\ |.. \HHN‘llhllWl \ WI 133 263 T8 QLAMUM MECHAMCS AND THE. ..:.1\;2<::r»-3A;.0L23 ZEEMAN EFFECT '7156515 for the Degree- c-{ M. S. NUCHEGAN STATE CCLLEGE ' \‘V CL’L’t'AJ A. HLIDJIlg 1 £4 3 8 IIIIIIHIIIHIIIIlllNIHllllHIIIIHIIllllllllllllllllHlUll 31293 01693 8809 LIBRARY Michigan State ' University .45 . ' _. . .. I! o ‘ .' 4 J PHYSICS LIBRARY . '37-. J QIILIHUII 11331.11? 3‘ and the 5*? r"'~“1"l~T w ‘ '-‘ f" . \ .ui m-.340U.4 4.25.52...“ 2.22.49; - w Tnesis for Jemree of ... :3. “’01 7161.011. 110111111 ‘ k) 19138 I wish to exgress xv aggreciation to 3:. u. directed th problem; and to Prof. oounse ; of the laboratony. '7" YT..‘,.1Ia V‘f') ".INH .Jo ...,...»O "4.0 .' r...) “N -- '\ Tr‘._~_ a) .p-..‘,.... . 1 t0 yr. 4. u. ngual lor 0.1““ 39 32a "V ‘ "n ’1‘ 4—1 V .1: .. "‘:man for use 0. DEC fgciLLLlos 1‘ ' . \Jo Id. V...“ 11590? II III IV V VI II T V" .5.“ ~- uAL ILII CCL.I‘..L."..'.J 7?. '- . (‘1 2‘ I‘- f4“ '1' ..1lbert 335.363, 331132111 ‘0 L 1:2 ' '- , . ° ,-: '1..- 11.3033 0.1011 C >31' tors. 111 ...lu-31't “$.08 T . 1 -v . ..3 0“, , 3 .,;_ {Jr-11:13.17; 3-10. ---3:2.--t1..1 C; 33.3.01‘3 . - . '\ “L... ‘1‘$'\ $.19 .3‘1'03‘I 1.3 :0.) 3.. ..1 J. 0 r 79.1” .1. 1.‘ ‘. . -.0.-.tlon 0.. -1.-.) 3 -L. 31.33 ~17... ' . 'h . .. . - .. . -- ...-‘ s; :31 rogt.._lzztes of .1‘..‘.;‘.t.‘v-‘.‘. --3C .....103 d . . . .D ‘ - .. ' ' . .. . . n . 30.13 0.:10J 0. ...r1\;111-.r --oz -331. .33. 1.“:"01‘1I7'G11V’Nfl «'7' ‘11 '171‘“ 7‘ ' Jt‘fi ’\~1j+ wrx 1J4.“ LL ~v—-»‘.~4h) 4.. 4*“.3“ ~43. .. . M4 .M—‘A 33lection 2.73103 for 311313.131‘ 2102.3;1tvm Portucrb .ntio 2:3"Z:ocl \’\‘ . L-JG :.M«~1 .LJJ- fJ—JCt I 113311 ‘DUGL‘IOII Quantum mechanics is the lust develoyed important branch of physics. From 1900 to 1925 veriOLs special problems in quantum.mechanics were solved, in which the names of Bohr and Sommerfeld were as conspicuous as any. In 1925 it emerged as a coherent subject, due largely to the efforts of Heisenberg and Schroedinger. some of the names of present workers are Dirac, von he'munn, Slater, Born, Condon, Uhlenbeck, nigncr. Its growth continues to be rapid. Quantum mechanics, as its name implies, is the theory [—Jo of the motion of bodies. But this statement, true as it 3, is not very illuminating. For the more obvious enameles of motion, indeed the motion of all bodies large enough to be individxully seen, are quite adequately dealt with by Sextoniun mechanics, which is a syeciel cuse of quantum mechanics and simpler. Also, it is a result of quantum mechanics that certain kinds of motion are always accomgunied by the emission of light. Thus in quantum mechanics the ideas of motion, light, and electricity are all fused tOgether, so that it is a theory of one almost as much as the other. Its results affect all branches of yhysics. Perhngs the most striking and largest upglicution of quantum mechanics to date is the eXplenation of sgectru. But it uls now has important results n kinetic thGOff, -1- “If 1'. (3‘3-.. 1") ”L v ‘5"' LJU- J. .. ‘u-‘. L‘ bl \I. 00 Q U.L-A~‘JIJ’ J. ., 3 die .. .'! f) --|.4 t l 7"' Wv-\. V .-n —\I L 11' “.'.l" ’1 1., .J Us -~,.,\. ,A. ._4. \ -‘ . .. I‘k'iai" l1 .tee . ‘I 4’.‘ J. S 3" " .34—3‘- ‘1 k4 3t 1" Dllt (a -1»), (u U :1 ;l ‘fiii‘c i 0' - l . U I“ 4' . "" 'I‘I .fi—L V-J . U 0.3 “ u-..» .\.. 3.. I. J.“ v .“(x‘ 14‘ .1. 54-- my :1— r'“"r'\" ..\_v\ 3 u 0 7. . J. 30.10. .5 .1. . .5v . 2 ~' ~- ‘ 53V}. (‘l'_r'.' .._I\- — .3 u 3.3 L... . ~_ g .- . ‘de ’ (“1' Vs C a. pm a. -. mu mu .1. W. a . 2 ~4 . .4. .l u... u . . ..,. .U .15 um av .r - nu .- 3 H... \.v I .L. . 1. me A. .n .1 e .i I. t :3 ,i. .I. ,._ r v ”-2 ~. .. J 1 . . , L ' .cip h J .1 f1 . g...... . . ,. ‘ . v. 7.. ‘lo .. -3 .12 .J «i. 1. 3.. r5 _. N. «u .w. 1. .3 Lb mu owl- my .. . .I . I .U Ian p. f. .. l. t.» W. flu mu 3 no nu .v. f .- . .u U‘mul __'| as a liriting case. In engineering and in many parts of pure science, an ertrf, common—sense sort of ixysicnl concefivt is verv useful. he bothersor.e co flesmi ies of surface of such a concegt are not brought out in the dim light of the s~e30‘3010us° only its bare silhouette appears to View, with satisfiflng simglicity. But quantum mechanics is more exact than clsssicul ghysics, enu this detail, obsc u red by the usual conce;t, is there n3ed37. Ehis is not surprising. Eor quantum meehunics 0“i;inutwu as an attempt to explain cert.3in nienomena of li;ht and heat which for ; weneration deiied forts of the n03t chs‘le men alive, 3» when they limited themselves to the truiitionsl concents. ConceItu allvt hen, aunt‘w mechanics is differ nt. But as cos, it is var? similar to the theori3s wnic b preceded it; it Eust t.kes over‘the Im.themntics of grec ding theories and adds more. ”he m tiemstics used in quantum mechan- ics which is not used much in other ysrts of ghysics is the theory of Hilbert 3:303, linear algebra, matrix algebra, theory of determinants and linear transf01nntions, esgecially the theory 1. (.8 H) o grougs of linear transaorm tions. 