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By Brain Judd (Auth.)

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Alternatively, we can set u p a triangular struc­ t u r e b y extending t h e ordinary spherical-polar scheme: Xi = r sin χ sin θ cos φ, x2 = r sin χ sin Θ sin φ, Xz = r sin χ cos 0, X4 = r c o s x . 15) If we t a k e θ a n d φ t o lie within their traditional bounds (0 < φ < 2ir, 0 < θ < *·), t h e n t h e surface of t h e hypersphere is covered in j u s t one way provided 0 < χ < π. T h e t r i a d (χθφ) is abbreviated t o Ω. W e can now evaluate L a n d A in t e r m s of these coordinates. I t is a t once found t h a t L = /, where I is defined as in E q s .

T o begin with, we notice t h a t t h e fundamental commutation relations between ρ and r are as well preserved b y t h e substitutions . , as b y t h e more familiar replacements 34 Momentum 51 Space etc. I n fact, there is a complete reciprocity between these t w o points of view. 15) where t h e integration is carried o u t over all space. ) T h u s φ(ρ) is t h e Fourier transform of u(r). I t is supposed t h a t b o t h ψ a n d u are normalized. 16) (2π) δ(Γ), where t h e delta function possesses t h e property, for a n y continuous func­ tion/(r), //(r)5(r-r')dr=/(r').

W e can now evaluate L a n d A in t e r m s of these coordinates. I t is a t once found t h a t L = /, where I is defined as in E q s . 6). As for A, we get [ 2 2 ] ( d sin θ θχ d d\ h cot χ cos θ — ± i cot χ esc θ — J , ΘΘ θφ/ Az = i ( c o s θ — - cot χ sin θ — J . 16) I n three dimensions, t h e operators Ρ a n d lt form a c o m m u t i n g pair, a n d t h e corresponding q u a n t u m n u m b e r s (I a n d m) a r e available for labeling 2 t h e spherical harmonics F i m( 0 , φ). I t is straightforward t o show t h a t k (of E q .

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