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A simple relation between critical temperature (Tc ) and boiling point (Tb ) is shown to be valid for many organic series. The relation is as follows: 1/Tc = α/Tb + β, where α and β are constants. It has been applied to a total of 24 homologous series and the results for Tc are quite accurate, the average percent error over all the compounds is 0.20%. Predictions are made for critical temperature and boiling point of certain compounds.

A linear relation between critical temperature and boiling point is proposed for elements belonging to the same group in the periodic table. The validity of the relationship is demonstrated for the alkalies and the groups 16, 17 and 18. From this the critical temperature of tellurium is predicted as 2325 K and that for polonium as 2277 K.

Some structural properties of energy eigenfunctions of two electrons in a simple harmonic potential are analyzed. Simple expressions are obtained for the energy spectrum based on a model potential, and on a model wave function. These expressions give accurate values for the energy eigenvalues and provide a physical insight into their structure.PACS Nos.: 3.65.Ge, 73.21.La, 78.67.Hc

The hydrogenic system, confined in an off-centre oscillator potential, is separable in terms of elliptic coordinates. Its general properties are analysed and energies are obtained for some states, for some values of displacement and potential strength. A model wave function is developed and used to obtain the energies and polarizabilities for the ground state. PACS Nos.: 03.65.Ge, 73.21.La, 78.67.Hc

A simple description of the spectrum of confined hydrogen atom is obtained in terms of an scheme involving interpolation between the expressions for large and small values of the confining radius. Satisfactory agreement is shown to exist between the eigen energies thus calculated and exact values. The scheme is extended for helium and lithium. With the simulation of interaction between the electrons by a screened nuclear charge, some energy levels are deduced for the confined helium and lithium.PACS Nos.: 03.65.Ge, 73.21.La, 78.67.Hc

A hydrogen atom in a high-density plasma is simulated by a model in which the hydrogen atom is confined in an impenetrable spherical box, with the atom at the centre. For the proton–electron interaction the Debye–Huckel potential is used. Variational wave functions are proposed for the 1s and 2p states. Energies calculated from these for different values of the radius of box (r0) are shown to be in good agreement with the exact values. The variational wave functions are further employed to calculate the absorption oscillator strength for the 1s [Formula: see text] 2p transition and the dipole polarizability for different values of r0. PACS Nos.: 03.65.Ge, 32.70.Os, 31.70.Dk, 52.20.–j

In this paper, we apply the semiclassical Wentzel–Kramers–Brillouin (WKB) approximation to an electron in a central force potential, confined in a two-dimensional disc, and we obtain the quantization rules for such a system. As explicit examples, we consider the two most widely studied potentials, viz., the parabolic potential (the harmonic oscillator) and the Hydrogenic impurity state (Coulomb potential) in two dimensions. Both the systems are confined within an impenetrable circular wall of radius r0: In particular, we determine the energies as well as eigenfunctions by this approach. On comparing these energies with those from exact numerical values, the agreement is found to be quite good. Moreover, the WKB approach is found to give a good estimate of the wave functions as well. These results suggest that the WKB approximation works well even for such rigid wall spatial confinement and the present approach can be applied to other confined systems such as those which are encountered in mesoscopic physics. r 2002 Elsevier Science B.V. All rights reserved.

The effect of high pressure on a hydrogen atom has frequently been simulated by enclosing the atom in an impenetrable spherical box. It is shown that for such a confined hydrogen atom placed at the centre of a spherical box a simple scaling relation exists between the energy and the radius of the confining box for 1s, 2p, 3d, and 4f levels, and another similar relation exists for 2s, 3p, and 4d levels.

The energies of ten excited states of screened two-dimensional excitons in a semiconducting quantum-well structure are calculated as a function of the screening parameter qs by the WKB method. The screening of the Coulomb interaction between the electron-hole pair by the free carriers present is taken into account by using the screened potential obtained by Stern and Howard (1967). The energies of the excited states are found to decrease with increasing values of the screening parameter qs in all cases. Further the levels become unbound at finite values of the screening parameter. Critical screening parameters are estimated for the ten states. PACS Nos.: 73.21Fg, 78.67De, 71.35-y, 71.35Cc Resume : Nous presentons un calcul WKB pour les energies de dix etats excites d'excitons blindes (ecran) en 2-D dans un puits quantique de semi-conducteur, en fonction du parametre d'ecran qs. L'interaction de Coulomb avec ecran entre les paires electron-trou par les porteurs de charges libres est decrit par le potentiel avec ecran de Stern et Howard (1967). Dans tous les cas, nous trouvons que les energies d'excitation decroissent lorsqu'augmente le parametre qs. De plus, les etats excites cessent d'etre lies pour des valeurs finies de qs. Nous donnons les valeurs critiques du parametre qs pour les dix etats. (Traduit par la Redaction)