21:3se suogects are all gui -.- ~ .. .. - ..~ 4. . ._ “I i .' u... _ 1.2 ~ ‘ :9. ... .. closely r31-.t.3 b0 chC'l other, 12.111151 rnC. . r the new-in“ of M HJSLC a slgeurn. . '3 1 1 '. . _“ h". . a c‘ _ r ‘ - ‘_ 3“. q o a n ‘ in the fOIlOHIHU 13.03 c 33113;, of the o sic ioeis or ‘4. 1., - 3-1 .. -. -. r quantum neC33nics till be same, lN‘ and viewpoints will be included. 30 illnet“;te how these .iOl‘I, they will tnen be &"'2".-lied. t0 " + . r-‘ 11‘. \ ~ ' I, .A- “A“ 3. ~39 we33 liel‘ neemIQ i~ct. C)“ H v A. *4 II YII?~:.~1~\~1 - 1~1 [’1‘ v “u ‘. IT hwrul") . ~71 ‘ -. i 34.1.3.3»; S -3344, Mtd‘hL—J 43¢”. 33.... .431 ~ A Luge part of classical '=‘1.‘1;I'sics d3zzls with vectors in three diz‘nension'zl Zluclidenn syncs. In all yrts of quantum mec': enics we levl with v3etors in n dimens ' 0:131 Hilbert secs. 1 C”? O O H) d' 5.: ._. 0 es to t3os H Liese vectors bear rut-11y striking sizrzile cormon t1: Ie, but still tZejj are more t 1331 Just a Stl“l\;§lthl‘.’.'3.I‘d generalization of the common type. If 3,) a,)- ' are rr1er1be's of some class, demueruble or not, £3 1d :3.- is s. tyeicsl member, we use tne not.-tion [3Jto {unote tne 01:.sz as a. xmole. Let [ejbe t‘11e clues of conyxlezz llLL‘leeI'S I and [in] some class unless elements are the f... [f] is called .Zilbert enamel/if for any tTIO elements of the syncs f; and £1 there is defined a. sum 1“. f ff, and an inner product (fluff)! and if for any coztzglezz nuanber a. and elergent 15‘] there is defined C a. scalar pro; not 8. ff , and if these Operations satisfy the following relations: :9 nff/fi] 9 a.‘i}{[f.] I (f.‘, ff)((/a.] 9 offifz (1) fi. £1=r7+f , (3+ f,)+f1= f¢-+/f1+f;) (2) a. (f+fl)=s., rifle f , (a‘.+aj)f;=~al.f;+ef r, ( f1 (:1 (a7)£,‘=a,.(a:f‘)‘ , 0f=0 , If =7 (f’f)303.ccording as 53:3 (4) (1339,11) =[—g.,f‘)+(£f,rk) , (2111., f7)=/fc-)f ) (12., 9% W where the symbol { is to be read "belongs to ", and amere a bar over a coupler lumber denotes its conjmge t3 complex. we define Au- the 371-2301 //f‘// 1-1;,r //f,-//=”&,-, 1-?) ; i of the vector fl'. If [f] is 3 true TEilbert 5 {:1 (D :J ('cl. L.- is 0311133. t3~ (1' U) 1 c L . (D Ho cf ’5. L C T H ( r‘ O J a 4-. . 'L: .... ....‘, -,.». £1.40 OO-1L-llU-LO.LLLJ, s—LJL‘... b—A . Cr F (f. r i .J k.) 0 O -' 1‘. .0 - .’-«-. satisfij t-1e 1.3110111; for: 11/4 3:111 2" ;_:r3;.,t1;r tlizmxm , 6 3:13? _1o;itive roe. ; If for any 30-13131 33 ff} out of [11-] 2:3 3:213 //’:',; fV//< e (5) then/f} converges. There ea: ate 3 3e .--c==)/f )0: t o: [fgh 31321 11121:; 52301-1 (5) point of [1 Z33 a li1 it mint. Elie conditions (1) t1 133011le (6) 23.330 tin; 3:101:13; let us now: P3 vet :1 £32; conclusions. 3,} (H-(H, f‘Ol f13€7/= \J //1':‘//:‘ //1://{Z-"10 (1‘..I1‘)=G‘,f)+ (3,3) - 6?,5‘1') ~ (3.1“): G~.3,f=s)?o :1e(f,3)sz //-1?//i//.;//') (7' 'L “j -r‘ -3” J- ,4 .- J.‘ 113 r3..o., evil ..~.-.rt 01” and "me .5 \ '1 . ‘ - J. 2111332.?) .1‘3 Stilt- If: 1.1.3 U0 . "1‘.“‘6 " ‘p‘.~ ‘ - '. \"J H 1 ‘1‘ - ‘~ J.‘ . - 1 I" -.. ‘ ' A‘- 1111-111 1,11,: _1.1.rt of". ...e_1l.-..ci11g I .....11 ' o; :11 3113 3 :g, ..1t.1 a. 113.11 and.>0 , equation (7) bec01;1cs' -;e(.;;,f)s{ 61/17/1133 ////‘) Or, replacin; tie rid-1111 of (8) by: its 1.1ini1.:1u'1, obtained by dif’e11nt13t1n. to a: Re (f,g;)‘~< If//'//;/ (9) .- . 1 {d . In (J), 1'e*‘1l.?.01n~; 1‘ 2.1.11 3 by e 5, :Jitncx real, .93 '01:: u, u lie é" (f,3)‘cosos11e(f,;)— sinvs Iz-1(f,g)-$ //1?///;'/ (10) Replecin': the s-::conr;’.~_ 113111111: of (10) by its 1-11::i111'1m, obtained by differentiating toor : [(le (1?, 3»); (Izz1(1;‘.3))t_lé 3‘ Kfrjl§ //:"////_j// (ll) If the earls Si_;n holds in (ll), it must 11016. in (7} 31:10, 1 fives, in the not:::tion of (7), 13—“ f~;j;)=o , or J! n . . 1k list egubtion 13 e 31 Ho (.‘1 f= In ”-5113 notation of (11) t3: 2 (a . . - - 3 3, or .. =3 e f=cf 1:33:33 c 1:; .3 com-31.3: n13.1112r. 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J _ .1 b 4 . 4 J . -. .unU rid 1” H». v ._ mu . I ” id P. .9 U o ... f . 2. . 9 .U . .U .i - .w 1. < ( . w W. 0. a u. ., .. 1. v .. ._.~ .2 .n... .l 2 .u r 4 v... 1 S .. ... \I. ./. .Q K. S. r f w , n... . J. J. a S C . . a a” v“ S O O ..._ 0. O w v . - I m C .... A.” H i.“ Z 3 .. o .1 r n V 3 t“ f. mm .3 U n /o {x /. G. v w 0 p, S O I e O ......_ / .7 y m; w . n w a O .x ) .0 " vi.“ t .\ WU .N} m a .fl .3 f ) y t L. S .0. . . Q I.\ / P. 3 r . .1 . x... . j / 4 ./ . . . v. L . . e ,L u . a I 2 /. ._ 4 .l .1 fl. 0 .C 1; 1— U 1. no 5.“ . u ‘a U .I . 1 n n .x 0 u /\ LA, .. L 0 a m s a .u e . _ . fl 3 .r. .r. v. 1).. m 4t [1: 5' //+// b .1— +v .p 3 .nL an. ) / o .«u .b . + 3 “n u u a A .J I, o. 1 .1 an M a. .. ‘ o I,» \PL .* 7, o .3— ! Q C l .3 2 + / u 0 3 3 f O f t O f. l T 5 / <\ . ~ . J c s 2“ av 4; II /I e a; q. r o 0 mg : w .l .. 4. .L. .a p «l I. L O Q I .. I t m . fl .H a J. . S d 5b .C \i 4. c .7. J . A l .01" <\ w . sf“ n «U «L 3 f f 1 0 n1 .3; r «I C n; R i. .M . .1 O I.\ pl 0 0 fl .3 n...“ n .3 .7.