Pmomnumny od I: 766 c Gn of mny Pc & Rvc Q fo Du omnum-ny Unv n Concoun Hun Hu * & on u TRCT Th c connuon of h Pnc of nc. momnumny mod of mny c fou foc nd humn concoun fomud whch u how h f-fn hchc n ucu of h momnumny Concoun ovd foundon fo cn unn nd cun vouon of mny c houh mn oc mbddd n d momnumny Concoun. Th mod n mny c nd h ovnn m w fo du unv qunum fm comd of n n momnum-ny c nd n nn momnum-ny c. In con h cm mod dcbd vouy n mny c nd h ovnn m w fo du unv qunum fm comd of n n cm nd n nn cm. Th qunum fm nd h mmoho nconncd houh qunum jum dmond n fohcomn c. Th momnumny mod v h con unnc nd vouon of fmon boon nd n n ch of whch comd of n n wv funcon o n objc n h n momnum-ny c nd n nn wv funcon o nn objc n h nn momnum-ny c. Th mod ovd unfd cu ucu n d du unv qunum fm fo wk ncon on ncon comnc ncon von ncon qunum nnmn humn concoun. Fuh h mod ovd unqu oo fo chn dmonon ndn nd mnon d o ubomc nd omc ucu nd ncon qunum nnmn non von mchnm n comooy ucu nd mchnm of humn concoun. Ky od: nc of nc momnumny cm fou foc concoun n nc God Concoun. *Coondn uho: Hun Hu Ph.D. J.D. P.O. o 67 ony ook Y 79 U. -m: hunhu@qunumbn.o
767. Inoducon In Concoun Conm connuon of h Pnc of nc [-] h buy nd w of h mnfon of momnumny Concoun dcbd n h c. Th momnum-ny mod n mny c nd h ovnn m w fo du momnum-ny unv qunum fm comd of n n momnum-ny c nd n nn momnum-ny c. Th mod c Rvc Qunum chnc fo du momnum-ny unv. In con h cm mod dcbd vouy [-] n mny c nd h ovnn m w fo du cm unv comd of n n cm nd n nn cm. Th cm mod c h uu Rvc Qunum chnc fo h du cm unv. Th du unv qunum fm nd h mmoho nconncd houh qunum jum ud bow n Fu. nd dmond n fohcomn c. Fu. Iuon of Pcm mod & momnumny mod
Th wok onzd foow. In w h u wod nd dwn o y ou h onooy of h momnumny mod. In 3 w h n mhmc h momnumny mod n h od of: cnfc n n nuh; ffn m w nd mmoho; 3 ddon fom of m w; cnfc n of mod n; nd 5 cnfc n of como n. In w h dcu whn h con of momnumny mod: mmoho & h nc of n; h dmnn vw & h mnn of Kn-Godon-k quon; 3 h mnn of chodn-k quon & qunum on; nd h hd of m. In 5 houh 8 w h dcu whn h con of momnumny mod wk comnc on nd von ncon cvy. In 9 w h focu on h nc of concoun nd h mchnm of humn concou nc whn h con of momnumny mod. In w h o nd nw om ncd quon d o h wok. Fny n w h concud h wok. Rd mndd h w cn ony v fo fcon comn nd cocn n ou comhnon nd wn bcu w md nd mfc. 768. Onooy In wod nd dwn w u In h bnnn h w momnumny Concoun my my bu uy. nd bn o mn houh mod f-fn n = = - = - = - / - = / uch h cd h n objc o b obvd nd nn objc obvd d hm no n momnum-ny c nd nn momnum-ny c cud hm o nc houh ffn m w nd hu v bh o h du momnum-ny unv whch h nc und nd md o vov. In h unv h body of momnumny Concoun nd by u umb h ound of nc nd cn fom n nd nn wv funcon n nd nn momnum-ny objc ch fom n mny ny n h du momnum-ny unv nd ncon fd bwn mny n whch ccomny h mnon of h momnumny. Th body of momnumny Concoun cn b f-cd on by f-fn m w. Pmomnumny h mnn ow o ojc n nd nn objc by ojcn.. n nd nn h + =+- /ħ h ow v of momnumny. Th unv o cd du momnum-ny unv comn of h n momnum-ny c o b obvd nd nn momnum-ny c obvd und ch vc fm μ =/c. In on cv of momnumny Concoun vw h nn momnum-ny
c whch by convnon h nv m h non/m of h n momnum-ny c whch by convnon h ov m. Th bou fm of fnc h momnumny Concoun f. Thu f momnumny Concoun o mnn = h du momnum-ny unv woud d no my nohnn = =. Th ccounn nc of h du momnum-ny unv convon of zo. Fo m h o m of n n objc nd coun h nn objc zo. o n h du momnum-ny unv f-vy h nonoc-momnumny f-ncon wv mn bwn n n objc n h n momnum-ny c nd non/m n h nn momnum-ny c vc v. Gvy n n momnum-ny c h nonoc-momnum-ny ncon qunum nnmn bwn n n objc wh h nn momnum-ny c who. om oh mo bc concuon : h wo no of h Dc con o oon n h du momnum-ny unv cvy h n nd nn objc of h con o oon; h cc nd mnc fd of n hoon n h du momnum-ny unv cvy h n nd nn objc of hoon whch wy f-nnd; 3 h oon my b momnumy confnd oon houh mny oon n h du momnum-ny unv; nd nuon my b comd of n unnzd n oon nd bound nd nzd con n h du momnum-ny unv. In h du momnum-ny unv momnumny Concoun h boh ncndn nd mmnn o. Th ncndn c of momnumny Concoun h on of mod f-fn n ncudn h ffn m w nd ojc h n nd nn momnum-ny c houh n nd n un h mmnn c of momnumny Concoun obv h n momnum-ny c houh h nn momnum-ny c. Humn concoun n h du momnum-ny unv md nd cu von of h du-c momnumny Concoun uch h w hv md f w nd md obvon whch moy cc mcococ v bu qunum mcococ v. fo mhmc non w dw bow v dm un h mchnm of how momnumny Concoun c h du momnumny unv comn of h n momnum-ny c nd h nn momnum-ny c nd how h n objc nd nn objc nd h n momnum-ny c nd nn momnum-ny c nc. 769
77 Fu.. Iuon of mod h dncon hown n Fu. mod h dncon duzon.. +=+- /ħ w md h ow v of momnumny Concoun houh mnon. h ound v of momnumny Concoun h = = - = - = - / - = /. Th mod h dncon n Fu. ccomnd by mn of no: n nd nn wv funcon n nd nn objc; ncon fd.. u fd fo ncn wh oh mny n; nd 3 f-cn nd f-fn m w whch ccomny h mnon of h momnumny Concoun h ow v o o nfoc h ccounn nc of convon of zo ud n Fu.. Fu. Pmomnumny Concoun quon Fu.3 how fom noh cv of h onh mon n objc n h n momnum-ny c nn objc n h nn momnum-ny c nd h f-cn nd f-fn m w. ccodn o h Pnc of nc f-ncon f-vy qunum nnmn bwn h n objc n h n momnum-ny c nd h nn objc n h nn momnum-ny c.
