separable

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21: 13.28 Physical Applications

… ►The reduced wave equation

\nabla^{2}w=k^{2}w

in paraboloidal coordinates,

x=2\sqrt{\xi\eta}\cos\phi

y=2\sqrt{\xi\eta}\sin\phi

z=\xi-\eta

, can be solved via separation of variables

w=f_{1}(\xi)f_{2}(\eta)e^{\mathrm{i}p\phi}

, where …

22: 25.21 Software

… ►References to research software that is available in other ways is listed separately. …

23: 30.18 Software

… ►References to research software that is available in other ways is listed separately. …

24: 12.17 Physical Applications

… ►Setting

w=U(\xi)V(\eta)W(\zeta)

and separating variables, we obtain … ►Miller (1974) treats separation of variables by group theoretic methods. …

25: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►With

e

denoting here the elementary charge, the Coulomb potential between two point particles with charges

Z_{1}e,Z_{2}e

and masses

m_{1},m_{2}

separated by a distance

s

V(s)=Z_{1}Z_{2}e^{2}/(4\pi\varepsilon_{0}s)=Z_{1}Z_{2}\alpha\hbar c/s

, where

Z_{j}

are atomic numbers,

\varepsilon_{0}

is the electric constant,

\alpha

is the fine structure constant, and

\hbar

is the reduced Planck’s constant. The reduced mass is

m=m_{1}m_{2}/(m_{1}+m_{2})

, and at energy of relative motion

E

with relative orbital angular momentum

\ell\hbar

, the Schrödinger equation for the radial wave function

w(s)

is given by ►

33.22.1

\left(-\frac{\hbar^{2}}{2m}\left(\frac{{\mathrm{d}}^{2}}{{\mathrm{d}s}^{2}}-% \frac{\ell(\ell+1)}{s^{2}}\right)+\frac{Z_{1}Z_{2}\alpha\hbar c}{s}\right)w=Ew,

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\alpha$ : fine structure constant, $\ell$ : nonnegative integer, $Z_{k}$ : charges, $m_{k}$ : masses, $s$ : separation and $E$ : energy
Referenced by:: §33.22(i)
Permalink:: http://dlmf.nist.gov/33.22.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §33.22(i), §33.22 and Ch.33

…

26: 10.73 Physical Applications

… ►and on separation of variables we obtain solutions of the form

e^{\pm in\phi}e^{\pm\kappa z}J_{n}\left(\kappa r\right)

, from which a solution satisfying prescribed boundary conditions may be constructed. … ►On separation of variables into cylindrical coordinates, the Bessel functions

J_{n}\left(x\right)

, and modified Bessel functions

I_{n}\left(x\right)

and

K_{n}\left(x\right)

, all appear. … ►The functions

\mathsf{j}_{n}\left(x\right)

\mathsf{y}_{n}\left(x\right)

{\mathsf{h}^{(1)}_{n}}\left(x\right)

, and

{\mathsf{h}^{(2)}_{n}}\left(x\right)

arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates

\rho,\theta,\phi

(§1.5(ii)): …

27: 28.33 Physical Applications

… ►The separated solutions

V_{n}(\xi,\eta)

must be

2\pi

-periodic in

\eta

, and have the form …where

q=\tfrac{1}{4}c^{2}k^{2}

and

a_{n}(q)

b_{n}(q)

is the separation constant; compare (28.12.11), (28.20.11), and (28.20.12). …The general solution of the problem is a superposition of the separated solutions. …