“ n... t r_‘ 111311 319 C I“ . -LU ‘2 1 \J-;~ “B (A? u. filing 6.1. n V.‘ 13 C:.' .L 31 is -7- 111.11."; £th 1' S ELS 0'; u n.o.s.) J. h 4 \‘AI .- ‘l' — ‘5 x .. a .rtr . x .L: '53" a. 00:1: .\ .‘nh Uni comm} v‘c ‘ J. l u -C “1Q; chl a .L r" d. ?01 '11". l 3 “n U-al u ).\~‘w U... "f\' 11m; $4510 a O - ' ‘ -I ' 1 ~* l'2 - .. J. _ .7. 3 ‘ V aw ~' ‘ - -. 1.113313? 11000-4. b“3.LO;L _.‘. ; DOW o L .. 1L: ._.‘. 0032113th 11000300 , J-" i.“ ,. .‘.,. .r inn, -I, ,. -,'-‘ ,-, . .,.' ;. § ‘ ..., .4. .. : 5' ‘ ,.,, /.: // r. .4,- AlfJJ; -l~.4U‘ ¥ :ubv’a .x. 0+ LI ~U ’U~~'-Ll DO ¢|-'—'.;JLI Vu.§ll.i. 1.211, {Ia-:41} f .1 J. _ -., “ '1. ._ ' ,., . .-.-—.,- A. . ' ‘ . .4. ., , - '. 4 .- fl ,. ”'1 ,. ‘ ._ .., ”HIM-la U3 110335;)“; :1; -:,.'-'.-. Liib ‘ C‘Cb' .“ 111-4}..-3; v0 1194.217. Cdlil 10b? 0.". « ~— . -. J.- —-. 1.00:). l 4.. 3.7.1.. U_.IL'-~IL1 &. , Al.‘ ‘ D H. U) i F.- D- ,4 F1. Ct- (u r. :3 H H ,4 O C C 0': . O ' 3 f... t’. '__l C (f L‘ 1 r... ‘W CL‘ 13' C ' J 5 J Ci“ 1- v 1 I P p H H ‘ ‘3 :I G 0 O L. O *1 Ha ”J J‘ 1- r 'I ‘\ '...‘ (I -r’ 1 L. in. mid: 1211 hOIIuCl‘o, 3-; Spud?“ Ck... , 9.10: ‘J r. we v 3499,20 (. (‘2; w t. a J'- . v . '. ‘1 - .‘1 , 2 :0 .‘ . ‘ ‘ ' - ~ _ _1 o no. mtlgiplging my cfl ...ims a. 0 i0: 3.3.1 I ; in“. Luis i means 716‘ is of grefiter number of 01318113101123 thin 11. 111“ is .- . . ,. n u. “ =4- '... .. .- a . . ' . _. an; mu”. V013; :1 AA? U..r.1.L:3..~v- in, .mo :1 Livn‘sudfq .- 9 Last; iv: .4. P. f it 1341 m4; or /g,%)ffl?,)/\<§zf///,¢.)/ ‘1 //,,qo,)/‘] (19) Zelutions (18) and (19), taken tore-them show that any series IN I..-l ( 3' V of form % [f'fv)(j27r) (:30) yz. is absolutely converge ent. For the sfi'of an n.o.s. we have the relation: ’fl+4’ I ‘ggll/ /2 //2 Irflfl:(ZI¢$V'§Ir¢V)~E/y‘Xk/fin%)R»:Af in (21) or, by 3.17.311'1' 3 cxiterion: the seriss 2.1342; converges when l and 0111:? anthem the series é/J'v/ converges. 2;; (1'7) 3:161 (2.3.) we have the identity: 2‘ “‘§(7‘"m"p' (2:) 4-4- whexe 1313 series on the ridht is convergent 110 I: 1.....31' =.'.'11.:~e.t -9- element of% the letter f may be, one whore there are as many terms as dimensions. In (:33) the 79r- are c:=.lle:3. unit vectors in the various mutuallv orth03onal Lirectione, and the (3443») are called. the cork-.gzonents of the vector 1; the o analog with tlr: ordinqr;r {'3 air-Jemeionel vector is ap parent . Iget OMI‘fZ) ‘ ' I be all 1100030 O‘WO 301‘.Sl£i¢‘:1‘ tile H) our statements: I. the systerin:‘1L; "’ is complete, (Lo) for every 31‘ offl, f "4% (f, f,)<}7¢r (BIL) the linear r mi 01‘. generated by«¥,24§,*-- coincides with;yp (25 for even",r f 2.1m. .3; 01‘2/ , //;j)=%(f"f°r)/49~j’)‘%//flp”l/f73(25} If (2:?) holds, then [f ‘é‘fiflJ‘flr/is ortho 30113.1 to each <2; ter and hence equ: ls zero; that is, c...- ..) (IE-’3.- . Tut if (25) form raggfiyfi, or (3.7,) .). 9“?) holds, then each I? in 3/ is o ( 5). But if’ (25) holcs , t .911 any f orthox-cavl to all the pr IQ is ortho jfoziel to7/ . and to itself, so twat (fl) )0: ;/= 0; or (.35) .), (25). In swzmzry <23) ,,, (3.1.) «1' (25) '1' (as). ' (27) from (26) it follows tint if f is ortho 70113-1 to "111 the 6?: Also (.25) -). (24) which 137.131.; in turn: (/Hfl (éa’¢»)fl,é ,, %%)%=) 27/ wand.) Or in 3133.37 (2‘5) 9- (2‘) 91(2)) (:8) Relations 27 and 23‘ combine? tell no that relatiors I (23)-(25) are all legicelly e~ cribs the method of :3. .-chmi./ in l s colurrn nnc. 7" t21 rom. J. Consirer 1.11013. ‘ .- - . .:. 7’57» 1”} flewZ~"'/NJ’3"1~""*"‘*’~"‘Pé’e3“/¥) for .111c11 I , Z I I [fac'f/‘P(Z)ZA)/ 6!], "1;... is finite. let us trf to construct :1 ilbert sync-3 with the i (J) O P]: {.3 FJ [—1 H, F. (5 C H. 0 D 'v (D O F .1 LL J O "4 various functions fies elements. Addition end. seal-1r multi— . I. I" rsr - w -‘.‘ l\' .‘ “‘. .“y' I‘.\A..-. ”.11 on 1.1111 vs 13.11.} 23.1-3 s... 032111.14? :-...-...'.i 10118.1..1 or..1.....;r F1. 1...! cf [—1. p H. 9 1—1- 1"“ O ,‘3’ O “a 3 :‘P C) tion 76 b' n comelex nnmoer. Inner multigliCdtion is defined by: C/v'v’r) r éf’f} “if (:55 'th these definitions of tne fundamental 01erwt10ns, (l)—(4) are plainly sntisiied. A proof of (5) and (6) is possible, but involves the wee of Lebesrne integrals; it is given by von Lr‘ .' ‘ ~r‘f‘.‘ “'«. - -. ‘ . .. -. '- -‘ lieu:1.m1" from wnose DVULx 1,1113 section 18 lag-331; 1,332.... T ",-,q ‘4‘ r l '. ' I’F U .. nann v. 1.011. 111111, Quentcnz 1.30. 1.3.1 1..., p .4.) 1"7’1T'1 ‘ ‘ l' ‘ T- "'T v “ 1 ‘ 5 1 Y III fiA~U('AJV-&ILLI OwaJLL‘-(au—~U J-‘U 4—..J-AL—Jo-L-L U—fi-oaJaJ ' ' ..‘ _‘ 1-,“ -‘ .A. —-\-... >q --‘ \,, ~. manifol of W co... 1033;. of .:-Ll 1.: el::...:--t.. ori...o ,,o:-..-.l to ell ’ I ’ 4’ up .~.!;1_ f -_- / _' _‘ ',_.. ‘ :, .. W~ ;. .- , / 2.: 9. seem-.. 13oz- j 1‘ , 1..777,..~..1 1.1 y}! ; t..e.1 ] f / X- 4 1-7! ‘11:: 1: '\ : '. “r‘~‘l‘“\ 2 .~.-1,‘ W” (‘1‘I1 ‘ v ' ’1 ‘ “~‘ '7 3“. ‘JL ,L._J .-_. ‘Jg/IV.- géALL LLL.\L \(Ok'uJ-(L Ukl “Ell-0. ~, . _ in J. , ,.,..,..J. .-. . ,3.