Hilbert Book Model Project/Slide F12

hbmp	Template:Center topHilbert Book Model ProjectTemplate:Center bottom	F12
	The solutions of $◻\psi =({\mathrm{\nabla }}_r{\mathrm{\nabla }}_r-⟨\overrightarrow{\mathrm{\nabla }},\overrightarrow{\mathrm{\nabla }}⟩)\psi =0$ offer waves as part of its solutions
	This can be shown by Template:Font ${\mathrm{\nabla }}_r{\mathrm{\nabla }}_r\psi =⟨\overrightarrow{\mathrm{\nabla }},\overrightarrow{\mathrm{\nabla }}⟩\psi =\omega \psi \Rightarrow f=\exp (2\pi i\omega x\tau )$
	The Template:Font considers the field Template:Font
	Template:Center top $\psi (q_r,\overrightarrow{q})=A(\overrightarrow{q})T(q_r)$ Template:Center bottom Template:Center top $\frac{⟨\overrightarrow{\mathrm{\nabla }},\overrightarrow{\mathrm{\nabla }}⟩A}{A}=\frac{{\mathrm{\nabla }}_r{\mathrm{\nabla }}_{r}T}{T}=-k^2$ Template:Center bottom Template:Center top $⟨\overrightarrow{\mathrm{\nabla }},\overrightarrow{\mathrm{\nabla }}⟩A=-k^{2}A$ Template:Center bottom Template:Center top ${\mathrm{\nabla }}_r{\mathrm{\nabla }}_{r}T=-k^{2}T$ Template:Center bottom
	For Template:Font the solutions have the form $A(r,\theta ,\phi )=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }(a_{\ell m}{j}_{\ell }(kr)+b_{\ell m}{y}_{\ell }(kr))Y_{\ell }^m(\theta ,\phi )$
	Here $j_{\ell }$ and $y_{\ell }$ are the Template:Font, and $Y_{\ell }^m$ are the Template:Font These solutions play a role in the Template:Font
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