Rotation of a rigid diatomic dipole molecule in a homogeneous electric field: I. Schrödinger equation. Quantization conditions according to phase-integral method

1994 ◽  
Vol 347 (1682) ◽  
pp. 1-22 ◽  
Keyword(s):  
Field Strength ◽  
Schrödinger Equation ◽  
Electric Field ◽  
Electric Field Strength ◽  
Integral Method ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Base Function ◽  
Dipole Molecule ◽  
Phase Integral

After a brief historical discussion of the energy quantization according to quantum mechanics of the rigid diatomic dipole molecule in a static electric field, the Schrödinger equation for that system is recalled. Previous attempts to obtain the energy levels clearly indicate that there is a need for a reliable method yielding very accurate eigenvalues for all values of the electric field strength. This is accomplished with the aid of new quantization conditions obtained by means of a phase-integral method involving a general phase-integral approximation of arbitrary order generated from an unspecified base function, which is chosen in two different ways such that, when the electric field strength is equal to zero, simple limiting forms of the quantization conditions give the exact values of the energy levels. The two choices of the base function are expected to be appropriate in the cases when the absolute value of the magnetic quantum number m is sufficiently large and sufficiently small respectively. For every value of m at least one of the two base functions should be useful.

Rotation of a rigid diatomic dipole molecule in a homogeneous electric field IV. New phase-integral quantization conditions, very accurate for arbitary values of the magnetic quantum number, expressed in terms of complete elliptic integrals: survey of their accuracy

1994 ◽  
Vol 347 (1682) ◽  
pp. 65-81
Keyword(s):  
Electric Field ◽  
Quantum Number ◽  
Energy Levels ◽  
Elliptic Integrals ◽  
Magnetic Quantum Number ◽  
Homogeneous Electric Field ◽  
Base Function ◽  
Dipole Molecule ◽  
Phase Integral ◽  
Complete Elliptic Integrals

New quantization conditions for the energy levels of a rigid diatomic dipole molecule in a homogeneous electric field of arbitrary strength, obtained by means of a phaseintegral method involving phase-integral approximations of arbitrary order generated from two particular choices of the base function, are expressed in terms of complete elliptic integrals in the first, third and fifth order of the phase-integral approximation. Previous results, derived for one convenient choice of the parameter £0 in the base function, namely £0 = 1/2|m|, where m is the magnetic quantum number, are used, and new formulas are derived for the other convenient choice £0 = 0. The accuracy of the eigenvalues obtained from the quantization conditions is demonstrated in a number of diagrams.

Rotation of a rigid diatomic dipole molecule in a homogeneous electric field III. New phase-integral quantization conditions, expected to be most accurate when the magnetic quantum number is large or equal to zero, expressed in terms of complete elliptic integrals: survey of their accuracy

1994 ◽  
Vol 347 (1682) ◽  
pp. 37-63
Keyword(s):  
Electric Field ◽  
Quantum Number ◽  
Electric Fields ◽  
Energy Levels ◽  
Series Expansions ◽  
Magnetic Quantum Number ◽  
Homogeneous Electric Field ◽  
Base Function ◽  
Phase Integral ◽  
Complete Elliptic Integrals

Some new quantization conditions for the energy levels of a rigid diatomic dipole molecule in a homogeneous electric field of arbitrary strength, obtained by means of a phase-integral method involving phase-integral approximations of arbitrary order generated from two particular choices of the base function, are expressed in terms of complete elliptic integrals in the first, third and fifth order of the phase-integral approximation. One choice of the base function is especially useful for large absolute values of the magnetic quantum number m . The case m = 0 is considered with another choice of the base function, expected to be useful for small values of | m |. The great accuracy of the energy levels, yielded by the quantization conditions, is demonstrated for arbitrary electric fields in a number of diagrams pertaining to different values of the quantum numbers. For very weak and very strong electric fields explicit series expansions for the energy levels can be obtained from the quantization conditions, and these expansions are compared with previously obtained series expansions. The investigation confirms that the phase-integral quantization conditions yield very accurate eigenvalues for all values of the electric field strength.

Optical Measurement of the Electric Field Strength in a Gel Insulator

2016 ◽  
Vol 136 (10) ◽  
pp. 1420-1421
Author(s):  
Yusuke Tanaka ◽  
Yuji Nagaoka ◽  
Hyeon-Gu Jeon ◽  
Masaharu Fujii ◽  
Haruo Ihori
Keyword(s):  
Field Strength ◽  
Electric Field ◽  
Electric Field Strength ◽  
Optical Measurement

Effect of external magnetic field on lane formation in driven pair-ion plasmas

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Swati Baruah ◽  
U. Sarma ◽  
R. Ganesh
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Field Strength ◽  
Electric Field ◽  
Electric Field Strength ◽  
Coupling Parameter ◽  
Critical Parameters ◽  
Coulomb Coupling ◽  
Lane Formation ◽  
Parameter Values

