Analytical Wave Functions Research Articles

Reply to "Comment on 'Analytical wave functions for atomic quantum-defect theory' "

The preceding paper, by Martin and Barrientos [Phys. Rev. A 43, 4061 (1991)], comments on our supersymmetry-inspired model of atomic physics. We relate this to other Comments [Rau, Phys. Rev. Lett. 56, 95 (1986) and Goodfriend, Phys. Rev. A 41, 1730 (1990)] and point out that, although the mathematics and physics of the 1974 Simons model [J. Chem. Phys. 60, 645 (1974)] is similar to ours, it is not identical. This is elucidated by explicitly comparing our symmetry-based approach to the more phenomenological approach of Martin and Barrientos.

Comment on "Analytical wave functions for atomic quantum-defect theory"

The atomic quantum-defect theory proposed by Kostelecký and Nieto [Phys. Rev. A 32, 3243 (1985)] is proven to be basically identical to the earlier published quantum-defect orbital formalism.Received 3 April 1989DOI:https://doi.org/10.1103/PhysRevA.43.4061©1991 American Physical Society

THE l-MIXING CROSS SECTION OF RYDBERG STATES OF ATOMIC Rb AND THE SCALING LAW

On the basis of impulse approximate merhod, a kind of analytical wavefunctions based on a potential model was used to calculate the l mixing cross section of thermal collision of Ryd-berg states of atomic Rb with rare gas (He, Ne). The results were compared with the experimental results and other theoretical values. These results show that there exists a kind of scaling law for the l mixing cross section of Rydberg alkali atoms.

Laser-induced detachment processes in an electric field.

An analytic momentum-space wave function for an electron in both laser and static uniform electric fields is presented. It is used to obtain analytic multiphoton detachment cross sections in a static, uniform electric field which include effects of static-field-induced electron-photon interactions. These general results are not restricted to weak laser intensities or to weak static-field strengths and depend only on (a) the electric dipole approximation and (b) the approximation that final-state electron-atom interactions are ignored. Four specific predictions of our general formulas for the most interesting case of linearly polarized light polarized along the static-field direction are presented for weakly bound electrons initially in an s state. First, the effects of a weak static electric field on N-photon detachment cross sections near threshold are shown to be described by two modulation factors, one for odd N and one for even N, which depend only on a scaled energy. Second, for photodetachment in a static electric field, effects of static-field-induced electron-photon interactions are demonstrated. Third, the lifetime against field ionization is presented. Fourth, the cross section for electric-field-induced stimulated emission is presented. Numerical results for the latter three effects are presented for the ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ ion. The simpler case of circularly polarized light directed along the static-electric-field direction is treated briefly. In particular, we show for this case that the static- and laser-field effects are uncoupled and that the near-threshold, weak static-electric-field modulation factors for the N-photon detachment cross sections are dependent on N.

Multiphoton detachment in a static uniform magnetic field.

Multiphoton detachment cross sections for a negative ion in a static, uniform magnetic field are obtained in the approximation that electron-atom interactions are ignored in the final state. The final state is described by the analytic momentum-space wave function for an electron in the combined field of a laser and a static uniform magnetic field. The effects of a static-field-induced electron-photon interaction are stressed. Three consequences of our general formulas are discussed. First, the ponderomotive shift is modified by the magnetic field. It goes through a resonance at the cyclotron frequency and becomes negative for smaller frequencies. Second, for N-photon detachment of an s-shell electron by a laser of arbitrary strength, the effect of a weak magnetic field can be described near threshold by two modulation factors, one for even N and one for odd N, which depend only on a scaled energy. Third, cyclotron resonances are shown to exist under certain conditions, as demonstrated here for the specific example of multiphoton detachment of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ by a weak laser in an arbitrarily strong magnetic field.

Reply to "Comment on 'Fine-structure and analytical quantum-defect wave functions' "

The preceding Comment by Goodfriend (Phys. Rev. A 41, 1730 (1990)) contains three criticisms of our model for analytical quantum-defect wave functions {ital vis}-{ital a}-{ital vis} the atomic Fues potential of Simons (J. Phys. Chem. 55, 756 (1971)). We rebut the first two criticisms explicitly. This makes the third criticism moot. We stand by our results.

Comment on "Fine structure and analytical quantum-defect wave functions"

Comment on "Fine structure and analytical quantum-defect wave functions"

New analytic wave functions for superlattices including band nonparabolicity

Using an effective-mass Hamiltonian and the transfer-matrix method we evaluate analytically the exact normalized wave function, the current density, and the band structure associated with an electron in a superlattice. The effective-mass difference between wells and barriers and the nonparabolicity of the (conduction) band are taken into account.