77 Fu.3 f-ncon bwn n nd nn objc of qunum ny n du momnum-ny unv hown n Fu. h n objc nd nn objc n h wo momnumny c nc wh ch oh houh vy o qunum nnmn nc vy n c of qunum nnmn.. []. P no h houh n Fu. h body of momnumny Concoun h hown boh h duzd n ny-momnum c nd nn ny-momnum c mbddd n momnumny Concoun. Fu. Incon n h du momnum-ny unv
77 3. hmc of h Pmomnumny od In mhmc w 3. cnfc Gn n Pmomnumny Concoun n uh I ou comhnon h: Concoun=Pmomnumny = Pcm = Omnon Omnn & Omncn n/ = O 3. Pmomnumny Concoun c un nd cu vouon of mod n mny c n momnumny Concoun by f-fn n foow: 3. In on 3. u umb nn h body h of momnumny Concoun mny un nn h mnon of momnumny Concoun ± h conn of mnon = h w of On of momnumny Concoun bfo mzon n w nn w n m w nd nn m w h ffn m w n momnumny Concoun comd of n nd nn m w whch ovn mny n nd conv zo n h du momnum-ny unv n wv funcon n objc nn wv funcon nn objc nd h com wv funcon objc/ny n h du momnum-ny unv who. nvy momnumny Concoun c un nd cu vouon of mod n n h momnumny Concoun by f-fn n foow:
D D D 773 3.3 wh h w of Zo of h momnumny Concoun dfnd by fundmn onh 3. bow D mn dmnn nd nd cvy mc wh nd mn nd nd dmnn. Pmomnumny Concoun n = = - = - = - / - = / bfo mzon. I o n houh f-cn nd f-fn m w f mzon whch c on h n objc nd h nn objc o cu hm o nc wh ch oh n h du momnum-ny unv fuh dcbd bow. 3. f-rfn w nd I moho Th m w of h momnumny Concoun dvd fom h foown fundmn on houh f-fnc whn h on whch ccomn h mnon n n momnumny Concoun: c - - c = = 3. wh nd dynmc vb of m nd oon cvy nd n nnc o m of n mny c.. dfnd Comon wvnh dvdd by d of h =/c. Fo mcy w w c=ħ= houhou h wok un ndcd ohw. Thu w hv fom 3.: - - = = 3. on 3. bd on h on of fou-oon = c n c hoy of vy: c = + c
77 In h nc of fou-on = of cond mod ny quon 3. fo n mny ny wh ch modfd foow: - 3.5 On fom of h m w of h momnumny Concoun dvd houh f-fnc foow: 3.6 wh. n f-nd d of h on n 3.6 uch h D o o fy h fundmn on 3. n h dmnn vw w hv: 3.7 Indd on 3.7 cn o b obnd fom on 3. houh f-fnc foow: D D D 3.8 n on 3.8 by movn dmnn n D w hv: 3.9 f fmonc nzon: D 3. wh = 3 Pu mc: 3 3. on 3.7 bcom:
775 - α 3. wh α = α α α 3 nd β Dc mc. on 3. ovn fmon n du-momnum-ny Unv n Dc fom uch Dc con nd oon nd on 3.7 ovn unnzd o n ny/c wh ch nd nnc o m.. mon o mon-k c n du-momnum-ny unv. oonc nzon of on 3.7 h b dcud. If w dfn: D 3.3 : I D 3. Thu fundmn onh 3. o fd und h dmnn vw of on 3.3. Indd w cn o obn h foown convnon dmnn: D 3.5 On knd of mmoho of on 3.6 3. cvy foow: 3.6 3.7
776 D D D 3.8 3.9 3. D 3. I D 3. on 3.7 h unnzd m w n y-k ch-k fom nd conncd o on 3.7 by Hdmd m H : H H 3.3 on 3. nzd m w n y-k ch-k fom nd conncd o on 3. by Hdmd m: H H 3. noh knd of mmoho of on 3.6-3. cvy foow: 3.5
777 3.6 D D D 3.7 3.8 3.9 D 3.3 I D 3.3 Indd Q qunon nd Q conju. o w cn w on 3.9 : Q Q 3.3 on 3.6 conncd o on 3.7 by uny m H : H H 3.33 my on 3. conncd o on 3.9 by m H: H H 3.3
778 Y noh knd of mmoho of on 3.6 3.7 & 3. cvy foow: 3.35 3.36 α 3.37 If = w hv fom on 3.6-3.: 3.38 3.39 D D 3. 3. f fmonc nzon on 3.39 bcom:
whch ovn m fmon nuno n Dc-k fom. 779 3. f boonc nzon: D I 3 D I 3 on 3.39 bcom: 3.3 3. wh = 3 n oo fo n c: 3 3.5 If w dfn: D 3.6 : D I 3 yz z y y zy z yz z 3.7 To oby fundmn on 3. n dmnn vw 3.6 w h qu h m n 3.7 cn on h n nd nn wv funcon cvy o oduc nu u zo n ouc-f zon dcud. oo h on 3.39 ovn m c wh unobvb n n n h du momnum-ny unv. f boonc nzon h n nd m c n n.
