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The visualization of the angular probability distribution for the angular Teukolsky equation with m ≠ 0
Author(s) -
Chen ChangYuan,
Sun DongSheng,
Sun GuoHua,
Wang XiaoHua,
You Yuan,
Dong ShiHai
Publication year - 2021
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.26546
Subject(s) - eigenfunction , wronskian , eigenvalues and eigenvectors , mathematical analysis , physics , function (biology) , mathematical physics , mathematics , quantum mechanics , evolutionary biology , biology
We present the exact solutions of the angular Teukolsky equation with m ≠ 0 given by a confluent Heun function . This equation is first transformed to a confluent Heun differential equation through some variable transformations. The Wronskian determinant, which is constructed by two linearly dependent solutions, is used to calculate the eigenvalues precisely. The normalized eigenfunctions can be obtained by substituting the calculated eigenvalues into the unnormalized eigenfunctions. The relations among the linearly dependent eigenfunctions are also discussed. When c 2 = c R 2 + ic I 2 , the eigenvalues are approximately expressed as A lm ≈ l l + 1 +c R 2 + ic I 21 − m 2 / l l + 1/ 2 for small | c | 2 but large l . The isosurface and contour visualizations of the angular probability distribution (APD) are presented for the cases of the real and complex values c 2 . It is found that the APD has obvious directionality, but the northern and southern hemispheres are always symmetrical regardless of the value of the parameter c 2 , which is real or imaginary.