The local orthonormal basis set (r,θ,φ) is related to the Cartesian system by:
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1 TIS in Sica Cooinats As not in t ast ct, an of t otntias tat w wi a wit a cnta otntias, aning tat t a jst fnctions of t istanc btwn a atic an so oint of oigin. In tis cas tn, (,, z as a t Coob otntia an t gaitationa otntia. Bfo w go ft wit tis, it is iotant to a a goo woking ationsi wit sica cooinats. iw of Sica Cooinat Sst T oca otonoa basis st (,, is at to t Catsian sst b: z cos,, cos z,, z tan cos Tn, to fin t ation btwn t nit cto an t Catsian nit ctos, w can wit t cto in ts of bot: zz ( cos ( ( cos z ( cos ( ( cos z So tat t nit ctos in sica cooinats n on osition (as on can tiia s. Bcas of tis, it so not sis o tat t gaint is o coicat tan in catsian cooinats:
2 z z an, bcas t nit ctos a aso fnctions of t aiabs, t c is n o coicat: W wi s ts ations as w o t TIS in sica cooinats. Back to Cnta Focs Sinc t foc is at to t otntia ia: F T foc as on a aia coonnt. Anga Mont Tis aia nat of t foc as on iotant consqnc. t s ain t baio of t anga ont fo a cnta foc: & T otion of t anga ont in ti is gin b: F t t t t & & & T fist t is zo c an cto coss wit itsf is zo, t scon t is zo (fo t sa ason a c t foc on as a aia coonnt. Tis ts s tat
3 fo a cnta otntia, anga ont is cons. Now, ca wat tis ans in qant canics: If an obsab is cons, tn its associat oato cots wit t Haitonian oato: H, an tfo aso: [ ] H, If two oatos cot, tn w know tat w can scif an ignfnction tat sitanos satisfis an igna qation fo bot oatos. W wi tn to tis iotant fact at. TIS in Sica Cooinats In aation fo tis ct, I a fon sa was to aoac t ont of t TIS in sica cooinats. I a going to snt two iffnt tos. Staigtfowa Saation of aiabs Sinc t ont oato is gin as: i tn, t TIS bcos: H Ug t sica cooinat sntation fo t c: H Now, if t otntia is cnta, tn it on ns on t istanc fo t oigin an w ook fo saab sotions:
4 ,,, an w a: Now, iiing b an tiing b gis: [ ] As bfo wn aing t to of saation of aiabs, on t (t t in t fist st of c backts is a fnction of on innnt aiab (, in tis cas an t st is a fnction of jst an. Tfo, ac of ts ts st b qa to a constant: [ ] T fo of t saation constant is aing to soting, bt in gna, can b an co nb. Now, w a two iffntia qations an w st a wit ac saat. I wi coos to a wit t anga qation fist, c t is no otntia o ng t t, an w can so it fo t gna cas. Mtiing t anga qation b w a: Now, t fist two ts ino o iatis wit sct to, an t ti t inos iatis wit sct to, so w onc again attt t to of saation of aiabs b assing tat t anga fnction (, ((: an tn again ii b :
5 Onc again, w a scc c t fist t is on a fnction of an t scon t is on a fnction of, so ac st qa to a constant: t s again attack t as qation fist: i W t aso b ngati an tis cos t ot sotion wit t ngati onnt. Now, is t azita ang (b? an tfo wn aancs b π, w a back to t sa oint in sac. Tfo:,K,, ± ± i i i π π π So tat o saation constant, st b an intg. Now w tn to t o iffict qation: [ ] witot oof, I gi t sotions to tis qation as: cos AP w, P P a t associats gn fnctions an
6 P! oigs foa a t gn Ponoias, t fist fw of wic a sown bow (fo Giffits: T gn fnctions a tn i fo t abo foa. It so b not fist tat st b a nonngati intg (otwis t oigs foa osn t ak sns an tat in t finition of t associat gn fnction, c w tak t t iati of t gn Ponoia, st b iit to as ss tan o qa to [t gn onoias a onoias of o ]. Tfo t a ( aow as fo : (, (,,, K ;, K, is ca t azita qant nb, is t agntic qant nb. T associat gn fnctions a ist bow wit sktcs of a fw. T sktcs so b otat abot t tica ais. Now, w can t togt t an nnc back into an noaiz to gt t sica aonics, t fist fw of wic a ist bow:
7 T noaiz sica aonics a gin b: (, ε (, ε, ( ( 4π (!! i P ( cos, OK, so now w a t anga at of t wafnction witot intifing t otntia, ct tat it is cnta. T sa of t sica stic otntia on affcts t aia at of t wafnction, wic w now tn o attntion to. T aia at of t TIS is gin b: [ ] ( if w fin: tn t qation siifis sowat:
8 [ ] [ ] [ ] [ ] [ ] Now, tis is act of t fo of t on-insiona TIS wit an ffcti otntia of: ff W wi tn to t aia qation, bt I fist want to go o anot to fo soing t TIS in sica cooinats wit a cnta otntia. t s instigat t otis of t anga ont in sica cooinats, c w a aa sn tat t anga ont an t Haitonian so a sitanos ignfnctions. W stat wit t Haitonian: H bt, b tat in sica cooinats, i i so tat o finition of t anga ont oato bcos: i an t sqa of t anga ont oato is:
9 o, soing fo t ont oato: ( ( bt, w saw wn w w soing t TIS abo g saation of aiabs tat: an w can s tat t fist t in t qation abo ating to is jst t aia at of t ont (sqa aning tat t scon at, st b t anga at: ( ( So, now o Haitonian can b wittn as: H Not: Tis Haitonian is not saab, c a t ts now n on. Bt, if o b back a itt, w a anot too b wic w can attack tis ob. b tat w fon tat an t Haitonian cot, an tfo t a sitanos ignas sc tat: HΨ Ψ (,, Ψ(,, (,, Λ Ψ(,, Now, a cos ook at ts two igna qations wi sow tat t a intica (to witin a facto of ћ to t aia an anga qations w i abo, wit t conition tat: Λ ( W wi nt o t anga qation in ts of t anga ont oato to s if t is anot wa to so it (b t Haonic Osciato?.
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