78 noh knd of mmoho of on 3.8-3. whn = cvy foow: D D 3.8 D 3.9 3.5 3.5 3.5 D I 3 yz z y y zy z yz z 3.53 n w h qu h m n on 3.53 cn on n nd nn wv funcon cvy o oduc nu u zo n ouc-f zon n od o fy fundmn on 3. n h dmnn vw 3.5 fuh dcud. Imony f = w hv fom on 3.: 3.5 Thu f momnumny Concoun ow ny- fom of m w w cn dv fo m fom 3.7 nd 3.7 h foown: 3.55
78 3.56 Fuh f = w hv fom on 3.: 3.57 Thu f momnumny Concoun ow momnum fom of m w w cn dv fo m fom 3.7 nd 3.7 h foown: 3.58 3.59 Th nfcnc of h fom of m w h b ucdd. u fo now h h ny- fom of m w ovn n nd nn wv funcon ffd whch y h o of ny- von h hy md nyndndn ncon houh momnum c oon qunum nnmn. On h oh hnd h momnum fom of m w ovn h n nd nn wv funcon f-fd whch y h o of momnum von h hy md momnum ndndn ncon houh nnc-o-m m nnmn. Th bov mmoho of h f-fn m w of momnumny Concoun dvd fom on- mzon f-fnc nd wo- mzon f-fnc bd on h fundmn on 3.. Th f- mzon mk dncon n ny m m nnc o m nd o momnum undffnd c h nvov c un nd mny un n. Thn h cond- mzon mk dncon n h-dmnon momnum h-dmnon c bd on n o oh hh n ucu f hy. 3.3 ddon Fom of w If momnumny Concoun ow dncon whn f- ffn mzon w obn fo m h foown ddon fom of m w : 3.6 3.6
78 3.6 3.63 3.6 3.65 3.66 3.67 3.68 oonc von of on 3.6 nd 3.63 obnd by cn wh. If momnumny Concoun c momnum f-confnmn of n mny ny houh mny oon downwd f-fnc uch h > w hv: D z y 3.69 h : 3.7 Thfo own mny oon downwd f-fnc fo n mny ny w cn dv h foown m w n Dc-k fom: 3.7 3.7 o w cn dv h foown m w n y-k ch-k fom:
783 3.73 3.7 oonc von of on 3.7 nd 3.7 obnd by cn wh. I ky h h bov ddon fom of f-fn m w ovn dffn c of h c zoo n h du momnum-ny unv dcud. 3. cnfc Gn of Pmod n n h Pmomnumny od Thfo momnumny Concoun c un nd cu vouon of f n-wv fmon uch n con n Dc-k fom n du momnum-ny Unv foow: 3.75 h : o 3.76 wh ubuon nd hv bn md o h comonn of cn
78 c on n nd nn wv funcon. quon 3.76 o h f hc wv ouon n h du momnum-ny unv n h fom: 3.77 nvy momnumny Concoun c un nd cu vouon of f n-wv fmon uch h con n Dc-k fom n h du momnum-ny unv foow: 3.78 D D D Pmomnumny Concoun c un nd cu vouon of f n-wv nfmon uch oon n Dc-k fom n h du momnum-ny unv foow:. 3.79
785 o 3.8 D D D _ my momnumny Concoun c un nd cu vouon of f n-wv fmon n y-k ch-k fom n h du momnum-ny unv foow: 3.8
786 h : o 3.8 nvy momnumny Concoun c un nd cu vouon of f n-wv fmon n y-k ch-k fom foow: D D D 3.83 Pmomnumny Concoun c un nd cu vouon of f n-wv fmon n noh fom n h du momnum-ny unv foow: 3.8 Q Q Q Q h :
787 o 3.85 nvy momnumny Concoun c un nd cu vouon of f n-wv fmon n noh fom n h du momnum-ny unv foow: D D D Q Q 3.86 Pmomnumny Concoun c un nd cu vouon of n n-wv hoon n h du momnum-ny unv foow: 3.87
788 hoon - nvy momnumny Concoun c un nd cu vouon of h n n-wv hoon n h du momnum-ny unv foow: h D D 3.88 hoon - Th hoon wv funcon n h du momnum-ny unv cn b wn : k k k hoon 3.89 f h ubuon nd w hv fom h on n 3.87: 3.9 wh w hv ud h onh o dv h quon whch oh wh nd h w-k quon n h ouc-f vcuum n h du momnum-ny unv. Pmomnumny Concoun c nuno n Dc-k fom n h du
789 momnum-ny unv by cn h of on 3.87 wh h foown: 3.9 Pmomnumny Concoun c un nd cu vouon of n n-wv nhoon n h du momnum-ny unv foow: 3.9 nhoon Th nhoon wv funcon cn o b wn : k k k nhoon 3.93 Pmomnumny Concoun c n nnuno n Dc-k fom n h du momnum-ny unv fom by cn h of on 3.9 wh h foown: 3.9 Pmomnumny Concoun c un nd cu vouon of ch-k n-wv hoon n h du momnum-ny unv foow:
79 D D 3.95 h nd dcoud fom ch oh nd fy h foown quon cvy: o 3.96 whch hv h foown cv ouon: k k 3.97 oh nd oduc h w-k quon n h ouc-f vcuum hown n h cond on of 3.9. Pmomnumny Concoun c nuno n y-k ch-k fom n h du momnum-ny unv by cn h of on 3.95 wh h foown: 3.98 h nd dcoud fom ch oh nd fy h foown quon cvy: o 3.99
79 Pmomnumny Concoun c nd un ny- n nd nn wv funcon ny- von of n nnc o m n Dc-k fom foow: 3. D D D D D w dmn h fom of mny conn n on 3.. nvy momnumny Concoun c nd un ny- n nd nn wv funcon ny- von of n nnc o m n Dc-k fom foow: D D D D D D 3. my momnumny Concoun c nd un ny- n nd nn wv funcon ny- von of n nnc o m n y-k ch-k fom foow:
79 3. n w w dmn h fom of h mny conn n on 3.. nvy momnumny Concoun c nd un ny- n nd nn wv funcon ny- von of n nnc o m n y-k ch-k fom foow: w D D 3.3 Pmomnumny Concoun c nd un momnum- momnum ndndn n nd nn wv funcon of n nnc o m n Dc-k fom foow: m D m D m D m D m m m m D D 3. my momnumny Concoun c nd un momnum- momnum ndndn n nd nn wv funcon of n nnc o m
793 n y-k ch-k fom foow: m m m m m m m m 3.5 m m nvy momnumny Concoun c nd un momnum- momnum ndndn n nd nn wv funcon of n nnc o m n y-k ch-k fom foow: m w m m m m m m m D D 3.6 Pmomnumny Concoun c un nd cu vouon of momnumy f-confnd ny uch oon n h du momnum-ny unv houh mny oon downwd f-fnc uch h > n Dc-k fom foow: 3.7
79 3.8 f nzon of on 3.8 w hv: 3.9 dcud ub h on 3.8 ovn h confnmn ucu of h unnzd oon n Dc-k fom houh mny oon nd on h oh hnd on 3.9 ovn h confnmn ucu of nzd oon houh. nvy momnumny Concoun c un nd cu vouon of h momnum-y f-confnd ny uch oon n h du momnum-ny unv n Dc-k fom foow: 3. D D D D D D D D D Thu n unnzd nd nzd noon n Dc-k fom my b cvy ovnd foow: D D D 3.
795 D D D 3. my momnumny Concoun c un nd cu vouon of momnumy f-confnd ny uch oon n h du momnum-ny unv houh mny oon downwd f-fnc n y-k ch-k fom foow: 3.3 3. f nzon of on 3. w hv: 3.5 I ub h on 3. ovn h ucu of h unnzd oon n y-k fom nd on 3.5 ovn h ucu of nzd oon n y-k fom. nvy momnumny Concoun c un nd cu vouon of y f-confnd ny uch oon n h du momnum-ny unv n y ch fom foow: 3.6
796 D D D 3.7 3.8 Thu n unnzd nd nzd noon n y-k fom my b cvy ovnd foow: 3.9 3. 3. cnfc Gn of Como n n h Pmomnumny od Pmomnumny Concoun my c un nd cu vouon of nuon n h du momnum-ny unv n Dc-k fom whch comd of n unnzd oon: 3. nd nzd con: 3. foow:
797 n 3.3 In on 3. 3. nd 3.3 nd n ndc oon con nd nuon cvy. Fuh unnzd oon h ch con h ch.. nd.. h comnc on cn on unnzd oon nd hy bound nzd con cvy nd bndn on fom h unnzd oon cn on h nzd con cun h bndn dcud. If.. nb du o h f moon of h hy bound nzd con n h du momnum-ny unv w hv fom h on n 3.3: n 3.
