another way to solve eigenfunction/eigenvalue equations allows solution without any prior knowledge of answer

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1 Lenng gos Chpte 3. Mt Quntum Mehns nothe w to sove egenfunton/egenvue equtons e.g. H^ = E ows souton wthout n po knowedge of nswe b f most popu nume ppoh In mt geb we n sove n egenfunton/egenvue pobem b dgonng mt! Empe: We n wte the sstem of equtons = = = s mt equton 3

2 o A = If we wnt to know the souton.e. vues fo we hve to tnsfom the mt A nto dgon mt B nd the esut veto nto : b b b 33 Wht does ths hve to do wth QM? We n onstut n of numbes jk usng  jk = jk = <u j  u k > Mt A s the mt epesentton of the opeto Â. Mt A enodes the opeton of  on the bss funtons u j.

3 Menng of mt eements: <u j  u j > = <Â> fo sstem n stte u j > <u j  u k > desbes how two sttes u j > nd u k > e ouped b opeto  If the u j e egenfuntons of  the mt s dgon.e. eements jk = fo jk. Gven: Stte funton > bss {u k } opeto  Needed: Fnd egenfuntons of  nd oespondng egenvues Gven: Stte funton > bss {u k } opeto  Needed: Fnd egenfuntons of  nd oespondng egenvues 33

4 34 Stteg:. Set up mt A wth A jk = jk = <u j  u k >. Epnd the geneed wve funton of the sstem n the bss {u j }: > = u > + u > + 3 u 3 >... = j j u j. The j epesent the oumn veto =  > s equvent to A: d d.e.  > = d u > + d u > +... = j d j u j Poof: d j = <u j  > = <u j  k k u k > = k <u j  u k > k = k jk k

5 35 4. We seek spe soutons to the mt equton A = these j epesent the spe ne ombntons of u j > tht ed the egenfuntons of Â. The est on fo spe whh e the oespondng egenvues. 5. Fom wte thngs dffeent: A- = Note: ths equton hs the fom M = M mt oumn veto

6 f M - ests then M - M = = = tv souton fo n non-tv souton M - does not est! detm = deta - = 6. Set of equtons fomed b det A- = n-th ode eds n soutons fo =egenvues n 7. Now tht we hve egenvues we n get the egenvetos j puggng the egenvues nto A j = j j The egenvetos j epesent egenfuntons wth egenvues j. 8. Constut mt S tht ontn the egenvetos k of A s oumns: s jk = j k 36

7 37 AS = A S jk = m jm s mk = m jm m k Beuse the j e egenvetos of A: jm m k = k j k = j k k A S = S S - A S = Bud unt mt ontnng the egenvetos of A tht tnsfoms A nto dgon mt ontnng the egenvues j

8 38 Empe Let s ssume tht ou stte spe s spnned b on two bss funtons e.g. spns nd the mt epesentton of  s gven b j u k A u Note: A s Hemtn but not dgon so u nd u e not EF s of Â. A deta- = - = - =! = ± Egenvues e sne A s Hemtn! SKIPPED Look fo egenfuntons: =-: = - = -

9 39 >= u > - u > u u =+: = = Nomton: u u Chek:. Othonom?

10 4... OK. Do the egenvetos fom unt mt? det S S S S S S S... OK 3. Does S tnsfom A nto dgon mt wth the egenvues? O S S... OK END SKIPPED END Letue /5/

11 Lst hou: Mtes n QM Go: Use mt geb to sove EF/EV equtons Most ommon omputton ppoh n QM ode We n onstut n of numbes jk usng jk = <u j  u k > Mt A s the mt epesentton of the opeto Â. Mt A enodes the opeton of  on the bss funtons u j. Menng of mt eements: <u j  u j > = <Â> fo sstem n stte u j > <u j  u k > desbes how two sttes u j > nd u k > e ouped b opeto  If the u j e egenfuntons of  the mt s dgon.e. eements jk = fo jk. Set up mt A wth A jk = jk = <u j  u k > Sove det A- = to obtn the EVs Pug EVs nto A j = j j to get egenvetos whh epesent the EFs of  4

12 4 In ou empe ou mt ws nd hs EVs = ±. tnsfomed mt s S - A S = = Note: T =T = ndep. of tnsfomton! det =det =- ndep. of tnsfomton If mt s dgon then opetng on one of the bss sttes w esut n nothe pue bss stte. If mt hs offdgon eements bss sttes w beome med b opetng.