The analytical wavefunctions of the SU(3) limit in the interacting boson model

The Su(3) limit intrinsic states in the interacting boson model (IBM) are given by the group theoretical method. A method to calculate the physical states from the intrinsic states is given. Using this method, the physical states of the SU(3) limit ground state band are given completely. Some preliminary results are also obtained for beta and gamma band states. The wavefunctions obtained are expressed in terms of the five building blocks of the IBM wave functions belonging to Sun, Zhang, and Feng. In the SU(3) limit it is found that the building block T+d, which represents three d bosons coupled to zero angular momentum, appears even in the ground state band wavefunctions. The wavefunctions have very interesting physical implications.

The Analytical Wavefunctions of SU(3) Limit in the Interacting Boson Model

A projection method to calculate the physical states from intrinsic states is put forward. Using this projection method, the physical states of the ground state band in the interacting boson model (IBM) SU(3) limit are calculated. The wave functions are expressed in terms of the building blocks of the IBM wave functions.

Fine structure and analytical quantum-defect wave functions.

We investigate the domain of validity of previously proposed analytical wave functions for atomic quantum-defect theory. This is done by considering the fine-structure splitting of alkali-metal and singly ionized alkaline-earth atoms. The Land\'e formula is found to be naturally incorporated. A supersymmetric-type integer is necessary for finite results. Calculated splittings correctly reproduce the principal features of experimental values for alkali-like atoms.

Wavefunctions and oscillator strengths of high Rydberg alkali atoms by a simple atomic potential model

The analytical wavefunctions and oscillator strengths between highly excited states of alkali atoms over a wide range of principal quantum number n(10<or=n<or=40, and for Na, n<or=60) are calculated straightforwardly by using a simple, one-parameter, exactly soluble model for atoms. The valence electron of the atom is assumed to be in a potential of the following form: V(r)=-e/r1+ delta + alpha '/r+ gamma '+ beta '/(r+ gamma ')2 where alpha ', beta ', gamma ' are the parameters to be determined and delta is the ionicity. Analytical solutions can be obtained by solving the Schrodinger equation containing V(r). The parameters in the potential and the wavefunction, which are dependent on the atomic states, can be expressed in terms of a single parameter, the quantum defect. The wavefunctions of the alkali atoms were obtained, and their general behaviour and the number of nodes agree with Hartree-Fock-Slater theory. This model is well suited for studying the behaviour of atoms that are mainly dependent of the outer region of high Rydberg states.

Correspondence between the fermion dynamic symmetry model and the interacting boson model in wave functions.

The analytical wave functions for the SO/sub 7/, SU/sub 3/, and SU/sub 2/ x SO/sub 3/ limits of the fermion dynamic symmetry model are constructed. These wave functions along with the already known wave functions for the SU/sub 2/ x SO/sub 5/ and SO/sub 6/ limits of fermion dynamic symmetry model are cast into such a form that their relation with the wave functions of the interacting boson model are vivid and transparent. According to the correspondence relation, the intrinsic wave functions for the interacting boson model SU/sub 3/ limit are constructed from its fermion counterpart and analytic expressions are given for the SU/sub 6/containsSU/sub 3/ isoscalar factors. The correlation between the D pair abundance in the ground states and the interaction strength is discussed.

Spin-singlet wave function for the half-integral quantum Hall effect.

We present an analytic wave function describing a spin-singlet incompressible fluid of electrons confined to a single Landau level, which exhibits a half-integral quantum Hall effect. Our numerical studies suggest that this state may be responsible for the recently observed effect with \ensuremath{\nu}=(5/2. .AE

Analytic Sturmian functions and convergence of separable expansions.