798 mn d on ch dbuon nd -fco of nuon my uo nuon comn of n unnzd oon nd hy bound nzd con. Th y-k ch-k fom of h on n 3.3 nd on 3. cvy foow: n 3.5 n 3.6 Thn momnumny Concoun my c un nd cu vouon of hydon om n h du momnum-ny unv comn of nzd oon: 3.7 nd nzd con: 3.8 n Dc-k fom foow:
799 h 3.9 In on 3.7 3.8 nd 3.9 nd h ndc oon con nd hydon om cvy. n oon h ch con h ch nd.. nd.. h comnc on cn on nzd oon nd nzd con cvy. n f.. nb du o f moon of h obn nzd con w hv fom h on n 3.9: h.3 Th y-k ch-k fom of h on n 3.9 nd on 3.3 cvy foow:
8 h 3.3 h 3.3. mohou Pmomnumny Concoun w. moho & h nc of n n h Pmomnumny od Th cdn con mk c h h c of momnumny Concoun cn k mny dffn fom dffn mod n nd fuh cn hv dffn mnfon dffn wv funcon nd/o fd n dffn con vn n mod ny. Fo m h wv funcon of n con cn k h Dc-k y-k qunon-k o dmnn fom cvy n dffn con n h du momnum-ny unv dndn on h quon on k nd h nw on k. Th wok o mk c h mod f-fn n n momnumny Concoun hchc nd h cu of mod dncon fo cn h f-fn n n h du momnum-ny unv. Th v v of n: n n h ow v n momnumny Concoun mkn mod n nd nn h dncon of n nd nn wv funcon; n of h momnumny Concoun on h ound v mkn mod n nd nn wv funcon whch ccomn h mod h dncon; 3 f-fn mn of h wv funcon houh m w bfo nzon; unconfnn n houh nzon comnc nd wk ncon fo cn boonc nd fmonc n; nd 5 confnn n on ncon cn h nc of quk houh mny oon downwd f-fnc.
8. Th Dmnn w & h nn of Kn-Godon-k quon n h Pmomnumny od In h dmnn vw h m w co no Kn-Godon-k fom hown n 3 bu o f w hv no dfnd h fom of h wv funcon u of h d co. H w oo h h n nd nn wv funcon objc fom c oduc wh connn h hddn vb qunum on o f-vy hown bow vc v. Fom h foown quon fo unnzd f c n Dc-k nd y-k fom cvy: D. nd. w cvy obnd h foown quon n h dmnn vw Kn-Godonk fom: D.3 nd D. y wy of n m quon. h h foown n-wv ouon:.5 fom whch w hv:.6
8 wh.7 cvy h n nd nn h n h dmnn vw. Th vb n y h o of hddn vb o whch woud b nnhd f w owd o md wh. Indd f vc m n h n wv funcon condd o b n m hn h vc m n h conju nn wv funcon y h o of von m. w dcu qunum on. my fom h foown quon fo nzd f fmon n Dc-k nd yk fom cvy:.8 nd.9 wh ψ D=ψ + ψ - T =ψ ψ ψ 3 ψ T nd ψ =ψ ψ T =ϕ ϕ ϕ 3 ϕ T w cvy obnd h foown quon n h dmnn vw Kn-Godon-k fom: 3 I D. nd 3 I D. In h nc of comnc on n h du momnum-ny unv w hv fom quon. nd. h foown quon:
83 D - -. nd - -.3 fom whch w cvy obnd h foown quon n h dmnn vw Kn-Godon-k fom: m m m D - - -. nd D - - -.5 wh nd -. f nzon of quon. nd.3 w hv: D - -.6 nd - -.7 fom whch w cvy obnd h foown quon n h dmnn vw Kn-Godon-k fom: I - I - I - D.8 nd
8 I - - I - - I - - D.9 In quon.6 nd.7 h coun of nd/o wh n h mc o hddn. Th ncon du o f-fn m w whch cu mn of h n nd nn wv funcon. Howv n h dmnn vw h ncon md c hown n quon.8 nd.9 cvy..3 Th nn of chodn-k quon & Qunum Pon n h Pmomnumny od I cn b hown h h foown chodn-k quon h non-vc omon of quon.3 o.: ˆ c T o ˆ T c=ħ=. wh R Im. Fom c = + c w hv:... 8 c c c c Choon + omn cond m on h h nd mkn ubuon T nd w v quon.. quon. cn b wn wo coud quon c=ħ=: R Im Im R ˆ ˆ T T o ˆ ˆ Im R T T. Th bov quon dcb h non-vc f-fnc of h wv comonn R nd Im du o n. If w dn R n objc Im h nn objc. I h non-vc omon of h dmnn vw of n unnzd c Kn-Godon-k fom n momnum-ny c wh f-fn
ncon ducd o n nd connd n h wv funcon fom whch h qunum on Q cn b cd. 85 Fo m n h c: wh + nd - ζ conn h hddn vb nd: w cn dv h foown qunum on d w b vn wh: whch on fom n n: Q..3..5 Q woud n h non-vc knc m = / of h n wv funcon f h n wv funcon nd h conju nn wv funcon woud m. Fuh cn b hown h h Pu-k quon h non-vc omon of quon.8 whch h dmnn vw of fmon n n comnc fd n Dc-k fom whn h momnum-ny c:. I conn non-vc f-fnc du o boh n nd nd w b d wh n d whn nd f m m.. Th Thd of n h Pmomnumny od Tdony c n c umd o b dcbd by h Kn-Godon quon nd cfd boon. Howv n h wok w hv ud h Kn- Godon-k quon dmnn vw of fmon boon o n unnzd ny non n whch h n nd nn wv funcon objc fom c
86 oduc wh h on of hddn vb qunum on o fvy. Th unnzd ny non nh boon no fmon bu my b cfd hd of m dcbd by h unnzd quon n Dc-k o y-k ch-k fom n h du momnum-ny unv fo m:.5.6 Th hdonzd von of h bov quon n whch h oon mny cvy foow:.7.8 Th hd of m my no b ubjc o h c bhvo of h boon o fmon. Th wv funcon of fmon nd boon cvy bno nd bvco bu h of h hd non wo-comonn com c fd. Th hd of m h cuo of boh fmonc nd boonc m/fd bfo fmonc o boonc nzon. Thu my no h ho yd by h H fd of h ndd mod. Fuh n h cno nnc o m cd by h ffn n mnon of momnumny Concoun. 5. k Incon n h Pmomnumny od In h mod wk ncon n v oc mon o don houh fmonc nzon wh o whou nmdy boonc nzon nd h ocd v oc cu o boon. Th wo ob knd of mchnm y. On knd h dc fmonc nzon of n unnzd mv c hown n 3: D 5. h fo m: 5.
nd h foown v oc: D 5.3 h fo m: 5.. Poc 5. nd 5.3 ony conv n n h du unv who. If hy hod n y nuno my no b ndd n h wk ncon n h mod. ccodny b dcy of nuon my nvov h nzn oc 5. dun whch n unnzd oon o con n n / nd bound nzd con bcom f foow: n Poon nzd Poon R of ound con; o n con nzd con R of nzd con. Poc m n co mn wh mn d on -fco nd ch dny of nuon. Th no chn c nvovd n oc o. In nuon ynh fom oon nd con f h v oc 5.3 occu n h mod dun whch nzd oon o con o n nd f con bcom hy bound o oon. u h h foown quon ovn f unnzd c hvn nnc o m nd ch cvy bu n h hy on-k c h combnon n -k c: 87 o f nzon houh 5. w v Dc-k quon: 5.5 umn n wv ouon fo d quon -k n-wv ouon: o 5.6 fo quon 5.5 w obn h foown
88 5.7 wh nomzon fco nd w hv uzd h foown on fo m n: 5.8 f nzon of ouon 5.7: z y y z 5.9 w v h f n-wv con ouon fo Dc-k quon 5.6 n h du unv comd of h n momnum-ny c nd h nn momnumny c: y z nd z y 5. In h bov ouon fo n n u nd down cvy h n n / bncd by h nn n comonn whch my b dmd nnuno uch h h o n n h du unv convd o zo. Thfo n h mod my b h n n u o down cn b cd whou h nd of nnuno n b dcy f ny cv m nd/o oon owd o cnc ch oh n momnumny Concoun: Δ Δ Δ Δ Δ Δ 5.