‘ , °-. _ ' t.18 $111.1 0.. L...O 01:....J:;o5, (111.: 1.1%, 9.1“ 0110 11. 2/ w . f 13 r, tiie "‘t~:'o’ectio:'1 0;.1f;:;'11b01"' 7%? sntisries t-.e follouvin 2425443: vé' lat/r (/31 f’Zf) 2 702’09) _ (3.; gag/r2”, 0°; 87%.) / Wm} 7»%/fl%¢»/fl £7/gror 1” 1 C) '1 1 H H O H w (.4 F1 *— ('1‘ (I Let f— ‘oe .:.;g ogerator (1.;1‘ 1:3.icii (c‘/,;)=//»Ff) 1 52:5 (5:3) Let 7/? be 1-19 :‘ 31.15015. 33:1 r31: .1 “q; 3.11 no 5/ . e nave (LP/JjV-[fl : (if?) ([7,£;)=({;7;)‘(F72j)’0 or, j-[f is .L £111£f.31.00 i‘rc1.. . KW. fz57+(f‘€f) ) Efba% , ]‘[j % L- A. 2 :£ "~ ‘-.. .1 -- .. ."‘-"" A . L~~.. ‘ ,~ ,' .-_‘ 1‘1 ') Y’e SJ: Ll bib ”7 i f . *A*\.U’ 1‘1 3": J ‘3“ ‘l , 3.1-. .’ .VLZJ.-LJ \l.‘\)} g: .3 '5- L .a 1m") ’3" ‘r '1' -‘ '5- ave-"“7. n: 'r “ \- -. ‘71:." ' " \ “ ”‘1‘ F HM) (H ' I) 7"-" '.—~,,—.4-~' "'3 dLuCl~JQ5J~.lU' L’ailyL v‘...LJ.—V..‘4&LLI LJ-.L.a.le.'a. DJUVU “[4 us}...\. *ILJDguVlvsl -15.... g§3 II T QR) § u \ . I {j '3 $3 \/ n h, 3%) T“ C.) f t‘ :3.» yrs action 04331393 //E//’~ a (5/, 5/) = (5/, f) (at-,1) .1130 /E///‘r//;»§z//7=/é7//u ////~E)///2 = ff/rf) %(//—¢‘)j,/): (éii) 9070”,”? //F/// “M :et 5 and. F be t~..:o 09:1“501‘3 54.13-; :muect all vgctcrs of 7/ 011120 tars-é yogi-3215.5 %"17.C;7Z . Lula-11 5F ‘..r:'-.11 aha}; ti first relation of (as) if ;n£ 031; if: (FF/qFOCA t})=C/I F57) = (FE/7) that is, ii‘ 5F =F£ . gut if EF=FE than (w) is entirely .,4.° ..r" '1 blulbl.lGU.3 U 699’: EFEF = EEFF 2 6%“ EF find them 55+ F7 :ill alJJJS sxtisfly_tge first relation of {53}. It will 2150 23313:; $13 unconi if and aux; if: (“4F)’=€‘+P‘+£r+r5 = €+r+ (9” PS) =€rF that its, if EF9‘FC . "0:: 51:30 -)- EF is a grogection Operztor -)' EFL‘FE . '-'o;1ce EF>0 -)' EP“FE . "Wt .LJ-‘w tua converse also molds lS erVEn 3:: -14- E/EFJFE) :P‘F-ré—‘FE =EF+ CFE -E(€F+ F€)(: =5‘pé‘ x FFE‘ :EF€+ EFC: =2€FE :0 And then E'F is jurojootion 0;;er:;-.t01' ix? ”“6. only i:;‘ I- (E —F) :(/-g) +F is one. 3v.» '1) Elgve 5110.51 t;-.;Lt: (/«t‘)+F a grogoction o_.-£:1'x.t01‘ I (/‘€)F:o I €F=F, 5113,) so, for $22] f in 2/ , €9=FEJ ; tum the elerxont 67:] 11310213: to the cross-out W of 7% 12:25. 77 . Conversely, if} belongs to?7 , 5/3/5121 or [Qt/=1 , so tout 5F”? . ' . Suppose in the socono‘. place 5F =F€20 . lot be t1.1e Iranifold generatol by 77K and 77 . ivy j in .1; 1.13.3 be ‘-.‘:ritten fZZ‘Ff. ,4{2(,f{7f. Also {12/ ,F{=F€(=o , grf , ‘7" _ "73.109: Gog/€7.20 ° (E4F)(1¢f'} =€1¢F[ 7L6}°fF/" 1‘ [*f or, 5:” F: I? . LZLI;'>=_?O;=e ‘5 mt ENC/APE 2F . Then 5 ‘ 7% a - 2 \ 3 a: “F 2 P ‘ . FF: 2(_7;1 9 P F E C F £0 ) ’7 , "more W 13 tile cross-01.7.1; of 74 and 7‘7? , that is, W:%~7Z . If? Li 111131? 'Qf? 77"‘TTI II-;f (lfwlflfufb Lo Before Join“; furt 1-3:, lot r73 sic-2.}: of tile .. 7-:1131‘31 coixccigt of linear Operator. Lot 7.73 use the 377777331 6 to canoe-3 Lizzcli 7327.171 1111011.”) 0:77.11 4/ also 173.213 over all of f, or W rcs,-ec‘oiv:31:7, so thzit tier: is a one to one correcgogccgcc between tnc vectors / £7,116. 4/. 5.31.3033 further tint, for/ and] 37.1137 17.30 17.1311‘03173 of €0r7o/z 417/7) :4/f’fj ’ 4/: 3 4f r (r the i::;z;3i:1:;ry unit (41) Then 0 is called a lizc'r vector fanction 0.11;: 4 is 0.1130. 57.. 1 Hair operator. STLm (41): or Cziw (Tu—MM for any COWUIWY Vilvei c o Conci'c“ tjic linear V30 ctor fLJlCLlOIl 831'1123, "3;; In 0333 of V. c ,2 Y7 ) '1 Q g 1 7L: C) LJ :1- O H, .3: c: d O "3’ D) [—1. p .7 {2. C‘.’ i: A ,1 N any second out, 33131 in noirz‘o.;.-17. In err-.33 of K to: azure 3.11:; set of vectors: 237:lr:7.7_'___-i: it to WA. 236. cm; fl 53:13 3.21: 7331771 in -177 111.717 "\ b 71:311'7111;;~ ’60 a second E’il‘uort 3533.03, zillion we 0:311 74 , tile 1 p 3‘ .‘ ‘ 1 " ' fi. (1317.1 04. K. In once of f tuc 701: 33:31 _71;;.tcs 1pc 017741113137 scalar product, in 7/ it doughy-.333 tic inner 371737111013. Zach .7 . 77.-.: ..,: é - - 7 .-'7 a .1 .1 1713...: 52.7 ffi .1th 1 M) term {72- of f/yg of (-15) is 32.11.; is called. 7.7. 61.37.713.10. Let 4 desi;11;.-te 13117.3 67.37.107.10 224]? ; than; by definition “’1 is 111713 li11e:7.r i‘zmction (13) 1.7.1151 4’4 tips li110.-::.r function 1/7, */’/zf,—/--- . 1114/, 4 is said; to be used as a. 3170372313017, in 1% :7.s .17. ;_:-osti‘.i7.ctor; in _;;011sr:;1 £17414 . If 4": f/fg , then 1:313 conjzgtzto gradic 4. is defined 1.334..- : fjtflfi in 057.33 01‘ 5 2117.73. 13;: 4(2)}; in 0:133 of7/ . 2918,1111] fl/ rx'é in 03.33 of z: , and, filam—in Case of? . Two dummies 1117-3 03.11731 (31.1.1511 if 47/ = A"! (44) for 7.7.11 4 in 12-13 $378.00 in guestion. 3.11 aquivslont com-.ition is that 174 r [6 (£35) for 7.7.114 . Iffl‘sd , than ¢1=fl'( for 7.7.11 111.1111 taus 7"4/‘777'4’7/ (45) for all f‘and 1; convarsel;.r ifffl'W 4 for 27.117“ 1.7116. ( , then 7 4‘7/ 673017 31.11% , of (7171-), 1.7.1111 hence q): 01 , by (/15). Huppose the value of a linear vector function is gnovn Jor an 11.0.8. of vectors 79.7 ; t11:;t is fl‘p‘.21‘- ,7 I':I,2,--- (-1-?) In case of g: tne<4? may b3 any three mutually 4L vectors; in case of 7/ any complets 11.0.57. tell, any vect017f¢.7997 , (.73. real number in 0333 of 5—, and com313x in case OfIZV , is trans- formed. by ”into Z4 ; but it is sizzilLLrlJ trsxmforztcd by -17- the dyadic<é;{*ft used as e urefector. QsLe any dyadic and trans— form the ff in terms of the 4?