On the hydrogen wave function in Momentum-space, Clifford algebra and the Generating function of Gegenbauer polynomial

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2 O he hoge we fco Moe-sce ffo geb he eeg fco of egebe oo M Hge-Hss Uesé Lbse Fcé es Sceces Seco Hh-Beoh bsc R R Usg he c sfoo he geeg fco eho we efo he Foe sfoo of he we fco of cooes of hoge o we f he c eesso of he we fco oe sce We ee he eees bewee he bss o -esos eg eeseo of he geeg fcos of egebe oos We f eosh bewee css of ffo geb he geeg fcos of hese oos -Ioco The obe of he hoge o hs e ce oe he eeoe of echcs Schöge soe hs eo fo he we fco of he cooe eeseo The obe oe sce hs bee efoe b Fock whch e o eg fo of he Schöge eo Ths eo s soe b oecg he hee-eso oe sce oo he sfce of foeso shee he egefcos e he ee es of shec hocs Dese he oce of Fock s wok he ees of hos -8 o s he we fco oe sce s o he h he ec cco of Foe sfo of he we fco of cooes s ow oe o hs wok s o f hs g The we fco of cooes 9 s he foψ R θϕ / Whee R s he Ω s he shec hoc Ω he so ge The ffc fo he eeo of he we fco oe sce coes fo he ece of he e he eoe of he We oose o cce hese obes b sg he c sfoo R R he geeg fco eho whee / s cos fo he eees of he bss fe cco we fo he eso of he obe fco he Foe sfo of he we fcos oso sce wh he we ob he c eesso of he we fco of hoge o oe eeseo

3 To o cbesoe he e b he cos we osees fo he ssge fo bewee he shec fco Wge s D- eees of SU We f Ieg eeseo of he geeg fcos of egebe oos s f s I kow hs s ew fo We f so h he Bg s eg of css of c fos ee o ffo geb ges s soo he geeg fco of egebe Poos Ths e s oge s foows I we cosc he geeg fco fo he bss of he hoge o The e seco s eoe o he eseo of he coeco of R hoge o R hoc osco I seco we ee he we fcos of hoge o oe sce I seco we ee he ssge fo bewee he bss of R he eg eeseo of he geeg fcos of egebe oos I he s seco we ese he eo bewee he ffo geb geeg fco of hese oos eeg fco fo he bss of he hoge o The we fco of hoge o oe eeseo 9 s ψ e ψ / Wh R s he R e L Whee L s he ssoce Lgee oo oc e se hogh he e The geeg fco of Lgee oo L The geeg fco of Lgee oo s: L e Fo he oe L L We ece h L e The geeg fco of shec hocs The geeg fco of shec hocs s: ϕ ξ Wh s eco of egh eo s cooes ξ η ξ η ξη

4 ξ η Wh ϕ ξ η eeg fco fo he bss of he hoge o We Ψ b ϕ ξ sg wh esec o ξ ϕ ξ Ψ / e L ϕ ξ Sbsg he boe eesso we ob: ξ e The coeco of R hoge o R hoc osco We w ee he c sfoo b se w he we eee he oe eee s of he coeco bewee he we fco of hoge o hoc osco s ge he foowg The c sfoo R R The c sfoo R R hs bee se fs b Ksheo-Sefe cees echcs ws se so b hos - fo he coeco of R hoge o R hoc osco We sh ee hs sfoo b se eho kowg h s eo c be oe b see ws - ose he eosh bewee he we-kow Wge s D shec hocs oos 7 D We we es of Ee s ges o e-ke eeo φ ψ φ ψ θ θ cos e s e D D ψθϕ / I s o o ehse h he eees of he D e soo of Lc wh he eeo of e-ke Δ If we we ob he c sfoo R R :

5 The oe eee We cose he sfoo θ ϕ ψ Wh θ ψ ϕ K J θϕψ The cco of he Jcob ges J /8sθ b θϕψ Theefoe 8 ψ f f ψ f The coeco of hoge o hoc osco ck cco shows h he eo of he hoge o h Ze Δ Ψ EΨ s he ece ss Th be we o he bss of hoc osco he fo h E Ψ Ψ Ze Wh cos o he egefcos: Ψ ψ 8E / Ze h The eeg s ge b: Ze E 7 h - The we fcos of hoge o oe sce We we fs he Foe sfo he eeseo wh he he of Bg eg we eee he geeg fco oe eeseo F he eeoe of hs fco ges s he we fcos of hoge o oe sce The geeg fco {} eeseo We eoe he geeg fco b ξ he eeseo {} B o eee he geeg fco we s b / o efec he chge he ese of ego We we ψ e ψ / To cce hs eesso we s we he eeseo sg he fo :

6 e / ψ ψ I he eesso ψ hee s he e fo h we cose ew geeg fco: / ξ e e e We sse h heefoe hee s o obe of coegece We we he ξ ξ Wh ψ ξ ϕ ξ The geeg fco of oe-sce We c o he ego of b ec cco wh he bes o oe ck sg he Bg eg 8 e e e / B X X B X Wh We he 7 We ob he X Becse we ece h: e X 8 We f heefoe he geeg fco ξ 9 I g he eo we f he geeg fco ξ ξ