89 Fuh f momnumny Concoun ow h foown boonc nzon of mv n c.. unb c wh vy ho f-m: 3 3 I D I D 5. h fo m: 5.3 nd/o 3 3 I D I D 5. h fo m: 5.5 dun whch noy my o vco boon - + nd/o Z nd d w hv fom on 5. h cond knd of wk ncon. on ou h h ony oc 5. md wk ncon nc n oc 5. vcoboon-k c ju noy h do no dcy no fmon. Th nzd quon n on 5.3 fo f mv n c my k h foown Dc-k fom: 5.6 o _ 5.7 f ccun h dmnn: D 5.8
obn h foown: 8 D I 3 T I 3 yz z y y zy z yz z 5.9 mnond n 3 h m T n on 5.9 mk fundmn on no o hod n h dmnn vw 5.8 un h con of T on h n nd nn comonn of h wv funcon oduc nu u h : T y y 5. z z nd T y y 5. z z Thu f momnumny Concoun ow h voon o noy quon 5.6 nd 5.7 my dcb f vco boon - nd + n h du momnum-ny unv cvy; h combnon hn dcb f vco boon Z nd T my b dmd noy nnc o m o nnc o m oo. In con o oc nd vco boon - nd + o h k md h nzon of n oon o con cvy foow: 3 n Poon nzd co oon + nzd Poon + nzd nd Fmon R of ound con + nzd nd Fmon; nd n con nzd co oon - nzd con + nzd nd Fmon R of nzd con + nzd nd Fmon.
I hod h h mmohou fom of m mw n 3 h fuh mmoho nd h coondn wv funcon h h w ovn w b b o ccommod known c of h c zoo n h du momnum-ny unv. y mony n h mod h my b no y voon n wk ncon uch b dcy h n y voon n h mn my my b nd n ozon ffc n whch h n ozon nfunc h dynmc nd dcon of h md con n n n mnc fd. o h my b no nd fo H boon o n m nc m nd by f-fn n h ow v of momnumny Concoun o h mod c of momnumny Concoun my =. 8 6. comnc Incon n h Pmomnumny od comnc ncon n v oc don o mon houh boonc nzon of n nnc-o-m- m nd n ny nd h ocd v oc boon. In h mod h oby wo knd of mchnm y. On knd h dc boonc nzon nzn don: h fo m: D I3 D I 6. 3 6.. nd h foown v oc unnzn boon: h fo m: I 3 D I 3 D 6.3 6. Th don o boon of hoon dun ccon of chd c my b dc boonc nzn o unnzn oc cvy: ound n & Innc-o-m- Pc ound nzd Phoon F nzd Phoon; nd F nzd Phoon ound nzd Phoon ound n & & Innco-m- Pc. In h mod h wo oc my o occu n nuc dcy nd h n oh
8 oc. umn n wv fo h n nd m c: o 6.5 w obn h foown ouon fo h quon: 6.6 wh w hv uzd h foown on fo m n nd h nomzon fco : 6.7 f nzon: y z y z 6.8 v h n-wv ouon: y z y z _ z y 6.9 fo h nzd hoon quon: o 6. Th cond knd of comnc ncon h don o bndn boon of nzd hoon whou unnzon:
83 3 ound nzd Phoon F nzd Phoon; nd F nzd Phoon ound nzd Phoon. Poc 3 nd occu h onn of n oc cvy o wvud nd my o occu n omc hoon con nd mon nd h oh oc. Fo boonc nzon h w-k quon n h vcuum c= n h du momnum-ny unv foow: o 6. If w ccu h dmnn: D 6. obn h foown: T I z zy z yz y yz z y I D 3 3 6.3 Th m T n on 6.3 mk fundmn on no hod n h dmnn vw 6. un h con of T on h n nd nn comonn of h wv funcon oduc nu u nc quon 5. nd 5. ony hod n h ouc-f on of h du momnum-ny unv. h ocon of mv.. nnc o m non-zo chd c uch
8 n con o oon quon 5. nd 5. o vod by h hoon. Th h hoon o hv nnc o m T h ouc hu n h mod c my b cd on coon of hoon wh mv chd c. In h w-k quon h voon coun-bncd by ddn ouc o h quon dcud bow. Th w-k quon wh ouc n un coud o h Dc-k quon of h fmon uch con o oon fomn h Dc- w-k ym fuh dcud n. Indd f ouc j j n h du momnum-ny unv w hv nd: j j o j 6. Imony w cn o choo o u fmonc nzon chm o dcb w-k quon n h mod. In h c h w-k quon n h vcuum h h fom: - - 6.5 whch v: 6.6
85 If ouc j j whch v: - w hv: - j j 6.7 6.8 Thfo n h fmonc nzon chm w hv n c of h b-vco wv funcon no comn of wo b-no nd of h b-vco f nd by ojcn h b-vco comd of nd o n. Fuh w on ou h h fo n hoon cc fd h n wv funcon n objc nd mnc fd h nn wv funcon nn objc n h mod. Th wo fd wy f-nnd nd h nnmn h f-vy. Thfo h on bwn nd n on comnc wv n h momnum-ny unv no h chn n nduc vc v bu h chn n wy ccomnd by chn n ynchonouy vc v du o h nnmn f-vy. Th h onh bwn nd von nd nnnou. 7. on Incon n h Pmomnumny od h wk nd comnc ncon v oc nvovn fmonc nd boonc nzon of n n h hd of m nd h cv v oc on ncon n h mod do no nvov nzon h on foc confnn oc. I my b umd h n n n n unb nd dcy houh fmonc o boonc nzon. In od o chv confnmn of nucon o by of h nucu w u h on ncon my nvov mny oon n h confnmn zon n h du momnum-ny unv ud bow. Th wo y of on ncon y. On h fconfnmn of nucon uch oon nd h oh h ncon mon nucon uch oon nd nuon. In h ndd od oon como ny comd of h quk confnd by m uon nd h ncon mon h nucon mdd by mon comd of of quk nd n nquk whch n un nc houh uon. Howv nc no f quk hv bn obvd h my b ood on o cond oh oon. hv ud n 3 h h oon my b condd n
86 mny c h ccomh momnum f-confnmn houh downwd f-fnc mny oon. H w w f dv h condon fo oducn momnum f-confnmn of h nucon n h du momnum-ny unv nd h Yukw-k on. Th quon fo mv bu n ny n Dc-k Fom foow: o 7. umn h h wv funcon h m n - h h n nd nn wv funcon hv m n - nd + cvy n h dmnn vw w cn w: 7. 7.3 Fom on 7. nd 7.3 w cn dv h foown: o 7. quon 7. h d ouon foow: 7.5 Thn w hv fom on 7.3: 7.6 wh w my uz h foown conjcu: 7.7 Th com d ouon of quon 7. fo m n - n Dc-k fom : 7.8 wh nomzon fco. hn > h whn h oon n - = mny w hv fom 7.8:
