Single-particle energy states for a neutron and a proton are obtained by solving the time-independent Schrödinger equation for the mean-field Woods–Saxon potential along with the spin-orbit term. The wavefunctions are expanded as a linear combination of simple sine-wave basis states, which are eigenfunctions of the infinite spherical-well potential. The requisite algorithm based on matrix diagonalization is implemented in Free Open Source Software (FOSS) Scilab. Initial values for the simulation were taken from model parameters given in the book on Nuclear Structure by Bohr and Mottelson, which were then optimized to obtain the best convergence with the available experimental energy values of various nuclei with magic proton and neutron numbers. The level scheme, as well as the energy values for doubly magic nuclei 82208Pb and 2040Ca, which are obtained using our simulation, is presented in this paper. Finally, energy level diagrams for neutrons and protons with respect to mass number A were arrived at, based on those obtained for various magic nuclei. The evaluation method, which is based on the sine-wave basis, is akin to Fourier analysis. When done with the aid of FOSS Scilab, this technique becomes easily accessible to students at the under-graduate (UG) level and may be studied through small projects.
The quantum anharmonic oscillator has been solved numerically using matrix diagonalization technique. The interaction potential consisting of quadratic () and quartic (λx4) terms is embedded within an infinite square well potential of appropriate width, ‘a’ and its sine eigen functions are used as basis functions ‘N’ for the employed matrix method. The energy eigen values for the resultant Hamiltonian are solved in a free open source software (FOSS), Gnumeric, a simple worksheet environment. The numerical parameters ‘a’ and ‘N’ are optimized to converge to the expected energies for harmonic oscillator and those for anharmonic oscillator from perturbation theory for small values of physical parameter, ‘λ’. The pure quartic oscillator is studied for both small and large values of λ and validated with results obtained from other numerical techniques. The breakdown of perturbation approximation for large values of λ is also shown.
Background:A high level of preoperative anxiety is common among patients undergoing medical and surgical procedures. Anxiety impacts of gastroenterological procedures on psychological and physiological responses are worth consideration.Aims and Objectives:To analyze the effect of listening to Vedic chants and Indian classical instrumental music on anxiety levels and on blood pressure (BP), heart rate (HR), and oxygen saturation in patients undergoing upper gastrointestinal (GI) endoscopy.Materials and Methods:A prospective, randomized controlled trial was done on 199 patients undergoing upper GI endoscopy. On arrival, their anxiety levels were assessed using state and trait scores and various physiological parameters such as HR, BP, and SpO2. Patients were randomly divided into three groups: Group I of 67 patients who were made to listen prerecorded Vedic chants for 10 min, Group II consisting of 66 patients who listened to Indian classical instrumental music for 10 min, and Group III of 66 controls who remained seated for same period in the same environment. Thereafter, their anxiety state scores and physiological parameters were reassessed.Results:A significant reduction in anxiety state scores was observed in the patients in Group I (from 40.4 ± 8.9 to 38.5 ± 10.7; P < 0.05) and Group II (from 41.8 ± 9.9 to 38.0 ± 8.6; P < 0.001) while Group III controls showed no significant change in the anxiety scores. A significant decrease in systolic BP (P < 0.001), diastolic BP (P < 0.05), and SpO2 (P < 0.05 was also observed in Group II.Conclusion:Listening to Vedic chants and Indian classical instrumental music has beneficial effects on alleviating anxiety levels induced by apprehension of invasive procedures and can be of therapeutic use.
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