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1.
The application of contact transformation theory to the perturbed harmonic oscillator is reexamined in the light of the harmonic oscillator tensors previously presented. It is found that the recasting of the formalism of this problem in terms of harmonic oscillator tensors results in great simplifications, most of which stem from the introduction of the additional algebraic quantum numbers (l, m). The order of magnitude of each fragment of the Hamiltonian is easily recognizable, and the diagonal and nondiagonal parts contained therein are readily identifiable. The determination of the contact transformation operator is reduced to a simple formula. First, an analysis is made for a single mode of vibration, and it is subsequently extended to a multimode case. The perturbed diatomic vibrator is presented as an example. 相似文献
2.
Pancracio Palting 《International journal of quantum chemistry》1993,46(2):257-270
The spatial symmetries of the harmonic oscillator and the recently found irreducible tensors constructed from the associated annihilation and creation operators are exploited to obtain new expressions for the elements of the matrix representatives of several examples of vibrational operators. Since all vibrational operators are expressible in terms of the irreducible tensors, their matrix elements reflect the angular momentum symmetry inherent in them, for the results derived here are in terms of the Clebsch–Gordan coefficients and the isoscalar factors that arise from the couplinig rule of the irreducible tensors. Familiarity with the mathematical properties of these quantities derived from the elementary theory of angular momentum facilitates the evaluation of many vibrational operators that may be of importance in the study of potentials in this basis. In particular, it is shown that the nonvanishing of matrix elements is governed by a law of conservation of angular momentum along the axis of quantization of the nondegenerate harmonic oscillator. © 1993 John Wiley & Sons, Inc. 相似文献
3.
Pancracio Palting 《International journal of quantum chemistry》1998,67(6):343-357
The step operators of the two-dimensional isotropic harmonic oscillator are shown to be separable into the basis elements of two disjoint Heisenberg Lie algebras. This separability leads to two sets of irreducible tensors, each of which is based upon its associated underlying Heisenberg Lie algebra. The matrix elements of these tensors are evaluated, along with those of some vibrational operators of physical interest. The possibility of other irreducible tensors are discussed and their usefulness is compared with that of those found here. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 343–357, 1998 相似文献
4.
Mohammad Moharerrzadeh Pancracio Palting Lue-Yung Chow Chiu 《Journal of mathematical chemistry》2006,39(1):57-72
The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole, or magnetic-dipole, etc.,
is derived through a translational transformation. The addition theorem of μth component of the angular momentum operator,
L
μ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an
arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric
and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF),
. The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients
and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination
of the matrix element of the electric multipole operator 相似文献
5.
Pancracio Palting Shan Tao Lai Ying Nan Chiu Jorge Ricardo Letelier 《Journal of mathematical chemistry》1997,21(1):31-49
Succinct expressions for the matrix elements of various vibrational operators have been derived in the basis of the nondegenerate
harmonic oscillator. Among these are the matrix elements of
and
, which are found to be dependent upon two quantities and their derivatives. Furthermore, the derivative property of the commutator
is used to obtain an explicit expression for the derivatives of an operator in terms of its nested commutator with the conjugate
momentum. It may be applied to any of the above cases to obtain the matrix representatives of expressions such as the mixed
products
, for example. In addition, a simple expression for 1/q is given and its derivatives may be evaluated by this commutator technique. Also the matrix elements of a Gaussian-type operator
has been evaluated.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
6.
Shan-Tao Lai Pancracio Palting Ying-Nan Chiu Harris J. Silverstone 《Journal of mathematical chemistry》1998,24(1-3):123-132
Two new methods to evaluate the sums over magnetic quantum numbers, together with Wigner rotation matrix elements, are formulated.
The first is the coupling method which makes use of the coupling of Wigner rotation matrix elements. This method gives rise
to a closed form for any kind of summation that involves a product of two Wigner rotation matrix elements. The second method
is the equivalent operator method, for which a closed form is also obtained and easily implemented on the computer. A few
examples are presented, and possible extensions are indicated. The formulae obtained are useful for the study of the angular
distribution of the photofragments of diatomic and symmetric-top molecules caused by electric-dipole, electric-quadrupole
and two-photon radiative transitions.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
7.
Pancracio Palting 《International journal of quantum chemistry》1991,40(4):457-473
It is shown that the Heisenberg Lie algebra of the nondegenerate harmonic oscillator leads to a basis {J+, J0, J?} of LASU (2). The Hamiltonian of the system is proportional to J0, and the basis elements give rise to irreducible tensors in the associative enveloping algebra of the Heisenberg Lie algebra. The construction of these irreducible tensors is studied with special attention being paid to the case in which they act upon a single vector space spanned by the harmonic oscillator basis functions. A tensor coupling rule is developed, and useful application is made of it in the calculation of general expressions for vibrational operators and their matrix elements. Throughout, the value of the additional algebraic quantum numbers (l, m) is emphasized. 相似文献
8.
Pancracio Palting 《International journal of quantum chemistry》1997,65(4):305-315
A mapping of 2×2 matrices into the space of single boson operators is shown to lead to the angular momentum operators that give rise to irreducible tensors for the harmonic oscillator. The mapping may also be used to define an axis of quantization. A rotation about this axis induces a wave function and Hamiltonian that may be applied to the study of internal rotations in molecules. The example of a molecule containing two coaxial symmetric tops is presented as a case in point. The case of a potential with a high barrier leads to the approximation of an internal rotation as a torsional oscillator and, consequently, to torsional oscillator tensors whose properties are the same as those of the harmonic oscillator. The possibility of studying more complex potentials is discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 305–315, 1997 相似文献
9.
Pancracio Palting Jorge Ricardo Letelier Ying Nan Chiu 《International journal of quantum chemistry》2004,96(1):56-68
Electronic geminals constructed as linear combinations of binary products of site functions are used to formulate a vibronic model of superconductivity in crystals that is based upon the approximation of independent correlated electron pairs obtained variationally from an electron‐pair Hamiltonian and the Jahn–Teller effect. The cyclic symmetry of the system is taken into account and the geminals are sorted into doubly degenerate pairs. The Herzberg–Teller expansion of the pair Hamiltonian in terms of vibrational modes leads directly to the Jahn–Teller effect. A contact transformation of the vibronic Hamiltonian containing only linear terms lowers the energy of the system by a second‐order term associated with the Jahn–Teller stabilization energy. A possible model for superconductivity in solids is proposed on the basis of the Jahn–Teller effect. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003 相似文献
10.
Mohammad?MoharerrzadehEmail author Pancracio?Palting Lue-Yung?Chow?Chiu 《Journal of mathematical chemistry》2005,38(4):641-656
The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole or magnetic-dipole, etc.,
is derived through a translational transformation. The addition theorem of the μth component of the angular momentum operator, Lμ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along
an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric
and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF),
. The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients
and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination
of the matrix element of the electric multipole operator. 相似文献