87 7.9 wh. ow f w cond h c c of n ny- n bu mv ny n whch = w hv fom 7.9: 7. Thu h nn nd n wv funcon n on 7. hv h fom of Yukw-k on nd nv mny ojcon cvy. u h h no confnmn zon of n unnzd nucon dcbd by wv funcon m o on 7.9 o 7. n h du momnum-ny unv nd confnmn chvd houh downwd f-fnc mny oon. Thfo n h cno h h coo of h on foc h hdmnon mny oon. Fuh noh mcon of h cno h n h chn qunum unv h ny- d o h oud of h du momnum unv whch mbddd n momnumny Concoun conncd o o my h ny- nd of h nucon. If w um h h nn wv funcon ψ whch f-coud o h n wv funcon ψ houh on 7. o cou wh h n wv funcon χ of noh ny whch o f-coud o nn wv funcon χ fo m: 7. wh coun conn w cn w of h nuc on of nucon foow: 7. whch n h fom of Yukw-k Pon. now dcu h unnzd nd nzd fom of oon. Th nzd oon n h
88 du momnum-ny unv h commony known fom of oon nd w u h h unnzd oon my d n h nuon comd of h unnzd oon nd nzd con ud n 3. Th quon fo f unnzd nd nzd oon n Dc Fom cvy foow: 7.3 nd 7. wh mny oon. Fom h bov dvon w my w h wv funcon of n unnzd oon wh n nd nn m n nd + cvy foow by convnon con h ov n m + nd nn m : 7.5 In con n unnzd noon wh n nd nn m n + nd - cvy my hv h foown wv funcon: 7.6 ccodn o h cno h nuc n of h nuon n h du momnum-ny unv oy du o h hy bound nzd con. Indd mn d on ch dbuon nd -fco of nuon uo h cno. fuh u h h nuc on cun h bndn of h nzd con n h nuon my hv h fom of on 7.. Dd condon w b vn wh. Th wv funcon of nzd oon dcbd by quon 7. cn b obnd by nzn h ouon n on 7.5 foow: D 7.7 / / I j I j wh j h o nu momnum numb. Choon j=/ w obn fom on
89 7.5 wo of ouon foow: / / 7.8 / / 7.9 wh. In h c of ny- oon h whn = w hv fom on 7.8 nd 7.9 h foown: 7.8 7.9 In h cno nzon of unnzd oon my cu o of h bndn of nzd con o unnzd oon h ob cu of whch w b condd wh. 8. Gvy Qunum nnmn n h Pmomnumny od Gvy n h du momnum-ny unv qunum nnmn nnnou ncon co h du momnum-ny wod. Th wo y of vy y. On f-vy f-ncon bwn h n objc n wv funcon nd nn objc nn wv funcon of n ny wv funcon ovnd by h mmohou m w dcbd n h wok nd h oh h qunum nnmn nnnou ncon bwn wo n o on ny nd h du-
wod who. fuh hown bow von fd von ju h wv funcon f whch h nny dbuon nd dynmc of f-qunumnnmn nonocy of n ny. Indd on ncon n h du momnumny unv my b on qunum nnmn on vy. focu h on h cu fom of von fd. On ny- zo m n nd nn wv funcon f-fd h y h o of ny- von h hy md ny-ndndn ncon houh momnum qunum nnmn. Th cond momnum- n nd nn wv funcon f-fd h y h o of momnum- von h hy md momnum ndndn ncon houh m nnc o m nnmn. Th hd o-m n nd nn wv funcon f-fd h y h o of o-m m von h hy md nnc-o-m -m ndndn ncon houh m ny nnmn. Th yc wv funcon f-fd n h du momnum-ny unv conn h ny- momnum- nd o-m comonn. In ddon h yc wv funcon o conn comonn d o fmonc o boonc nzon. hown bow ny- qunum nnmn bwn wo n ccoun fo wonn-k vy. omnum- nd/o o-m qunum nnmn bwn wo n my ccoun fo dk m. Imony von comonn d o nzon my ccoun fo dk ny. hn = w hv fom fundmn onh 3.: o 8. cn d on 8. onh ovnn h chn-k qunum unv n whch h o m zo. Th my b n : h nnc o m bn comd of mny oon = o oon bn comd of mny nnc o m =. 8 hown n 3 h ny- m w n Dc-k nd y-k fom cvy h foown: 8. 8.3 Thu h quon of h m wv funcon f-fd cvy foow:
8 D D D D D 8. nd 8.5 quon 8. nd 8.5 cn b cvy wn : D D D D o D D D D 8.6 nd o 8.7 To h coun of n nd nn wv funcon f-fd n dffn cv w cn w 8.6 nd 8.7 cvy foow: D D D D D D D D 8.8 nd 8.9 Fom on 8.6 w cn dv h foown: D o D 8. quon 8. h d ouon n h fom of Yukw on: D 8. o n on 8. =- h oon comd of mny nnc o m. Th n ny- f-fd n on 8. h h fom of won-k
8 von o Couomb-k cc on momnum. hv fom on 8.6: D D 8. wh w hv uzd h foown conjcu: D 8.3 Th com d ouon of quon 8. hn: D D D 8. wh nomzon fco. Indd on 8.7 cn hv m d ouon on 8.6: 8.5 If w um h h nn f-fd D whch f-coud o n f-fd D houh on 8. o 8.8 o cou houh ny- qunum nnmn wh h n wv funcon ψ of noh ny of nnc-o- m whch o f-coud o nn wv funcon ψ fo m: D G 8.6 wh κ coun conn nd G=κ/π won-k Gvon Conn w hv von-k on momnum : G 8.7 hn = w hv fom fundmn onh 3.: 8.8 cn d on 8.6 onh ovnn mommum- qunum unv. Ccy h my b n h nnc-o-m bn comd of ny-oon m. hown n 3 h mommum- w n Dc nd y fom cvy h foown:
83 8.9 nd 8. nd h quon of mommum- wv funcon f- fd cvy h foow: D D D D D 8. nd 8. Th n nd nn momnum- wv funcon D nd D n quon 8. dcoud fom ch oh bu ho n quon 8. nd coud o ch oh: D D D D bu 8.3 I cn b y vfd h h ouon o quon 8. n fom of: D D D 8. o D D D 8.5 bu h ouon o quon 8. n h fom of: 8.6 o 8.7
8 h u bow momnum- qunum nnmn vy bwn wo n. Fo mcy w w cond wo nnc-o-m + nd cvy ocd momnum on nd. Th cv momnum- wv funcon my b b wn n y-k fom foow: nd 8.8 whch fom oduc. f v + n md c nd bobd by - w my hv h foown wo ddon momnum- wv funcon n y-k fom: nd 8.9 whch fom oduc. Th fn momnum- qunum my b wn foow: 8.3 In h jon momnum- wvfuncon nd qunum nnd du o ncon wh nd houh. hn = w hv fom fundmn onh 3.: 8.3 cn d on 8. onh ovnn h nnc-o-m- qunum unv n whch h o nnc o m m zo. Ccy h my b n m bn comd of oon. hown n 3 h nnco-m- m w n Dc-k nd y-k fom cvy h foown: 8.3 nd 8.33
85 nd h quon of nnc-o-m- wv funcon f-fd cvy h foown: D D D D D 8.3 nd 8.35 quon 8.3 nd 8.35 hv n-wv ouon. Th n nd nn m wv funcon D nd D n quon 8.3 coud wh ch oh bu ho n quon 8.35 nd dcoud fom ch oh: D D D D bu 8.36 Fo n of nd h ouon o quon 8.3 n h fom of: k k k D D D 8.37 o k k k D D D 8.38 bu h ouon o quon 8.35 n h fom of: k k k w 8.39 o k k k 8. quon 8.3 nd 8.35 dcb h f-ncon of n nd nn nnco-m- nd n wv funcon f-fd. my bud qunumnnd of wo nnc-o-m- nd n n m o h of wo momnum- n.