In certain endohedral systems a metallic ion is trapped inside C 60 such that the energy minimum corresponds to an off-centre position of the ion. A simple analytical function is proposed to represent the potential–energy curve for such systems. The applicability of this function to potential energy curves of endohedral Na + and Li + ions in C 60 is demonstrated and it is then used to calculate the structure of spectra for the two systems.

A model is proposed for positive trions (X + 2 ) in bulk solids and another one for surface trions. These models are applied to calculate the bound-state eigenenergies for such trions in CuCI, CuBr and CuI.

Two parametric relations are proposed for calculating the level energies of an electron bound in a screened Coulomb potential in a plasma. The first relation is between the energies of levels which have the same l quantum number. The second relation is able to represent the bound-state energies for 1s, 2p, 3d...levels as a function of the screening parameter to a good degree of accuracy. These two relations taken together can give the energies of any level up to and including n = 8, and estimated energies for n = 9 and 10 for any value of ne and Te. These are relevant for at least three topics in plasma physics. The application of these relations for the calculation of the partition function of plasmas is discussed.

Confined quantum systems have been studied by various authors over the past decades, by using various mathematical techniques. In this work, we derive the WKB quantization rules for quantum systems confined in an impenetrable spherical box of radius r0. We apply the proposed method to two systems explicitly, viz., the confined harmonic oscillator and the confined hydrogen atom. The results are found to be in reasonable agreement with those obtained by other methods. PACS No.: 03.65

In this paper we formulate the shifted 1/N expansion method for constrained quantum mechanical systems with spherically symmetric potentials. As an example, we apply our technique to the confined Hulthén potential V(r) = –Zδ[e -δr/(1 –e-δr)] for different values of the confinement parameter b and the screening parameter δ. It is found that the agreement between our results and the exact numerical values is reasonably good.PACS No.: 03.65Ge

A hydrogen atom confined in a spherical box with impenetrable surface is investigated. The critical cage radius , at which a state is no longer bound is calculated for levels up to and including n = 10. Regularities present in these radii are pointed out. An empirical equation is proposed between and n for a fixed . The total number of bound states is also studied as a function of .

We present here the solutions of a bounded linear harmonic potential and abounded quartic potential using the modified Airy function (MAF) method. Resultsobtained by the MAF method have been compared with the analytical (numerical)results and with those obtained by the JWKB method. The comparison showsthat the MAF method gives very accurate results and is, in general, the moreaccurate of the two methods. The MAF method also gives an accurate descriptionof the eigenfunction. A perturbation correction when applied to MAF helps usto get very accurate eigenvalues. The method should be useful in determiningthe eigenvalues and the eigenfunctions of any smoothly varying arbitrarypotential confined by infinite walls.PACS No. 03.65

An Ansatz for the wave function is used to obtain an exact solution for the ground-state energy for the potential T Eo '@o n Ko n So in an impenetrable spherical box, provided a certain constraint is satisfied between the potential parameters, @cKc and S. Two types of solution are shown to exist. Results are obtained for several sets of potential parameters. R´ esum´ e :L 'utilisation un succedane pour la fonction d'onde permet d'obtenir l'energie exacte du fondamental du potentiel T Eo '@o 2n Ko 3 en So dans une boite spherique impenetrable, a condition que certaines contraintes soient satisfaites entre les parametres @cK et S du potentiel. Deux types de solution existent. Nous presentons les resultats pour differents choix de ces parametres. (Traduit par la r ´

We evaluate energy eigenvalues corresponding to different central potentials with nonpolynomial character and exhibiting broken supersymmetry. The calculations were carried out using numerical integration, the WKB rule, and a quasi-classical quantization rule for broken supersymmetry. It is shown that in most cases the quasi-classical rule gives far better estimates than the WKB rule applied separately to the supersymmetry partners. R´ esum´ e : Nous calculons les valeurs propres d'un Hamiltonien avec potentiel central de type non-polynomial et capable de briser la supersymetrie. Trois methodes sont utilisees: l'integration numerique, la methode WKB et une regle quasi-classique de quantification adaptee aux cas de bris de supersymetrie. Nous trouvons que dans la plupart des cas, la methode semi-classique donne de bien meilleurs resultats que la methode WKB appliquee separement aux partenaires supersymetriques. (Traduit par la r ´

Wavefunctions are proposed for the 1s, 2p and 3d states of the compressed hydrogen atom at high pressures. It is shown that these wavefunctions give highly accurate energies in the positive energy region.

