of GGG even though the representation TTT does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and Gamma\\GammaGamma action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in mathbbH\\mathbb{H}mathbbH. The scalar and vector coherent state methods will both be applied to Sp(nnn) and Sp(n,mathbbRn,\\mathbb{R}n,mathbbR).