We apply our recently proposed proper quantization rule, (x)]/h to obtain the energy spectrum of the modified Rosen-Morse potential. The beauty and symmetry of this proper rule come from its meaning-whenever the number of the nodes of φ(x) or the number of the nodes of the wave function ψ(x) increases by one, the momentum integralx Bx A k(x)dx will increase by π . Based on this new approach, we present a vibrational high temperature partition function in order to study thermodynamic functions such as the vibrational mean energy U , specific heat C, free energy F and entropy S. It is surprising to note that the specific heat C(k = 1) first increases with β and arrives to the maximum value and then decreases with it. However, it is shown that the entropy S(k = 1) first increases with the deepness of potential well λ and then decreases with it.
Impurity doping in semiconductor nanowires, while increasingly well understood, is not yet controllable at a satisfactory degree. The large surface-to-volume area of these systems, however, suggests that adsorption of the appropriate molecular complexes on the wire sidewalls could be a viable alternative to conventional impurity doping. We perform first-principles electronic structure calculations to assess the possibility of n- and p-type doping of Si nanowires by exposure to NH(3) and NO(2). Besides providing a full rationalization of the experimental results recently obtained in mesoporous Si, our calculations show that while NH(3) is a shallow donor, NO(2) yields p-doping only when passive surface segregated B atoms are present.
Key words Proper quantization rule, bound states, solvable potentials.In this article, we present proper quantization rule,/ and study solvable potentials. We find that the energy spectra of solvable systems can be calculated only from its ground state obtained by the Sturm-Liouville theorem. The previous complicated and tedious integral calculations involved in exact quantization rule are greatly simplified. The beauty and simplicity of proper quantization rule come from its meaning -whenever the number of the nodes of the logarithmic derivative φ(x) = ψ(x) −1 dψ(x)/dx or the number of the nodes of the wave function ψ(x) increases by one, the momentum integral will increase by π. We apply two different quantization rules to carry out a few typically solvable quantum systems such as the one-dimensional harmonic oscillator, the Morse potential and its generalization as well as the asymmetrical trigonometric Scarf potential and show a great advantage of the proper quantization rule over the original exact quantization rule.
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