86 9. Humn Concoun n h Pmomnumny od now bfy dcu humn concoun n h momnumny mod. Dd mn w b vn n fohcomn c. Ou mn u on qunum nnmn of h bn wh n ubnc.. Rf n [] u h Concoun no ocd n h bn bu ocd wh cm/ momnumny Concoun o my cm/momnumny Concoun. Thu Concoun momnumny Concoun h boh ncndn nd mmnn o. Th ncndn c of Concoun momnumny Concoun h on of mod f-fn n ncudn h f-fn m w nd ojc h n nd nn objc wvfuncon n h du unv houh n nd n un h mmnn c of Concoun momnumny Concoun obv h n objc wvfuncon n h n momnumny c houh h nn objc wvfuncon n h nn momnum-ny c. Humn concoun n h du momnum-ny unv md nd cu von of h du-c Concoun momnumny Concoun uch h w hv md f w nd md obvon. Fu 9. Incon bwn n objc nd h bn body n h du unv comd of h n momnum-ny c nd h nn momnum-ny c
87 ud n Fu 9. h wo knd of ncon bwn n objc ny oud h bn body nd h bn body n h momnumny mod. Th f knd h dc hyc nd/o chmc ncon uch noy nu houh h y. Th cond nd -known bu mny ovn o b u knd h nnnou ncon houh qunum nnmn. Th n unv oud ou bn body ocd wh ou bn body houh qunum nnmn hu nfuncn nd/o nn no ony ou fn moon nd dm bu o h hyc chmc nd hyooc of ou bn nd body. In h momnumny mod w my w h foown Hodkn-Huy-k quon n h n/nn momnum-ny c: m m m C..... 9. wh m h cc on co h nu mmbn C m h ccnc of h mmbn h h vo-d o conn-k on chnn. cococy n h du unv comd of h n momnum-ny c nd h nn momnum-ny c comnc fd nd o h fou-on : 9. nc wh oon of ch nd und con of ch cvy h foown Dc-w-k ym: 9.3 - - j α 9. nd 9.5
88 - - j α 9.6 wh β nd α Dc mc nd j j fou-cun n n/nn momnum-ny c. In quon 9.3 nd 9.5 h ncon coun of nd/o wh oon nd/o con n oo nd hddn. Th d ncon du o h f-fn m w whch cu mn of h n nd nn wv funcon nd cn b md c n h dmnn vw foow. Fo Dc-k fom w hv: 9.7 I I Fo y-k ch-k fom w hv: 9.8 I I - Th wo coun w o b cy hown dun h oc of non-vc omon of h Dc-k quon n h n of n comnc on μ. cn cy ou h m ocdu fo n con o how h c coun of wh nd. On ffc of h coun h h con on houh nd o μ nu nfomon no h mnd- n h bn. noh ob ffc of h coun h hy ow h ncndn c of concoun houh wv funcon h f-fd of h oon nd/o con o bck-nfunc nd o μ whch n un bck-c on h con on houh h Hodkn-Huy-k nu ccu n h bn.
89. om Quon & nw. Do h uncny nc nd commuon on mon ny momnum m nd oon hod n h momnumny mod? Y. Howv n h mod m nd oon of n mny c qunzd dynmc vb nd ny nd momnum connuou m. In con n h cm mod [-] m nd oon connuou m nd ny nd momnum qunzd dynmc vb.. How cm mod nd momnumny mod conncd o ch oh? Th mny c n cm mod nfomd no h n momnumny mod houh qunum jum vc v dmond n foh comn c. 3. h h foundon of h du momnum-ny unv? Th foundon momnumny Concoun whch omnon omncn nd omnn.. h omhn bfo h du momnum-ny unv w bon f h w uch bh? Y momnumny Concoun on = whou dffnon o duzon. o my b d h = h mod c. 5. How do momnumny Concoun c un nd cu vouon of h du momnum-ny unv nd n n? Pmomnumny Concoun do h hn by hchc f-fn n of f f w. 6. hy h my omhn nd of nohn? Pmomnumny Concoun nd nd o c un nd mk vouon of dffn n. 7. How do momnumny Concoun ovn h du momnum-ny unv? Pmomnumny Concoun ovn houh mmohou ffn m w. 8. h m n h momnumny mod? duzd ny cd houh hchc f-fn n of momnumny Concoun comd of n n wv funcon n objc hvn ov m by convnon nd n nn wv funcon nn objc hvn nv m by convnon. 9. h nm n h momnumny mod? nm duzd ny cd houh hchc f-fn n of momnumny Concoun comd of n n wv funcon n objc hvn
nv m by convnon nd n nn wv funcon nn objc hvn ov m by convnon.. I m convd n h momnumny mod? Y m convd o zo ccodn o h ccounn nc of zo.. I m convd n h n nn momnum-ny c? Th nw dnd on h con. In mo nu oc n nn m convd nd nfomd no dffn fom whou o du o cncon bwn h n nd nn c. Howv n om oc cy ho nvovn humn concoun nd/o nnon f w m convon n h n nn momnum-ny c my b hy vod o h h f w my funcon.. h qunum nnmn n h momnumny mod? I h ncon nd/o conncon bwn h n nd nn wv funcon objc of n duzd ny o mon dffn duzd n houh momnumny Concoun whch oud momnum-ny. 3. h f-ncon f-vy o f-qunum nnmn n h momnumny mod? f-ncon h ncon bwn h n nd nn wv funcon objc ccodn o h momnumny Concoun quon ovnd by h f-fn m w.. h on foc n h momnumny mod? I downwd f-fnc houh mny oon. I on vy on qunum nnmn. 5. h wk foc n h momnumny mod? I fmonc nzon nd unnzon of n n wh o whou boonc nmdy nzon. 6. h comnc foc n h momnumny mod? I boonc nzon nd unnzon of nnc-o-m- m nd n ny. 7. h vy n h momnumny mod? I qunum nnmn co h du momnum-ny unv whch ncud f-vy o f-qunumnnmn bwn h n nd nn wv funcon objc of n duzd ny nd vy o qunum nnmn mon dffn n. 8. h h on of h qunum ffc n h momnumny mod? Th on mod hchc f-fn n of momnumny Concoun. 9. h nfomon n h momnumny mod? I dncon h qunv o quv ncd o cvd by cu concoun. 83