The shifted 1/N expansion technique is applied to the radial Dirac equation. It is shown that the fixing condition E1 = 0 produces an enhanced accuracy for Yukawa and Hulthen potentials. The modified shifted 1/N expansion results are compared with the exact numerical results and results obtained by other approximate methods.

A supersymmetric version of the modified Airy-function method is developed. Explicit calculations for eigenvalues are obtained for the Schrödinger potential V(x) = x6 − 3x2. Comparison is made with exact numerical results as well as those of the Wentzel–Kramers–Brillouin (WKB), supersymmetric WKB (SWKB), and modified Airy-function approximation methods.

A method based on supersymmetric quantum mechanics is given for obtaining exact solutions of the potential V(r) = r2 + β/r4 + λ/r6, where β and λ are parameters, provided a certain constraint is satisfied between β and λ. Detailed results are given for the ground state as well as for several excited states. A method for determining the exact energies of two levels for the same values of β and λ is given and illustrated by examples.

We present a systematic perturbative method for calculating the binding energies of ro-vibrational quantum states for a finite depth potential. Using an asymptotic expansion in powers of 1/η, where η is the dimensionless physical depth parameter, explicit calculations are performed. Our analytic results lead to highly convergent series for the binding energies, and numerical results for realistic potentials such as Morse and Lennard-Jones potentials are found to be in excellent agreement with the exact ones.

Using two nonperturbative methods, namely, the shifted 1/N expansion and the SWKB techniques, we determine the energy levels of a system of two anyons in an external constant magnetic field and interacting via a Coulomb repulsion. Some exact results have also been found using the notion of supersymmetry.

The WKB approximation is applied to the rotating Morse oscillator. Results are presented for the ground state of the H2 molecule and are compared with those obtained by several other methods. The WKB method gives the best results for v > 5. At smaller values of v, the WKB results are only slightly less accurate than the best ones.

The relative convergences of the WKB and SWKB approximations are examined by calculating the eigenenergies for four potentials by one- and two-term WKB and SWKB approximations. Exact eigenenergies for these four potentials are also calculated by a numerical integration of the Schrodinger equation. Varied results are found for the four potentials. It is found that in general the effect of the second term in the WKB and SWKB approximations depends on the potential, the parameters involved and the quantum number of the state. No simple generalizations are possible.

By combining the analytic continuation of 1/N results with additional selection criteria implied by the related WKB description, Papp has obtained a closed analytical expression for the energy levels of the following potential: V(r) = De[(re/r)4 - 2(re/r)2]. The eigenenergies obtained from Papp's expressions are compared with those obtained from one-, two-, and three-term WKB approximations. The WKB approximations give better results than Papp's expression but no closed analytical expression can be obtained from them.

Computations of energy levels, expectation values of various powers of internuclear separation, and Franck–Condon factors for molecular potentials have been carried out within the framework of shifted large‐N technique. In contrast to other perturbative as well as semiclassical calculations, we obtain compact analytic expressions for these quantities. For illustration, our predicted results for perturbed Kratzer potential are compared with those derived by numerical integration. The method gives quite accurate results for a wide range of rovibrational quantum numbers. Further possible applications, in particular, to Morse oscillators are finally discussed.

We discuss the possibility of obtaining new quasi-exactly solvable systems within the framework of supersymmetric quantum mechanics.

A method based on supersymmetric quantum mechanics is given for obtaining exact solutions of the potential V(r)=\ensuremath{\alpha}/r+${\mathit{p}}_{1}$r+${\mathit{p}}_{2}$${\mathit{r}}^{2}$+${\mathit{p}}_{3}$${\mathit{r}}^{3}$+${\mathit{p}}_{4}$${\mathit{r}}^{4}$ where \ensuremath{\alpha} and the p's are parameters, provided certain relations are satisfied between the parameters. Detailed results are given for three specific cases. The potential in question gives rise to some very interesting shapes (double-well, etc.). The applicability of the shifted 1/N expansion method to such potential shapes is examined by comparing eigenenergies obtained by this method with the exact ones obtained from supersymmetric considerations. It is found that in certain situations, the shifted 1/N expansion method may give poor or erroneous results. Applicability of the proposed method to potentials involving higher powers of r is also discussed.