Lane formation dynamics in externally driven pair-ion plasma (PIP) particles is studied in the presence of external magnetic field using Langevin dynamics (LD) simulation. The phase diagram obtained distinguishing the no-lane and lane states is systematically determined from a study of various Coulomb coupling parameter values. A peculiar lane formation-disintegration parameter space is identified; lane formation area extended to a wide range of Coulomb coupling parameter values is observed before disappearing to a mixed phase. The different phases are identified by calculating the order parameter. This and the critical parameters are calculated directly from LD simulation. The critical electric field strength value above which the lanes are formed distinctly is obtained, and it is observed that in the presence of the external magnetic field, the PIP system requires a higher value of the electric field strength to enter into the lane formation state than that in the absence of the magnetic field. We further find out the critical value of electric field frequency beyond which the system exhibits a transition back to the disordered state and this critical frequency is found as an increasing function of the electric field strength in the presence of an external magnetic field. The movement of the lanes is also observed in a direction perpendicular to that of the applied electric and magnetic field directions, which reveals the existence of the electric field drift in the system under study. We also use an oblique force field as the external driving force, both in the presence and absence of the external magnetic field. The application of this oblique force changes the orientation of the lane structures for different applied oblique angle values.

scholarly journals Investigating the feasibility of cerebellar transcranial direct current stimulation to facilitate post-stroke overground gait performance in chronic stroke: a partial least-squares regression approach

2021 ◽  
Vol 18 (1) ◽  
Author(s):  
Dhaval Solanki ◽  
Zeynab Rezaee ◽  
Anirban Dutta ◽  
Uttama Lahiri
Keyword(s):  
Field Strength ◽  
Electric Field ◽  
Electric Field Strength ◽  
Chronic Stroke ◽  
Least Squares Regression ◽  
Significance Level ◽  
Gait Performance ◽  
Wilcoxon Rank Sum Test ◽  
Gait Parameters ◽  
The Mean

Abstract Background Investigation of lobule-specific electric field effects of cerebellar transcranial direct current stimulation (ctDCS) on overground gait performance has not been performed, so this study aimed to investigate the feasibility of two lobule-specific bilateral ctDCS montages to facilitate overground walking in chronic stroke. Methods Ten chronic post-stroke male subjects participated in this repeated-measure single-blind crossover study, where we evaluated the single-session effects of two bilateral ctDCS montages that applied 2 mA via 3.14 cm2 disc electrodes for 15 min targeting (a) dentate nuclei (also, anterior and posterior lobes), and (b) lower-limb representations (lobules VIIb-IX). A two-sided Wilcoxon rank-sum test was performed at a 5% significance level on the percent normalized change measures in the overground gait performance. Partial least squares regression (PLSR) analysis was performed on the quantitative gait parameters as response variables to the mean lobular electric field strength as the predictors. Clinical assessments were performed with the Ten-Meter walk test (TMWT), Timed Up & Go (TUG), and the Berg Balance Scale based on minimal clinically important differences (MCID). Results The ctDCS montage specific effect was found significant using a two-sided Wilcoxon rank-sum test at a 5% significance level for 'Step Time Affected Leg' (p = 0.0257) and '%Stance Time Unaffected Leg' (p = 0.0376). The changes in the quantitative gait parameters were found to be correlated to the mean electric field strength in the lobules based on PLSR analysis (R2 statistic = 0.6574). Here, the mean electric field strength at the cerebellar lobules, Vermis VIIIb, Ipsi-lesional IX, Vermis IX, Ipsi-lesional X, had the most loading and were positively related to the 'Step Time Affected Leg' and '%Stance Time Unaffected Leg,' and negatively related to the '%Swing Time Unaffected Leg,' '%Single Support Time Affected Leg.' Clinical assessments found similar improvement in the TMWT (MCID: 0.10 m/s), TUG (MCID: 8 s), and BBS score (MCID: 12.5 points) for both the ctDCS montages. Conclusion Our feasibility study found an association between the lobular mean electric field strength and the changes in the quantitative gait parameters following a single ctDCS session in chronic stroke. Both the ctDCS montages improved the clinical outcome measures that should be investigated with a larger sample size for clinical validation. Trial registration: Being retrospectively registered.

scholarly journals Optical properties of c-Plane InGaN/GaN single quantum wells as a function of total electric field strength

2019 ◽  
Vol 58 (SC) ◽  
pp. SCCB09 ◽  
Author(s):  
George M. Christian ◽  
Stefan Schulz ◽  
Simon Hammersley ◽  
Menno J. Kappers ◽  
Martin Frentrup ◽  
...  
Keyword(s):  
Optical Properties ◽  
Field Strength ◽  
Electric Field ◽  
Quantum Wells ◽  
Electric Field Strength ◽  
Single Quantum ◽  
Total Electric Field
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