- , 3.11 11.0.0. of 7/ , c.1111 tie/67 in terms of the <fl' , an 11.0.3. 0337/ ; the result is gig? :gq’r‘fl"; (4:2) - ‘i. I ~- ~r |‘- -" L1! 4"- ' ' r - 7‘ ' . J'- " . ‘0 4'.‘ l-‘ .r .~(.L-‘ In 03.533 01 g, ”’8 O--_‘~)0.nu. b01111 0112:? // nil-.:.} in bcrflb 04. 21.1.6 0.3.119 A The expression on the right of (43) or (43) is C3119; the nonion three d'mensions it contains nine terns. The xor: of this nnrzjrijh shows: there is a one to one corr083ondence between the elements of nny pair of the three classes, linear ogeretors in n dimensions, dyndics in n dimensions, and matrices of the n th order; no nutter whether the suzce to which the onerntors are a; lied is Z/cnr;?/. How for the ieen of Chen e of coordinates in.2%/ . snpgos ’\ ontsining the (7.97 .1110. 93'7.7.71110171 are L111 0 that in any engrcssion \ 11.0.5. for 21/ 3.110. for 7/ , t:1::..t7..7e reg-17.1013 t-1e 42701111 97330.7 tne \ yéunnf widch.sre a $000117“7 pair of n.o.s. forii/nndg%f , Lhd '.7 . '7. , u ' >7. 1 . rs -7 . j -'4 «9 :1 a 4‘“ ‘ 7" ‘ l‘ A This 317 03s 15 0.1-071 .. 0.1.1 e o. coorunetes 1-70-7. one 11.07.... .— 1 u .7 1‘ ¢\‘ “ .H -" '4" I-"‘-" "' '1 " - -F of coort.i17.._7..tes to .11othcr. e 17.3-17-3 t..-..t nevzr 7-1.0 17.8.1-0 one 1‘71 0337/ ,30—7 v“ (Q- f—Jo { H.) 1 k: a h '. J A k’ V t rd L") 4:» . h :.7-.nr.i 1117 r 1-.-tr1--r 4:}... J {7 ’ ' . ‘-.. ‘4..— '7,- ‘ 7 .0 .-~.. .-._7 .113 retrizc ( 03.11 0e nutter. «.:.b t1; 31177-7. 01. 7.. renl 7.1.7.. l.) , 1 L 'C‘J 6‘7 *1 11121337 Luitrizz 7.1:, 317mm ;.1177 two cor-13.13 -: . ""1 ~‘::: ll’L":-:DBI‘J 4rf’: ,‘r ’ {It Uswvi: 4L: iAL‘J ( fL'IKQ; + 5;} 1‘ [(7:33 7‘ {‘77} 191‘7DV‘J‘CACI ,. -- ' . .7? ‘ , .~- 47-. r ' r.1--tri:. o. the {30’397311 is J. bILUAi: U unit“. 7 0/7 = OAK/”U --.: ~54U‘— -as _--~'-— ‘ --',. . .I- _ ." , 3.- ' ‘U ‘ .t- 1.03 001.13.71.32- 1.1.-01‘1013; 0.} / - t1;- 1‘7._;.7_t :7" (7.., 017.3, 47747.... fl 50270 I .r. ’1 ,‘7 ‘. ‘V-- ,. — , . ..‘1. v.2. mix-3.013 o; -Jr..-.7.7.-.7.'s rule, 3-1103 b..’ o {913111117371 .7- 0:: .‘3 t.) 1 p11, J '7. C f—h O r4 *4 7c? O H k). I \ O (’3’ C‘ L‘. A 0? b4 r H L O r +— 1.. ( in. 4/:77/ 2 U 2 U‘- 3‘:- .-.. J- \ ‘011 U). '. -:.LUL t 15 r r7 -~~+-~ ~- LAJ—‘J ;IL‘_7‘J“O ' lt‘I‘u’JJ —- Jun“- J/ufi—‘IL l) Lat Aré‘f'fflfi bf. :1. riitiml for; with. tit-‘7. c’ \— r—m 11.0.3. of 7/ 117177? the ‘fi :-;-.11 11.0.2.0f 7 , \it'n 3.7.1:. i‘utaer F1; 0011;".itior: t.71;:.t 49;: 99, :33 t11r5t-(‘Par7f>=(‘7022¢&), let let 4 "13:33 “-777 .7.71;7.’31“'L.-. of O C) H b ’.J 61* u 53’ c+ ( i... ‘ c ’(L be a vactof in this bilinaar form A35 f a m trix, Lith z r05 Eisegt 'n first 17011.7, 51.7.11? with. tie xrth comgomant of tip: ector,(//79), in the /t11 (30173.11: .15.? first row. The e:-:.1;7.tio711 fl'f =A’f X a. 005131-33: 11‘0..1":i1_7r (Co) ...' 1 - - A“, ' - “,3 1 - .:7 . - ._, .. - .. uiil be ..t1.11ec l. “A Owl. l- 7M $ § {L w W :1. 3 C; :3 J 9. 131131011 of (£37) 7.7.11 b the if, , 1:11;; components f '3. vector, :3.; I, ,. \T;*ox’.7, by a. 1.6.1.1717‘t 317335.735, e:<._;-121.i17.e‘- nor-r equation I ' ' I u x a I (232-) , construct 57.11 11.73.57. soyflfiafiéf . 17.1 true o. 017-7 mat-:3 system of the ~6- : @‘%X (M 52:0 that .i‘orf-T/ 7.7.173 gut I‘f-‘z‘, , :_..1'1‘ X, 1.17.131; be I‘¢‘-'2.1. Ye may A -— / 17.07.71 write th: ‘oili1'1enr fora/751:7- 1,1,1, +2 47—911] . 2.01.7.7, 4‘ r regeatizg tho- .I‘OO{:SS for the 50171:. :2 4’41? 52,116. so on, o ‘36 obtain the iI’7';-,OI"t‘-"..Z‘.'t l‘-:f‘-J.lt: ) Jui— ( --‘— - Via-s .\ L L.~4'.~J., l". '0 A/ 3 o I l - w . ~_-‘- 5 ‘.I U I- 'i .I .e- ’ 0;.4- ..'..L.L qJJJ. v .17. “1'3 '1‘?‘ “I. 9 -u .. 2‘ J. {k}... a - "“ ~J if oovo ‘ Q ,. C) J- .- I I . .5 , _ r: ‘\ . ~. ‘7 '4. ‘~.:. V '. na— -x?0f) Eli/((1’) 7.1:1'. ‘ . ; n ...-\J .— 1 if V36)”. .3115). -, J. - Dim; l.‘; , I‘l-‘N;-'-h .._‘,_ ,7. I .L 1 A .r. \ ’.I‘ )fifn .. . . I |( JAK '-‘.\J-~ ;. _.l I 1);: :11 l ..“ » hJLL'.|-J 11-. oi :e;::-;p,7.ce -' 0.17 .71 ‘m 7 71 175.77. .7-L-7:-.. 1171 13:7. 4'7 - .2717. 3'17. 775 7.171 11:117-7- l . o . - . .. . n .- '1-) 6- - 1 _1‘ CK' ~ ~, w 3‘ ,- .. CJT: )1- U'-! A. -. ‘v ‘--__l 7-' AA. J3. - - Chairs” '- --' "‘7 r. ' ‘- ~-- 9 . . -7 _ .__- w . C 3‘ “'3‘" 2‘ ‘ '1 fl ('7 »‘ \-' '.;-L-J -03).; '.J». .. . .4- -T‘: ‘,\\\ -: I’) J. ‘. "r: 1 . J. .n- - . 4‘.L—.'g V._V *lL J 0 . L 1" ‘.. ' a. ' . r ) ‘ ‘V' ‘ -‘\" L u'xd Ll UJ. 'JL....L Lil: ‘.;... '4- 1, ~ r: | ‘- 7" 14-3..-. 0.4 . . bed L.Li\,L ' I 4. 1 ‘ P,-— y71r‘la J ' "- h, C A. y. - U --- ._ _ _ - :1 «7.. - - -- 7'. .. . =- .1---. 'o L311 '.;. 93.. O Cud 1-...1 74‘}..- 1.11 5.118 93.3,.1 I41. ”I, r .r 7 177 ' ' . - -A'U-l ')¥’.U *¢. — d — — A u .- , . -~~J wf- (- -.' - I ' . '- .u k) ALL) . . 