The difficulties associated with application of the shifted large-N technique to the Dirac equation have been resolved by applying the method to the Klein-Gordon equation in which a spin-orbit interaction term is included analogous to Pauli theory. Explicit analytical expressions for the relativistic screened Coulomb bound-state energies, radial wave functions, and normalizations are given. For the point-Coulomb problem, we restore exact results for the relativistic binding energies and almost exact wave functions. The 1/N expansion results are then compared with the exact numerical solutions as well as with those obtained in other analytical methods for a number of screened Coulomb potentials and for a wide range of atomic numbers Z. In general, excellent agreement is found. In contrast to the limited applicability of the usual perturbative methods, our technique is found to be flexible and may be extended to a more general class of relativistic potentials that has applications in atomic and quarkonium physics. Encouraging aspects of the present approach are also briefly discussed.

A detailed study of the bound-state properties of the Hulth\'en potential is presented. Accurate eigenenergies are obtained for the Hulth\'en potential by numerical integration of the Schr\"odinger equation. One-parameter variational calculations are carried out. The variational results are practically identical to the exact energies, except in the high-screening region. The critical screening parameter is calculated for various values of l for n\ensuremath{\le}10 by a numerical solution of the wave equation. The energy eigenvalues obtained by a variety of methods are compared and discussed. The variational wave functions are employed to calculate absorption oscillator strengths for 1s\ensuremath{\rightarrow}2p, 1s\ensuremath{\rightarrow}3p, and 2p\ensuremath{\rightarrow}3d transitions.

The conditions for the existence of an ion-acoustic double layer in a plasma with warm ions and two distinct groups of hot electrons have been studied using the Sagdeev potential method. A comparison is made with the published results of Bharuthram and Shukla for cold ions and a two-temperature electron population. Numerical studies have been made to find out the effect of a finite ion temperature on the Mach number of the double layers.

Some general properties of a hydrogenic system confined to a circular disc are analyzed. Based on these, simple and accurate expressions are developed for the energies, critical radii, and some other properties, for all the states and all radii of confinement. Model wave functions are considered for some states, which give accurate values for the energies and other properties, and provide an insight into their physical structure. PACS Nos.: 3.65.Ge, 73.21.La, 78.67.Hc

A simple asymptotically correct model potential is considered for the valence electron in an alkali atom. Superposed with the confining infinite barrier potential, this potential yields the energies of various states and the dipole polarizability of the ground state of the confined alkali atom. Superposed with the fullerene shell potential, this potential yields the energies of the alkali atom inside a fullerene cage. Numerical results are presented for Li, Na, K, Rb, and Cs under such confinements. PACS Nos.: 03.65.Ge, 73.21.La, 78.67.Hc

The WKB (Wentzel–Kramers–Brillouin) as well as the SWKB quantization rules are applied to a number of quasi-exactly solvable problems (which apart from being SU(2) symmetric are also supersymmetric). The results obtained by these methods are compared with the exact numerical results.

It is observed that the well-known shifted large-N technique proposed by Imbo, Pagnamenta, and Sukhatme [Phys. Rev. D 29, 1669 (1984)] fails to yield the correct behavior of the bound-state wave function for the Morse oscillator. To remove this inadequacy, we propose a modified approach which not only restores the exact analytic expressions for the energy eigenvalues and eigenfunctions for the Morse oscillator but also predicts quite accurate results for the rotating Morse oscillator

The shifted large-N technique (SLNT) has been applied to study the relativistic motion of a particle in the presence of vector and scalar interactions with special emphasis on the construction of both large- and small-component Dirac radial wave functions. Numerical results for the binding energy for a particle in the presence of the Coulomb plus linear confining potential compare very well with those obtained by the elaborate analytic approximation method using the Pad\'e-approximation technique. We illustrate that one recovers not only the exact analytic results for binding energies for vector and scalar Coulomb potentials, but also exact wave functions from the leading-order SLNT calculation. This motivates future applications of the same method to more realistic atomic systems governed by screened Coulomb potentials where the knowledge of the large and small components of the radial wave function is essential.

L'ecrantage dielectrique lineaire de Thomas-Fermi et ses diverses ameliorations sont utilises pour calculer l'energie de liaison d'un centre donneur dans le silicium et le germanium. Les resultats sont compares aux valeurs experimentales