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Zhong-Qi Ma

Publications and source records attributed to Zhong-Qi Ma.

At least 37 records · Page 2Linked to original sources
arXiv1999

Exact Solutions of the Schrodinger Equation with Inverse-Power Potential in Two Dimensions

The Schrodinger equation for stationary states is studied in a central potential V(r) proportional to the inverse power of r of degree beta in an arbitrary number of spatial dimensions. The presence of a single term in the potential makes it impossible to use previous algorithms, which only work for quasi-exactly-solvable problems. Nevertheless, the analysis of the stationary Schrodinger equation in the neighbourhood of the origin and of the point at infinity is found to provide relevant information about the desired solutions for all values of the radial coordinate. The original eigenvalue equation is mapped into a differential equation with milder singularities, and the role played by the particular case beta=4 is elucidated. In general, whenever the parameter beta is even and larger than 4, a recursive algorithm for the evaluation of eigenfunctions is obtained. Eventually, in the particular case of two spatial dimensions, the exact form of the ground-state wave function is obtained for a potential containing a finite number of inverse powers of r, with the associated energy eigenvalue.

quant-ph↗
arXiv1999

Quantum Three-Body Problem

An improved hyperspherical harmonic method for the quantum three-body problem is presented to separate three rotational degrees of freedom completely from the internal ones. In this method, the Schrödinger equation of three-body problem is reduced to a system of linear algebraic equations in terms of the orthogonal bases of functions. As an important example in quantum mechanics, the energies and the eigenfunctions of some states of the helium atom and the helium-like ions are calculated.

physics.atom-ph↗
arXiv1999

Algebraic description of anharmonic stretching vibrations

A U(2) algebraic model is presented to describe stretching vibrations of XY$_n$ (n=2, 3, and 4) systems, where anharmonic interactions between the bond modes are considered. This model in a limit corresponds to an anharmonically coupled local mode model. As an example, the model for a molecule XY$_4$ is applied to recently observed spectra of methane in both gas and liquid phases, and the results obtained are in good agreement with the experiments.

physics.chem-ph↗
arXiv1999

Fermi resonance-algebraic model for molecular vibrational spectra

A Fermi resonance-algebraic model is proposed for molecular vibrations, where a U(2) algebra is used for describing the vibrations of each bond, and Fermi resonances between stretching and bending modes are taken into account. The model for a bent molecule XY_2 and a molecule XY_3 is successfully applied to fit the recently observed vibrational spectrum of the water molecule and arsine (AsH_3), respectively, and results are compared with those of other models. Calculations show that algebraic approaches can be used as an effective method for describing molecular vibrations with small standard deviations.

physics.chem-ph↗
arXiv1999

Levinson's theorem for the Schrödinger equation in one dimension

Levinson's theorem for the one-dimensional Schrödinger equation with a symmetric potential, which decays at infinity faster than $x^{-2}$, is established by the Sturm-Liouville theorem. The critical case, where the Schrödinger equation has a finite zero-energy solution, is also analyzed. It is demonstrated that the number of bound states with even (odd) parity $n_{+}$ ($n_{-}$) is related to the phase shift $η_{+}(0)[η_{-}(0)]$ of the scattering states with the same parity at zero momentum as $η_{+}(0)+π/2=n_{+}π, η_{-}(0)=n_{-}π$, for the non-critical case, $η_{+}(0)=n_{+}π, η_{-}(0)-π/2=n_{-}π$, for the critical case.

quant-ph↗
arXiv1999

The Exact Solution to the Schrödinger Equation with the Octic Potential

The Schrödinger equation with the central potential is first studied in the arbitrary dimensional spaces and obtained an analogy of the two-dimensional Schrödinger equation for the radial wave function through a simple transformation. As an example, applying an ${\it ansatz}$ to the eigenfunctions, we then arrive at an exact closed form solution to the modified two-dimensional Schrödinger equation with the octic potential, $V(r)=ar^2-br^4+cr^6-dr^4+er^{10}$.

quant-ph↗
arXiv1998

Irreducible bases in icosahedral group space

The irreducible bases in the icosahedral group space are calculated explicitly by reducing the regular representation. The symmetry adapted bases of the system with {\bf I} or {\bf I}$_{h}$ symmetry can be calculated easily and generally by applying those irreducible bases to wavefunctions of the system, if they are not vanishing. As examples, the submatrices of the Hückel Hamiltonians for Carbon-60 and Carbon-240 are re-calculated by the irreducible bases.