3,-L J u-.. . _._-. -4 J 4 . “I , .1- . ”I a? [I . "7 7. f‘ ‘- ‘ . . . .4..LL2-L / v-1 .u;;~. .s— \I A.-- --\.«.~. ~_..-. II] (II 4.4— ,3 . ‘-. ' _ ‘ I '1 .— .--o lL-l /; 51-. C- -- 7..-), f; '0 rv;uleo ..) my I I ’3 "‘7\.-. "r7 .1 -- J--7 .ij.-. -r-v-~— ~‘ .A. at... Q q- 7. -r ‘-~ 7 ”U ‘ 7 .~ 5 ,_ . .4. ‘- l. 1 r... (757) a... \JA.'J.1. l 1 .y',' .L . ._ A. .1 f) Lil‘.‘ . 4.9 .- v 1ou7 ml- ‘1. J. 7.) ~9-"' uh I], .uk 70v .J-.. 3‘. 1 11:31.11 V"! C L." OI 27' cl QC 1‘1 . . _-A V. ‘— l "I V .1 .tion W 21.0.»;. ’ 4-. U i. -t ‘| 01 (1-. \"-VV 3.3 "i {7‘ r-'\ \'I 'x'* U v. ‘.1' 7“. ‘ “ .$.\J-.-- c | - 4J- --.-A-.¢ . ‘_l h .1) f .~‘u 4 ..-" H “\yiLu‘ OJ U 4 -3 l. .1 I ,3 ( 11(' i . -L.LLU ’r': '. v. o 1‘ 7117-1 U¢.g '-Y ‘8 1-: 10m cl s.) O «O +7f/7/f =0 . _ '.N/E 9.,LLuO .1: i but 7.8 UOl’lL' l J . SUo encefortil if X l 2 ad .. O 1/,(5'), $9 a a— .44 -—. ‘ W9 U ‘- i ‘ . A. \ I u I A- 1 R 50123 9 2‘ C L-_\.‘~}’-3117J-L5 Oil fl th- 2'. gent 01 ‘ [€(f)7‘/(0) , 2417,“) ’f/{j +[r((r)/‘.(r)a{r DrOVlQGL of COUPE? tout two .:quznoe jé.,(%{} conver,35. Lot 17.111: z.7.:::.:7.:: “((9 L773 11173-171 37;. wafi)/,= #- 6‘) 7/._ ,Ifl‘) “(1‘): / . " «31:31], -_, 1,3 C LOJC, ._;:'_"'LILI_I'_:'.‘.3§ L11 Cr 1'1: -‘.."_,'Ll‘ ii... the (:39) series 1:5 C(éfi) conv I a. grow tee definition it follows: 01’ r W): [/0’ /, (ma-.1 756.217.4(72wd7fl, (70) 30w to evaluate (70). Jrom o S (1‘27//t?/‘/41//<(.-/)L , 4’” ”"A" ’KQ’ numbers /24.747 7; é/ 3 /4.'/1/{,/‘ «7 //.7/'/4f/‘ it follo.;s: /§4 {/ ,11;-'in_; (7:3) to illte,'1‘:.l;s //a(/7(7)0(7‘/$£c//4(r)/c{7 T ' . fi-1 - ““7 "v *7. 7'1 “ -‘-‘ .' -‘7 . ‘n ' ' ‘.‘v r < Lettin, “ be .43 X-----.ll..1J.; o. 6&1) in. .1113 1.1t-31.r_.l 0 \ 1‘3? ‘- (7:42 and C'._'l?l‘."lll, (73) to (70): /V.Cr?/< (C, T)! so 13171:..t é 0,, COILVL’l‘J‘JS Liosolut 1.1;: for 3.11 r , by cori- 172123171-7. 01 7.7.7it.1 L. ~o*7tr.7. sezies 1_-.e.1‘-.. Let fa) be L72. v.7; etor in "‘il‘o..-rt SQLCG oxen-Lin, on the p-LLrsmet-or ll .2an1 let 06):?j76)16% be L71 ‘.e 711.1 .. :71" trizr. in tnis syuce, ..7here the J; L.7.7e ei,-env=ctors. If f“) satisfies (5.5) we have: gfaocyé 2/67) =fjr, [mag/wows/(zy SQfifWQE‘y/éflm miere ZZ is :7; v...7.ri.-1‘ole L7..;17L12777.131...-r, due/{[11:15 L.;'1L. “21:37.70 (/6) 5 20(5) is uniting 1121.1. 7-7217173 the 1:.st ee:1olit;_;r of (75) serves to define (9(5) . that is, C96) is that To .2.iti 11 f3??? '.‘il‘lOS-‘C.’ v.';.71'i7.-.tion produces the scare ei:;.n,-7e in ‘17. .e .31--..1311 f6) (NH/(1‘) 11min ,. the t ‘1“J‘.‘.l c‘ to (<45{Z‘ , err'wressims Z6)L 0.. .LL /(5‘)ale 1717-er eon t.:.t, 71.; 77.7.3. 177:7oj7J/3-‘ .1; e:‘__77;7~73;;.;i;;1 {374369147 , 1;; the 1-237.711 1,». (R?) is .:.-3117‘. congt Izt 2.111 /() 1.1;111/éz)v:1;r :7-Lceor'L‘zin, to (5:3). ‘ :2 1117.7; .7 ge/fézqéf Mo] = {at 72/19 710% 72/4 ) L 3‘6) (62;. . 162/11; ” if: 0(6) - wow!) ~ (wet) (to If Gé‘) .-..7.'7.1"1 ()é‘) eort.‘ 2:153, twn 331: 11-11:; g/{fA/)Q{f)/(({Af) _, . . :0 ’4. .._, - ‘- 11 b ‘ 0 ‘_ L. L. 3 ' o 0 No.1 7 . .1 1 l . o .—. - - -e- 1 w .-7. -...,—.-.‘ .. ,- .117.- 1:31.15. .1111 Hear , .J-‘U. ‘)2 mflt VAL TU‘u'J-Q J.1Lg~ JUL...- .U JAJ .kzdi L..¢&J.AL. VII FffglfléL PCMJILLJI_ OJ ,ELLDC: LSJT”;IJJ T ‘ I ' "' ‘ .~ . Q ‘ - .. 'I‘ ~ ‘ ‘ ' . r . " ' I 1‘ ~ .' U‘LlLt four 3-1.1,:alc-wl Oi)‘ ‘IJ_.L\_g\/\}u ’ UJ.‘ ¢‘_.U4LJ~‘ U'J' .‘I'JD U). — D03t11;te, gr: nocgcg in quantum necganics. Dhc; ;n be guita briefly stated in terms 0: “he pure mgthcugtics to which xe have confinsd ourselves on tag wrecegin4 DL‘GS. fiere tufi; gre: (I) Corroswonuin; to GVDr; st;t3 of ovary machwnICII 3;: am, there exist: a Vector in ; diuan;ions, n, as tnarc are ac;r 0“ of freeuom or Egg S'atem. find corr;spon¢in to every measurable {Agsical gruparty of tue i t1' ' ll «L - ‘.‘ ~ ‘ ~-“~ _. - ‘x. “. - ‘.- - .\ - ".1- ~_ 1.. - , ‘-. ratau CO‘JJSyOAu' : .e‘hltltJ I :L LA 3 L1¢aatlomu. -_l tme prop: t104 of thy 8;;tor1-axs fixeg us sea; :35 the stat: vector is fiyed, or, the state v;c“or uni4uely dfitcruine° tn: prog- crtiat of t¢e at to. (II) gay st;“3 of a agntcm varies :it: time Lccor414; t 13'1“ 82.11.11} 3'. on. . (7 ~ ' " _ g_ _ .v _ ‘ .‘. .' . a. ‘ wgcre~J4flis PlJnCn'S Constaut, f4 Lge VcCUOf ugfipuLfl- “A; 1:d¢;§t',.g;~ AK tgo T¢rmitiun xoru (r C 9‘ h '\ Ho :3 f" L (3 (J) Cf H. C H 'J C ‘ '. A- . - \ - '\ J-‘ . .J- . ‘ \ ' . - q - unsettnl L3 LCLHl ench; 0* U'3 :gutah. 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".4 U L L. ,~ , ' I’ if," ‘ 1.- ‘J_ .:.fi - '1”, . .p v . .. - _ ,. g. r' 1d \igx), Mo uLHu bge i *¢cv 0; 7’,0"4W )C j} / 3:0 *- —-. ~ .. - Ho .1) g ' .‘1,.' ,1 «“113 0-: .33 u-;uu_- mrflh fz’fz . “2.1;...319’. Wm 'J;-_w ”"70 a. a ,.. J.- c, ,. 2.. o..¢, 2 101-. 4‘1 Kfl \f?’ - A) Uni-‘3 U-.9 L) 0- :53 :1‘ Uilbue J 4- f/ o a #3 § - -c n -, ‘ -- ‘.ng 4'." \ -~. ~+ 9-— 7. - .‘ . I}. L 4.13.1]: 1.03 L111!” Can be 0.1. L: u_(l'.,u3.’3 *0: ' .N \ wgl fi;1;11' thT“éllu‘L I £1333 ~--‘, ... Vulgcs [33(?;réb/{ -q R‘- ‘\,.\., ' Llau Jdve one _’.