Timeline for What physically determines Bessel functions' orders?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 15, 2018 at 14:47 | comment | added | Paul Cwave | I see the connection now | |
| Jul 15, 2018 at 13:46 | comment | added | Emilio Pisanty | @PaulCwave They're not independent - the behaviour on the angular direction has a direct impact on the ODE for the radial direction, i.e. the $m^2/\rho^2$ term on that ODE, which comes directly from the angular part of the derivative with a suitable $1/\rho^2$ geometrical factor. This is the generic behaviour on general coordinate systems, and the systems where this doesn't happen (like cartesian coordinates) are few and far between. | |
| Jul 15, 2018 at 13:34 | comment | added | Paul Cwave | Different angular momentum states can have different spectra yes i get that, but I still don't see why the number of angular nodes would impose a certain index on the radial Bessel function (especially if they are independent)? How can they be independent and yet the Bessel function bear the exact index of the number of angular nodes? | |
| Jul 14, 2018 at 17:57 | comment | added | mike stone | @Paul Cwave Yes. So what's the problem? Different angular momentum states have different spectra. This is hardly surprising. | |
| Jul 14, 2018 at 16:34 | comment | added | Paul Cwave | but the index $n$ in $J_n$ (radial term) is the same as the one in the exponential (angular) term? | |
| Jul 14, 2018 at 16:24 | comment | added | mike stone | @Paul Cwave They are independant. If you impose a radial $\psi(r,\theta)=0$boundary condition at $r=R$, then that selects $\kappa_m = x_m/R$, $m=0,1,2,\ldots$, where $x_m$ is the $m$-th zero of $J_n(x)$, as the set of eigenvalues associated with the angular mode number $n$. | |
| Jul 13, 2018 at 22:48 | comment | added | Paul Cwave | Interesting. But I can't make sense of the connection between the number of nodes in the angular direction and what happens in the radial direction... | |
| Jul 13, 2018 at 14:44 | history | answered | mike stone | CC BY-SA 4.0 |