math-ph↗
arXiv1998

Irreducible bases and correlations of spin states for double point groups

In terms of the irreducible bases of the group space of the octahedral double group {\bf O'}, an analytic formula is obtained to combine the spin states $|j,μ\rangle$ into the symmetrical adapted bases, belonging to a given row of a given irreducible representation of {\bf O'}. This method is effective for all double point groups. However, for the subgroups of {\bf O'}, there is another way to obtain those combinations. As an example, the correlations of spin states for the tetrahedral double group {\bf T'} are calculated explicitly.

math-ph↗
arXiv1998

Schrödinger Equation with the Potential $V(r)=a r^2+b r^{-4}+c r^{-6}$

By making use of an ${\it ansatz}$ for the eigenfunction, we obtain the exact solutions to the Schrödinger equation with the anharmonic potential, $V(r)=a r^2+b r^{-4}+c r^{-6}$, both in three dimensions and in two dimensions, where the parameters $a$, $b$, and $c$ in the potential satisfy some constraints.

quant-ph↗
arXiv1998

Levinson's Theorem for the Klein-Gordon Equation in Two Dimensions

The two-dimensional Levinson theorem for the Klein-Gordon equation with a cylindrically symmetric potential $V(r)$ is established. It is shown that $N_{m}π=π(n_{m}^{+}-n_{m}^{-})= [δ_{m}(M)+β_{1}]-[δ_{m}(-M)+β_{2}]$, where $N_{m}$ denotes the difference between the number of bound states of the particle $n_{m}^{+}$ and the ones of antiparticle $n_{m}^{-}$ with a fixed angular momentum $m$, and the $δ_{m}$ is named phase shifts. The constants $β_{1}$ and $β_{2}$ are introduced to symbol the critical cases where the half bound states occur at $E=\pm M$.

quant-ph↗
arXiv1998

Levinson's Theorem for Non-local Interactions in Two Dimensions

In the light of the Sturm-Liouville theorem, the Levinson theorem for the Schrödinger equation with both local and non-local cylindrically symmetric potentials is studied. It is proved that the two-dimensional Levinson theorem holds for the case with both local and non-local cylindrically symmetric cutoff potentials, which is not necessarily separable. In addition, the problems related to the positive-energy bound states and the physically redundant state are also discussed in this paper.

quant-ph↗
arXiv1998

The Relativistic Levinson Theorem in Two Dimensions

In the light of the generalized Sturm-Liouville theorem, the Levinson theorem for the Dirac equation in two dimensions is established as a relation between the total number $n_{j}$ of the bound states and the sum of the phase shifts $η_{j}(\pm M)$ of the scattering states with the angular momentum $j$: $$η_{j}(M)+η_{j}(-M)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$~~~=\left\{\begin{array}{ll} (n_{j}+1)π&{\rm when~a~half~bound~state~occurs~at}~E=M ~~{\rm and}~~ j=3/2~{\rm or}~-1/2\\ (n_{j}+1)π&{\rm when~a~half~bound~state~occurs~at}~E=-M~~{\rm and}~~ j=1/2~{\rm or}~-3/2\\ n_{j}π~&{\rm the~rest~cases} . \end{array} \right. $$ \noindent The critical case, where the Dirac equation has a finite zero-momentum solution, is analyzed in detail. A zero-momentum solution is called a half bound state if its wave function is finite but does not decay fast enough at infinity to be square integrable.

quant-ph↗
arXiv1998

Levinson's theorem for the Schrödinger equation in two dimensions

Levinson's theorem for the Schrödinger equation with a cylindrically symmetric potential in two dimensions is re-established by the Sturm-Liouville theorem. The critical case, where the Schrödinger equation has a finite zero-energy solution, is analyzed in detail. It is shown that, in comparison with Levinson's theorem in non-critical case, the half bound state for $P$ wave, in which the wave function for the zero-energy solution does not decay fast enough at infinity to be square integrable, will cause the phase shift of $P$ wave at zero energy to increase an additional $π$.

quant-ph↗
arXiv1998

Overtone spectra and intensities of tetrahedral molecules in boson-realization models

The stretching and bending vibrational spectrum and the intensities of infrared transitions in a tetrahedral molecule are studied in two boson-realization models, where the interactions between stretching and bending vibrations are described by a quadratic cross term and by Fermi resonance terms, called harmonically coupled and Fermi resonance boson-realization model, respectively. The later is a development of our recent model. As an example, the two models are applied to the overtone spectrum and the intensities of silicon tetrafluorde. Those models provide fits to the published experimental vibrational eigenvalues with standard deviations 1.956 cm$^{-1}$ and 0.908 cm$^{-1}$, respectively. The intensities of infrared transitions of its complete vibrations are calculated in the two models, and results show a good agreement with the observed data.

physics.chem-ph↗