‘ J—‘ 0,. u. -1‘3 VCLLL;K3 3 []@f ,)/f z‘ ?1'_ :‘ {-ta.“ h_ -0" -‘, “ ‘ '5‘ 4:0 -4- 0.. “L1” pectlm am hie U113 11 n ' ic‘~~~c‘r~: ~‘ '~' {10.qu ‘* OH “I 1101"“1 e f0 4 Kay LI Ub-’.UOJ-U¢1 U1 V l‘wdlfil L“. _ .Jl. LI-L U: . \ 11‘ -'r 1 ‘ .o‘fJ. 1' O ‘W .&q u . Ir‘ .0 JJ‘. 1 V... 3 I “’.7 9 g 'V- .~-.~ *ALUU la l V & ‘ -41- J‘ ‘01101':in;; is has- 7=fr% firfi—o Mfl-fl/ time 123 :,. Lagl“ ‘ \J C: . ‘. ‘.1 A 4. .L’. u Lu. 1“ v.3” N ~‘. A..fi.'._‘1.‘Ty,-I-— -\.,_-.,1‘ 'er.‘ ‘.u‘.-~ .\ ~.,—“-.‘«‘.—..Tv-- L - ‘ ~ " { ‘ . - 1- ' , 4 4‘. yw_4U--Ka—- -.L,'--“J.-Jtd -“l J ‘-_.'-l‘A...~ L2.L< g. ...La, . 1 l n ‘I ‘ s . Let toe 0‘ 4 08 :3. comglete set of ooservaules 101‘ LEE] "- '3‘ . - " \ : rv ‘1‘ 1‘ '~‘ -, 3* . .- h . J—‘ __. ‘. at04.'1‘w;LlC;1 or illoO COlleCQlto 03. to, 1.10b10u. mien bile gree- “ . H . ' >r' "I I ‘ fl . . ' '0? ‘ ' A‘ ' -‘ ‘1 r‘.‘ -‘~ “A '. rv I“ 1“’ .. Swill "f 04. b*‘a.ilblt.‘.'011 .1...‘UI.’1 ‘3. b34133 C.Lt.;,i‘.i-C U-‘Jl‘l .48 '-1. .J: 12-1.9 e "envoluos o<” to a state or’ contains the factor /fl< ’AO /o('r)/ wlere14) 13 the tot;l electric moment of the atom. Unis result .y, ‘ fi" 1‘ 13'. . .' ‘4. .. w . 1.. ‘ .- 3 gnu go» o; Algae, uhb me Joe 1 more ——-x ‘3"\ i 2 é-QW (10":de 7.1".1-‘31‘81 is the c214..r‘;;e of an electron, m its ‘istiilce from the :1 nucleus, one the SL‘JmatiJn over all the electrons 01. the :tom. from ('33) one (1239): [Jelfl‘r/zfl7 , /J:,07 fr—A, ) [32,0z [:0 (150) O -. ‘.‘Iit'l Sifi‘lilfll‘ trill-etc D :‘ ezxuc-Ltimw for X and 7, aurilere fit: the ‘— r w vn- ,-- ”a -, J‘-~ - 4‘ -- r. C .L— .' -fi. |-\ ‘8 \ ‘r-."' . total “Alwalrr nelieawm o-~. tie .:.-Duels. ..e u-oe one notwtion [1)i]~«[4’172/1’41/(//])47]f¥/):%/[ (”Lb") wilere for instance [1, J;/ is 13-10 Eoisoon br:'r;c,.:et of tire taro SCiilCtl‘S I 3.11:1 flx one}. '.hzre (r is L‘i'lit vector in the * direction. Using this notation: ‘l g A ’73 M \s V N N a N N N. '~‘ Hr. [JD 0/~ J -/J,o/—/o,J/-J~ (j'OXj‘OXfl‘J‘) (m ** 03%) 4233(5) :— 2(}“x'3~cf A) wixere the letter J7 denotes the unit dyadic 017‘ 232d order, amid " v c‘- - ¢ .- x \r‘ - *1- . "(v -. ‘ ‘ u..." 1 ~ f -. .’ \ ~“- ‘ - 7“ ”-11.6336 the SCCOAU. eqiuzllt / Of (lofi) idliO'JS L‘...LJ. new; 371‘..:I.1 (1..-;O } . .‘.} .:N 1:,j2725//=~2131 zfxzwmzwm«im— dam/f 2 \Z('2:(J'5.~5J')+(/al’?(-fi<\f) O =~2{~2.-. 2: ix (ix3~c2‘23)~z1(r3 “579/ $31,339 ‘73 _ eve use '. the identity ,7‘x (52.?) =ZZ-F_:¢«ZF~/3.2/? Q I \3 N \J .- to (277.1116. t-1e 03:11‘538231011 J‘ X (IT XZJ). not also (1.5 ‘. l 113(rg/135//=/J;KJZ~5r9/2 Jvzflasvuzjf' (1.2.) (/33) 03") -1111, froz‘. (3*) 12.110. 5—1“? Jy3“2J‘5J11-2571: 21‘0'2123 J94” JV?“ (its) ‘K '- , -‘,\ 1-. f ‘. ‘ ~ ~ ,.‘v--‘ , - 7.. ,~ .‘L ill Lie set of o.-::;ei‘l.wies 7 ; 3‘2 , ’7‘ let 7? racemes» v_ _‘ J.~ '~J- Li‘. .:., ' _‘ ‘ - ‘ "‘ ,..—-- _..¢- . . o L.‘ “,‘ 4.1‘ - O seen a set tiot use 1o: oi set is C: globe. ”cluLov use tJPL011 '. ‘.. 5‘ .“‘..,.A. ,.. .4- w 4.“.‘ . .._. A. 1‘ 4. element of (1 1n tun CuUrLLu to sgst.m o uulS con;lete set: Xm/f /jH) “Zf/f#r)f’ G’H)+Ir ‘Kj’rfl /{“f’K/0/o(’f’m) “NH/W7“)*7’69/)/6<7»~/5/«7:._I) , f3éf/ .41.. H. -, b ,\ 1,,4. r“ '1. 1‘3““ :0. -110 scenic r.c.-..ev 11-1 t1-e cit Lo. #7”) 7 /7/"’)/ ‘ff‘/)///]~) smile that on the rid is: ”7"” */’/7’~>/ 3 %//f’/’*/)‘/ 7793/ *Tence: /6/) 31', J. ECOt [ff/7H)—////ff) /]f<74~\/0/a(’/r mj—ta (133} 11106 001;!le 2110. 7 1'0 /" O £3..‘Ld/f(f:’,t i'J Jll‘b‘d 01";3'3t 01 //r/f/)~ “/ff/(y,) /f_f,)LC//f4\/5/wffl\): 0 ov a " v a - ~- 1' v? “v ‘0 (1‘ J) C L11 LO u V5- 2.. ._'--L.S 11113 7 1 ’br'. .CJIAJO;1CALU DJ. 3.1.: we . . ~ q. . _‘_U L‘.\ . .-,‘ .- ‘: ._‘ . ‘ ' 1 a / . ." ‘ 1101-5 FIFO Cain," ll. v.18 LOCOS“ ‘I<,L1;LulL'./‘b 01‘ ii //f 1/. 1-4.113 7 [7:0) 25/ (1.1:?) o .. _ o ___ . -flVOo‘ L410 3:11: ~ct -311 .mle Barf . \‘.Vu~ ‘ are real: 2&129/‘8 , 2:307 =E~S , b27812/J52) 021*“.07jto7 ft'flr :‘c' jzg‘BJfiz :‘18 (1 130n3"""‘-" the l‘liii‘l‘l : 2 ‘:1 *(ZJ) . .n. \ --.. -\1 ,\ J- . v ‘—- -- ‘- .‘ “I ‘--. (u ~‘L- , Q , ‘-. 1 '1 ‘11-: 1311; 1.111011 1:11.1'1; 0311301113111 0-. 11.11., 131—” e. 11.131011: 424(«f/h—x /9/.')§/’/o/ LE? /3 4’, ¢4”/0) ‘2 Oi BIL- 4. , ’\ U .¢ L1.) 7 r: .4—0 U (s, ,»- V ,1 ...J 1 O m ‘1 ~ .1. q oruer ch ~ N/M/s’). 644/2 “fl nw~~7cv.’l' ”fl 1 [Va ‘1‘ us". I D‘AAA.~..3 “‘.;..Asl‘ (D H D 0 d *1 }. ’.; (D H OJ (a y. , J :3: C} ‘ x t .1 '— (D ct I.” O L" h]. C; ‘.J :1; O (.1. r J (v "'; O '1 (3 (D O :J 4.3 - . :- . W ‘. ' .. 1.x .~. :3... 4.: = j ‘ 1‘" , t. . ~ u 4 usual mil 1. 10: ueluvln _, 9.x. 0 11.13.104 .10ch 0.114 ~..:.3n tile n .r‘ C“ 43V J'. : l. a.“ f . 9 ‘ K 1orco lo €§yTGSSl 13 as 1 1430u101 bus» 01 -3,7,?v 01» (1:7) ostul .tes of the til-301:? so as to cf 0 L O (D "3 co get an ogerator 7V in toe ecgation oi‘ motion {/4 (74/ :0. .5 4 he will now get "uch an operato: f0: ‘7 which reduces “ “' '7 Mil I111 ‘.w“"{ a} " ‘ 1‘3!) ‘1“"31 “1:131" fin y$0 321" ‘ . DO bile .