7 The we fcos oe-sce We e he bss of oe-sce sg he fo / ψ ξ ξ ϕ I hs cse we s ke / o eece he ccos we ocee b se: - Deo wh esec o ξ We he We ece h ξ - Deo wh esec o Usg he f fo fo he geeg fco of egebe oos We we Wh / heefoe ξ P We ob Ths we f he sfoo oce b Fock / - Deo wh esec o ϕ ξ B sg he fo we ge he foowg eesso

8 / ξ ξ ϕ - The we fcos oe sce The cosos of ge s he es: / ψ 7 wh he he of he ececes fo We ee f he we fcos oe sce: ψ 8 I s ce h we ob b eee eho o o he we fco oe eeseo b so he hse fco - Pssge fos bewee he bss of R bewee he geeg fcos of egebe I oe o o cbesoe he e wh he cos I esce o o he cco of he ssge eees fo he shec bss o he bss whch eees e he g bewee he geeg fcos of egebe oos ψθϕ D If we cose he shec eeo θ ϕ θ ϕ θ cos c s s s s cos s s os The Lc he soos he shec fcos -9 Δ cos Wh Pssge fos bewee he bses of R We cose he geeg fco of Wge s D-: e U D ϕ ϕ B ecg he fo b we he 7

9 We f he he c sfoo R R Wh e ϕ D U ϕ seco fo of c be oe hogh he eeoe of he we fee he he geeg fco of egebe oos Legee co fo We eeo e o he shec hocs bss * e θ ϕ θϕ we we e e cos / * J / s θ ϕ θϕ k Wh he seco geeg fco of egebe oos cos e s J s We ob e * 7 cos 8 θ ϕ θϕ cos 9 Wh Legee co fo we 9 b θ ϕ Ω he we eece he ego We ob / / / ϕ cos k θϕ ϕ ϕ θ Ω D U Wh he he of he eesso 7 8 U D U D U D U / / / / / / 8

10 F we ge he we kow eesso U D Ieg eeseo of he geeg fcos of egebe oos Wh he eeoe of he e we he eesso we f h cos e { / / e } s We c eece he ego sg he Bg eg we ee he se cco of gh We ob e / e X Wh X e X The / e P cog he wo eessos we F he eg eeseo of he geeg fcos of egebe: s cos e cos e ffo geb geeg fco of egebe oos We oce eosh bewee he geeg fco of egebe oos he ocoos geb hs s o ese hs eosh Bg eg Le- sfoo The c sfoo s R R b og wh he eesso we we: 9

11 The Bg eg hs cse s e e The seco s he geeg fco of egebe oos We so we I Wh I Bg eg he eos I s we kow h I Whee e he eeseos ces of eo We f b ec cco he Bg eg I e e The seco s he geeg fco of egebe oos Bg eg he Ocoos The Hw sfoo of s ge b 8 R R We cose s eos: 8 s he sose The Bg eg hs cse s e 9 we f g h he seco ebe s he geeg fco of egebe oos

12 We so we We oe h { Δ e } whch e s o o he eso of 8 o he bss of SO o o he bss of SU he of ssge bewee hese bses s eee b he se eho of Bg eg he ffo geb Bse o he eesso of we c geee hese ess b wg: I I Usg sboc og we f so: Fo e 8 Fo e 8 We ece fo he boe eoe h hee s cose eosh bewee he ffo geb he geeg fcos of egebe oos o eho of ccg Foe sfoo of he oso cooes c be geee o oes 7 Refeeces V Fock Z Phs 98 9 M Be Ikso Re Mo Phs 8 9 E Wf I o Theo s cos E EM Loebe ew ok: cec Pess 97 H Behe EE See Q Mechcs of Oe- Two-Eeco os Sge Be 977 T D T Sec; Fcos go Theoec och Be ew-ok JR Lob Ph Re V Mee J-M Ros Moe esos Fock se of hoge o X:h-h/ 8 JS Dehes S Loe-Ros B Oos RJe Fshe foo of D- eso hogec sses oso oe sces J Mh Phscs 7 9 Messh Méce e - E Do Ps 9 PKheo E Sefe; J Ree gew Mh 8 9 M Boe R c Sc Sée B T ég These Lo 98

13 M Hge-Hss Ie eso coss oc -esos sce Pe h-h/ M Hge-Hss Hw s ces e sfoo he -We bss fo he ohogo gos Pe h-h/ M Hge-Hss M Kbe o-bece Qc sfoo he Theo of g oe Seece ocs ssc hscs Es: Logo Wo Scefc: Sgoe 99 D Lbe M Kbe J Phs :Mh e R Eos g Moe Q Mechcs Pceo UP Pceo J 97 8 V Bg O he eeseos of he Roo go Re Mo Phs 9 9 L Beeh J Mh Phs 9 Schwge J I Theo of g Moe E Beeh L V D H ew ok: cec Pess IEe Hghe sce fcos Vo Mc w-h ew-ok 9

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