Hilbert Book Model Project/Slide F12

hbmp	Hilbert Book Model Project	F12
	The solutions of ${\color {white}\Box \ \psi =(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi =0}$ offer waves as part of its solutions
	This can be shown by separating variables ${\color {white}\nabla _{r}\nabla _{r}\psi =\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi =\omega \,\psi \Rightarrow f=\exp(2\pi i\omega x\tau )}$
	The Helmholtz equation considers the field separable
	${\color {white}\psi (q_{r},{\vec {q}})=A({\vec {q}})T(q_{r})}$ ${\color {white}{\frac {\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \,A}{A}}={\frac {\nabla _{r}\nabla _{r}\,T}{T}}=-k^{2}}$ ${\color {white}\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \,A=-k^{2}A}$ ${\color {white}\nabla _{r}\nabla _{r}\,T=-k^{2}T}$
	For three-dimensional isotropic spherical conditions the solutions have the form ${\color {white}A(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left(a_{\ell m}j_{\ell }(kr)+b_{\ell m}y_{\ell }(kr)\right)Y_{\ell }^{m}({\theta ,\varphi })}$
	Here ${\textstyle {\color {white}j_{\ell }}}$ and ${\textstyle {\color {white}y_{\ell }}}$ are the spherical Bessel functions, and ${\textstyle {\color {white}Y_{\ell }^{m}}}$ are the spherical harmonics These solutions play a role in the spectra of atomic modules
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