-~. DJ- .11 Lu‘Llva. L) LI..LV ‘.U\‘-;v $k- bx) Lab) “..0. ‘which does at least satisfif the Cannouicul equatiCLs; it W111 thus be a very regs naolo ertension 0;: tgo tneorr. Jhe function :1- ,1 I 1 A ‘8 A Natl fies Lagran;ds BulfithXS, a tyjloul one of 1110n is; wow 1- -' IN ‘- . u V : ~ J“ ‘Ar‘~.~.vJ‘ .~‘ ‘ . ..‘ - . O. . I‘ but tuis last egoaolon 1b thSu mu you's 13¢ 01 30.10“ for a “ " ’: ,‘ v-o: .~ .' ‘ " ‘r L . (Q ~~ vv -«, “" r» ' n "r part1cla d1tu force (11/). 1» lo 30h Latnrul Lo ta13 ... 1.. . functlon />/= 441% WC)?— f/, ~§/4,)‘fy.~§»/, ””1 (129) a/x‘*;f‘ ’ ax «* z’ ‘- _ '. 4.. ,j.. 2 4.: .' g- . .- '. ‘ “on to latcocucc tJe fiffiOU¢3“lS o. electron len. -o “4“V‘v‘) L’ ‘3 ‘V'r fiY V?“,{)‘v‘:"fi1‘1'}‘ ' ‘.“1 :+' .:C' “ Qr‘r‘r."hr t .-"‘ru~<-s) '.;».L, .31L304.‘ 13.111 “'-;/.2; 1:01;“ o .23.; Cu, .L v 1-.) anu 9.3-3.3. y 0 (..o QLLLL that econ GlHCLTCJ 13 an QDuJ 13 -' 1t1AJ JUOdt an axls t1*o¢_1 ° 1. .... o. ,‘ 1‘.‘ . .-- r— -.4. rnfi‘»-sn-u 1".-. ‘.y“. Y‘Q 3,~.~« ..‘ .."y‘ J- ... "1"". - . 5,3,54- . p. ,‘ ya .. .. .. 1.. 3- A.AU--—‘¢.. .... H'L ..rJ w...‘ -L‘JJ. v A. \s-AL . A ' ¢. 4 . lo lo. ‘1 1' ‘ L’ C 10‘ L l J U]. Lublu C L ~ _ _' 1. J.‘.. .., . . .1 "v- , '. ,4. " _._- ° .‘ 4. ..n.:1v+-3~ dust UJLG Values —2 mm. t.;e 1 so swan. 11.131-9th 1.0:.ch t2. 0 , .. C‘ *— fivlc; \‘ +. “‘.w"‘\ .‘ ..‘..‘.. 1;) m. ‘5 '1‘ O 3‘ c C‘ .1 .. LVN-.4» l .513, 1:1 :L185J + c . *0 Ugv-L'J ~‘ICOOL’ALU 04‘ I)“ «It. 1U ll.) d‘vAK~AUI§-‘.Ju'_- 2;{‘;V’ 2444C O"‘ . V 1‘1“"“ '0'“: 1‘fl '- - ”Lore 0” lb 31 0c3e1u1o19, 1113c ~ '~ 1.- 1- V. . to can t1e term to (1;f} to jgt the total ”amiltonian, ‘ ‘ '= 0-” ~“V9 oolw -~ t...‘ ...-V \J -- L ,.J C“? I (D (W . 5 Ho p. t O I " (-3 ...-Luv szu a 30.31ulae ...-.Lu, 0 v1.0.4; ‘V _' Shem the totul dgmiltonian :or one 31:0trcn In a centrc1 11c1d is 2 //= ~;;7/, W t , , ~ Zmfl/ )ZW )fi/ ‘54.: ,cJZ/szuc) ‘7" c r .4» 44‘ _~ I \ 1 I t “12 C2; fi/g / z ) ‘JZ:£__ -£2&: (} - .* {LZ<2& or 4M 1‘ , or 7% 74H( 2 __ L ‘ 4 2 . “24.21% f/i fl) ”£2”: f‘:€—Z /%2¢0') M0 where 23113 the total charge of thc atom ca“c and (the t OHPOHOHt 0? rbitgl an ular Ochtts of he particle. ‘. -...L._-',‘.'O .. ' .( ‘\_ 3., .’.,_rs'v"* 33.0 .):;1‘o....‘o3.u10 .; in” 05.1,..331'. u. 11.00.1111 11:210. .. .5. -J ‘A n a l . I 1-. ,h',"" .h F‘ I J on LlAJle alccvrcn ix 1 cent‘gl 11311 11 cc 4 3:-“ \1'1) DO ’11 - K4 \‘01 [(2 f1“ ) ~ "I. .... .1 4. L“A_‘ 1‘ ._ H -_I '.‘u ‘1 a .38 let " . ..83 1’01 3; 0123.1 3.1., 111 $3.10.. 8. 3.1.41.9. .‘ . 1U ls: - 3’ :7cgfiszoz):; it (1%) /? fiKLfi) (1.3.2) 1. 32/»... ) S‘Zt‘ “f"; p j: :11 f5: ‘.;-3 hive, since <1"). 3} =3, SK 8.72? S. ($33,333) = J. (5.1. +3 1+5: 3. )Ms; 3 Wig); + (52 32"st) >177 2 I 1 1 ’\ 1.3:) 31311111”th J~3/1[)/77;\)3J7] ( ) m N u L) },=3,I‘_J,S., :3 L -1 s :L r’:Lth‘Lx S r1;Lx$\L53 ‘I Jorra 131301151111: .3 o *F‘H)pre 1132,23 fr 02:1 2,1.) : I ‘5 o '1 c" \ /\ ~ 2 z <2 ._ N —\ C .7 "p (1,. . 3“ 1”?‘ (\n E-r +‘lf _:\._ 1«.\ 1- ~= qr 31 3‘ 2 313‘- 2' ifilL-d , 1....) 1.1. $1.1 I, UAl—ho... U-~UAJ‘3 nu‘s. I‘A-Ju ‘.;..L'434 1.. ‘ .... q .._ _q ' ‘ #9,... - f, -? - x ': ‘ ‘ 3 A" 1 -"v all]. L29 1123 .. -Zd1‘0 Lug ”9.1.3 .111 fiffif: O I .7: / o :50..- (-L-JL); Lt LOLluuS ’-‘-~ -~ '-~ 7"1‘“ '- if? A'r-l‘ ./ :Ow-Tv. :9 L #0 b.1113 {.1 .1011-2'31 0 01.21.3311, 1:3 [lb/O ”div 1d- / 11...“: 1.1. —f o ' ‘ . - 4-.‘- ., A~ .r-.. 2" J... --.\ L : '- ,-l " ,1'.‘ 1‘ .5‘3 6 -11 b-u‘lLOuI' llUll‘- OJ. v.19 lA'Jv 91“}. 01.1“ LB‘LUU («74. //’., /"/f/”“’) I («fa/”#279,. ) (1.3-) are nor-zero azal‘f’:f ‘ . . -:"01' if ff:/ , tun“. frogs; l/{f/rd/7=Aflw7m‘it 313110223 4/“, = 44“; = 4M» :0 . r s 1 . . a '\ "1‘3 non-2321‘s 0-17.7 1.? 7 -/ ”~l; 3.111: 5111.123 1., J. J .11 nor/'3'”?- 1. . P -‘ 31., J-‘- — . 4— 1-- “I . —. . *r L‘. . ‘ .. . 2- a '1" j ' . ‘ I f‘ .‘> , “ . .3 n N .' f n . ‘v ‘ . ‘9 .- .. ...-LA]. J , 0-1.1.2] {1-50.23 -.. ,ui. J..- 0—2.... ‘41.t~‘ J..- 3 ‘ .:Cu‘il'J, 3‘ ..l O; ( .. ‘ .71 ‘1”‘1‘1 ...-..V... """»T‘ ....... _. +l .:.. n m. .. ,3 .'.1 ""1" .o If“; e l-u-.-..4‘v°.. J .LL .5 ‘:-.L\.n .‘. ; cf: ~ ._,.LIJ.CG '.;: C41. :51 ‘L; L.¢ VLyJ-LLe 0.1.. -21.)- , .4/ ... perturbln; 911011;;3‘ fivj’ /’ )7] “_.‘_ ‘ “G _n .- 1. 11....) 110 63;.-.00u L. ~ _. : - ~u ,‘ - - ‘- “,, '.s - .\ . -' " '~ ,r< .—-V f 1 on tud fLTLt oruer JeerTUJtLUdu of 34p chur . ldelb. ~ .. '.ar ‘ ‘. ,, - -- n‘ .J- ~-.---'-<‘~.-*- -'~ fleblcctlgg tilJ tJrL, tAe Llruu orWQr ::£u;‘uutluu lg: 2 I 1. JW£{/,JJ~ 7+? 2W%1( 2 4r: JMOSG eigenvalues "re: 2mc f/f/r) O (- Q’A‘-“‘—...l .r‘ f I 1 Av ...v .-5, (.:.; 2411 S 5 fgctor. the curly br:zcget 1:233 Luanda /'/4)~4’/2$() fS(/%’4) 7 / .fi J-‘ . - . an“. . ’ .w‘. bike 33.33:.J9105‘. 111 (157) ~ - I' \ ‘ i «W‘ q.“ 4-: " ..1. ..‘ - . I- . g . f.‘ 03 pa;e 5‘/ occur: a table 0* uxa LCGMQu PJtUUfflo of . -“‘ - .* t 'V 1 J- 0‘ . . : - ‘\ . ‘ r. - . .... - "~-'— 311 tAe co lflfl.lldeh Lue uO tAuJSICLOflo bequen COuDibb 0‘ 1,: ~ p - " ., — - - ‘.r .“. -: "3‘ ~ v 1.3-" '- w‘v-‘ Tm 3L, en Lrlglct lnvcla. It Lu 0Lu3*hc¢ Lg Ll‘ocu bun- , LPJ. . 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