A method for solving integral equations is developed and applied to the homogeneous Lippmann-Schwinger equation in momentum space. It has been used with Yukawa-type potentials, V(r)=\ensuremath{\Sigma}, and yields solutions that are analytic expressions rational in the variable ${k}^{2}$. More specifically, the principal-value form of the homogeneous Lippmann-Schwinger equation is solved by making an analytic series expansion of the integral, which is then summed using Pad\'e approximants. An application to the Malfliet-Tjon potential V in s wave is given. In a finite subspace of rational functions with fixed denominators, the solutions, referred to as Sturmian functions, are obtained corresponding to the energy -0.35 MeV, which is the physical bound state energy for this potential. With these analytic eigenfunctions as form factors and with the associated eigenvalues, a separable expansion, namely, the unitary pole expansion, is constructed for the local potential. The unitary pole expansion is then used for analytic k-matrix calculations. At intermediate energies through ${E}_{\mathrm{c}.\mathrm{m}.}$=666 MeV, and at ultrahigh momenta, as the rank of the unitary pole expansion approaches 13, analytic wave functions (or, equivalently, half-shell k matrices) and phase shifts are found that are in good agreement with exact results.This is in accord with our observation that separable expansions should not be regarded as low energy approximations, but instead are finite l approximations, for the following reason. For the class of potentials we consider, the partial wave two-body operators ${V}_{l}$(p,q), ${t}_{l}$(p,q,E), and ${k}_{l}$(p,q,E) are compact even though V(r), T(p,q,E), and K(p,q,E) are not compact. Therefore, in principle, these partial wave operators can be approximated by sequences of separable expansions which converge to them in norm. We show that the unitary pole expansion may be such a sequence. There is a practical need for analytic separable expansions that converge more rapidly. In this light we illustrate how our solutions can be used with the Ernst-Shakin-Thaler formulation of separable expansions, and we also discuss the application of our method to the calculation of Gamow states.

Approximate factorised plane-wave Born calculations of the dipole (e,2e) cross sections for the Ne 2s and 2p and Ar 2p and 3p orbitals

The factorised plane-wave Born approximation is employed with a Coulomb wavefunction description for the ejected electron in the calculation of single ionisation dipole (e,2e) cross sections of the Ne 2s and 2p and Ar 2p and 3p orbitals. These calculations are made using the frozen-core, or single-electron excitation, approximation with a final-state wavefunction both non-orthogonal and orthogonal to the initial-state wavefunction. Clementi-type analytic Hartree-Fock one-electron wavefunctions are used to describe the ground state. Results have been compared with the absolute experimental values of Lahmam-Bennani et al. (1984) and indicate reasonable agreement for the Ne 2s case. For the Ne 2p and Ar 2p and 3p orbitals, reasonable agreement is also obtained in the large-angle cases and in the binary lobe of the small-angle cases. However, there is poor agreement for the recoil peaks. For the Ar 2p case the binary peak and recoil peaks can be fitted to the data, but only with different values for the effective ion charge. The effective ion charge is found to increase with increasing scattering angle.

Ground-state variational wave function for the quasi-one-dimensional semiconductor quantum wire.

An analytic variational wave function is proposed for the ground state of a quasi-one-dimensional electron system as occurring in narrow inversion layers in metal-oxide-semiconductor field-effect-transistor structures. The ground-state energy and the charge density are evaluated as functions of the average channel-electron density and the width of the metal gate by solving Poisson's equation and Schr\"odinger's equation in the variational self-consistent Hartree approximation.

Proton potential and properties of hydrogen bond

1. A model potential of the hydrogen bond in water dimer can be represented in the form of a sum of long-range charge-dipole attraction, short-range attraction, and exponential repulsion. Here, by a proper selection of parameters, a good description can be given for the dependence of the bond energy on the interatomic distance R. 2. The presence of a finite zero energy of the proton and the coupled character of the atomic vibrations in the H2O molecule lead to a certain redetermination of the parameters of the model potential V(r, R). In particular, the depth of the potential well for the O−H bond increases, as does the parmeter C of Coulomb attraction. 3. The relatively low-lying excited quantum levels of the proton and its analytical wave function can be obtained with an adequate degree of accuracy by means of a simplified potential V(r, R) that has the from of a Morse potential with parameters depending on the interatomic distance R. This simplified potential gives a correct reproduction of the form of V(r, R) close to the equilibrium position of the protonr≈r0. 4. The dependence of the bond energy E(R) has the required form, with the radius of action of the forces of the hydrogen bond and their magnitude becoming slightly greater with increasing sequence number of excitation. In particular, the work of shortening the distance R from ∞ to Ro for the first excited state amounts to about 45% of the excitation energy.

Analytical wave functions for atomic quantum-defect theory.

We present an exactly solvable effective potential that reproduces atomic spectra in the limit of exact quantum-defect theory, i.e., the limit in which, for a fixed l, the principal quantum number is modified by a constant: ${n}^{\mathrm{*}}$=n-\ensuremath{\delta}(l). Transition probabilities for alkali atoms are calculated using the analytical wave functions obtained and agree well with accepted values. This allows us to make phenomenological predictions for certain unknown transition probabilities. Our analytical wave functions might serve as useful trial wave functions for detailed calculations.

The analytic wavefunctions of the interacting boson model

The analytic U(5) and O(6) wavefunction of the interacting boson model in terms of the building blocks ( s †, s , d †, d ) are derived. It is shown that there exist very simple diagrammatic representations for both of these limits. An example of the usage of these wavefunctions is given.