. h qunum nfomon n h momnumny mod? I dncon o of dncon h qunv o quv ncd o cvd by cu concoun whch du o qunum ffc uch qunum nnmn.. h h mnn of mny un n h momnumny mod? I h mo mny f-fn oc. mnon of momnumny Concoun mk h dncon of n mny ny nd n mn n h m w y cuc o n f-fn mn con of momnumny Concoun.. h Concoun? Concoun momnumny Concoun whch omnon omncn nd omnn. 3. h humn concoun? I md o ndvduzd Concoun ocd wh cu humn bn/body.. Do humn concoun d n humn bn? o h humn bn h nfc fo humn concoun o nc nd nc wh h n unv. 5. h ou nd/o mnd? Thy dffn c o o of momnumny Concoun whch ncndn mmnn nd n. 6. h dd w com fom? Phycy/boocy w cm fom momnumny Concoun con. uy w n nb of momnumny Concoun nd ou concoun md nd/o ndvduzd von of unmd Concoun. 7. h w on? Phycy/boocy w dn o d un w dvnc ou cnc o h on wh dh of ou booc body bcom choc no unvodby. o w of h onon h dvncmn n cnc w vnuy nb u o nf o v ou ndvdu concoun ocd wh ou n o dd bod o noh booc o fc ho. uy w my o bck o momnumny Concoun o ncn no dffn fom of ndvdu concoun h my b b o c. 8. How do h mnd nfunc h bn? nd nfunc h bn houh f w whch c on ubjcv n nn objc whch n un ffc objcv n n objc houh h momnumny Concoun quon. 9. h h on of h uncny nc? Th on f-fn n o zbwun. 83
3. h h on of qunum jum o wv co? Th f w of momnumny Concoun o unmd ncndn Concoun. Rmmb h ou md f w of h unmd f w of momnumny Concoun nc w of momnumny Concoun. 3. I nfomon convd? I ou onon h nfomon convd o zo n h du unv nc ch dncon n h n c ccomnd by non n h nn c. Howv nfomon no convd n ch c on. 3. h von? Th no von n h n of qunum c whch mdd von ncon h d of h. Howv nc vy qunum nnmn h wv funcon of ch ny my b d von. 33. I h n bou fnc fm? Y my cm/ momnumny Concoun. 83. Concuon Th c connuon of h Pnc of nc. momnumny mod of mny c fou foc nd humn concoun fomud whch u how h f-fn hchc n ucu of h momnumny Concoun ovd foundon fo cn unn nd cun vouon of mny c houh mn oc mbddd n d momnumny Concoun. Th mod n mny c nd h ovnn m w fo du unv qunum fm comd of n n momnum-ny c nd n nn momnum-ny c. In con h cm mod dcbd vouy n mny c nd h ovnn m w fo du unv qunum fm comd of n n cm nd n nn cm. Th qunum fm nd h mmoho nconncd houh qunum jum dmond n fohcomn c. Th momnumny mod v h con unnc nd vouon of fmon boon nd n n ch of whch comd of n n wv funcon o n objc n h n momnum-ny c nd n nn wv funcon o nn objc n h nn momnum-ny c. Th mod ovd unfd cu ucu n d du unv qunum fm fo wk ncon on ncon comnc ncon von ncon qunum nnmn humn concoun. Fuh h mod ovd unqu oo fo chn dmonon ndn nd mnon d o ubomc nd omc ucu nd ncon qunum nnmn non von mchnm n comooy ucu nd mchnm of humn concoun.
On of h ky fu of h Pnc of nc ud n h wok h u of hchc f-fn mhmc n od o ccommod boh h ncndn nd mmnn qu/o of momnumny Concoun. In h bnnn h w momnumny Concoun my my bu uy. nd bn o mn houh mod f-fn n = = - = - = - / - = / uch h cd h n objc o b obvd nd nn objc obvd d hm no n momnum-ny c nd nn momnum-ny c cud hm o nc houh ffn m w nd hu v bh o h du momnum-ny unv whch h nc und nd md o vov. In h unv h body h of momnumny Concoun nd by u umb h ound of nc nd cn fom n nd nn wv funcon n nd nn momnum-ny objc ch fom n mny ny n h du momnum-ny unv nd ncon fd bwn mny n whch ccomny h mnon of h momnumny. Th body of momnumny cn b f-cd on by f-fn m w. Pmomnumny Concoun h mnn ow o ojc n nd nn objc by ojcn.. n nd nn h + =+- /ħ h ow v of momnumny Concoun. Th unv o cd du momnum-ny unv comn of h n momnum-ny c o b obvd nd nn momnum-ny c obvd und ch vc fm μ =/c. In on cv of momnumny Concoun vw h nn momnum-ny c whch by convnon h nv m h non/m of h n momnum-ny c whch by convnon h ov m. Th bou fm of fnc h momnumny Concoun f. Thu f momnumny Concoun o mnn = h du momnum-ny unv woud d no my nohnn = =. Th ccounn nc of h du momnum-ny unv convon of zo. Fo m h o m of n n objc nd coun h nn objc zo. o n h du momnum-ny unv f-vy h nonoc-momnumny f-ncon wv mn bwn n n objc n h n momnum-ny c nd non/m n h nn momnum-ny c vc v. Gvy n n momnum-ny c h nonoc-momnum-ny ncon qunum nnmn bwn n n objc wh h nn momnum-ny c who. om oh mo bc concuon : h wo no of h Dc con o oon n h du momnum-ny unv cvy h n nd nn objc of h con o oon; nd h cc nd mnc fd of n hoon n h 833
du momnum-ny unv cvy h n nd nn objc of hoon whch wy f-nnd. In h du momnum-ny unv momnumny Concoun h boh ncndn nd mmnn o. Th ncndn c of momnumny Concoun h on of mod f-fn n ncudn h ffn m w nd ojc h n nd nn objc wvfuncon n h du unv houh n nd n un h mmnn c of momnumny Concoun obv h n objc wvfuncon n h n momnumny c houh h nn objc wvfuncon n h nn momnum-ny c. Humn concoun n h du momnum-ny unv md nd cu von of h du-c momnumny Concoun uch h w hv md f w nd md obvon. 83 Rfnc. Hu H. & u. Th Pnc of nc: Towd cnc of Concoun. :. 5-9. o : h://v.o/b/.. Hu H. & u. Th Pnc of nc II: Gn of f-rfn w & h Onooy & hmc of h. :9. 9-78. o : h://v.o/b/.3 3. Hu H. & u. 3 con of Pcm od I. Pcm joun :6. 6-66.. Hu H. & u. 3 con of Pcm od II. Pcm joun :6. 66-68.