13 43 Lenng gos Mt QM 4. The -Stte-Sstem SS The SS s ve mpotnt sstem n quntum mehns one of the smpest whh ehbts nteestng quntum mehn behvo. It s one of the few sstems whh we n most ws sove et. It so hs mn ppent qute dffeent phs pptons. Suppose we hve Hmtonn * H To fnd the egenvues of the Hmtonn.e. the eneg egenvues we sove the seu equton * * nd tte geb gves us the foowng ve usefu fomu we woth emembeng:

14 EVs e vege of the two eneg eves nd pus spttng tem whh s the sum of the sques of the vne n the dgon tem nd of the nom of the off-dgon tem. No unves EFs but smpe nswes n be onstuted. To smpf epe the ompe b e e Sove Ĥ Use pmeteton whh utomt gves nomed EFs s ed mng nge os sn e sn e os uppe EV equton beomes H os sn e Souton: tn Note tht the mount of mng between the two sttes depends on the dffeene of the eneges s we s on. Wte 44

15 45 * H st pt: vege eneg; nd pt: devton fom vege Let s = / Then tn o tn tn tn Lmtng ses: <<: Epnd oot of eneg EVs... ; Sne / s sm: + nd Note tht the gp ves s.

16 b >> : The two eneges e just the vege of the two dgon tems pus o mnus the off-dgon tem: Sne tn = / the mng nge goes to ± /4 fo / nfnt 46

17 Apptons te moe n depth: spn Zeemn effet voded ossngs of potent eneg sufes tme-dependent eve poputons dung n etton... 47

18 Lenng gos Chpte 4 5. Angu Momentum Css: L F v p L p m m dl dt p p F T Fo ent feds. e. F : T = F = L = onst END Letue /7/

19 49 Lst hou: The -Leve-Sstem Hmtonn * H = e e Consde s ogn eneg eves beongng to the ogn bss vetos now shfted b oupng mt eements * Use pmeteton whh utomt gves nomed EFs s ed mng nge os sn sn os e e Menng of mng nge: otte oodnte sstem of the ogn bss b to get to the new oodnte sstem Souton: tn <<: ;

20 Note tht the gp ves s. b >> : Sne tn = / ± /4 fo / ±nfnt Angu momentum L p m m dl dt p p F T Fo ent feds. e. F : T = F = L = onst. 5

21 5 In Quntum Mehns: e e e e e e e e e e e e p Wthout poof: A ngu momentum opetos e Hemtn. ] [ nd pemuttons ] [ Tnsfom to sphe oodntes : = sn os = + + /

22 5 = sn sn os = os tn SKIP Tnsfom ngu momentum opetos : os / sn sn os sn 3 3 /

23 53 sn os sn sn...aaargh! sn ot os os ot sn sn ot Tht ooks hob ompted! Wh bothe? In these opetos e funtons of thee oodntes But: In the e funtons of on nd! septon of Shödnge equton nto d nd ngu pts! Egenfuntons of Angu Momentum Opetos

24 54 j j h h g g f f Cn n of these EF s be the sme? ] [ ] [ ] [ ] [ nd pemuttons n Cke queston Wh : Phs ptue: If we know n two of the omponents of nd we know the pne of moton pese detemnton of poston pne eo momentum pne voton of HUP n fnd ommon EF s fo nd o but the nnot hve the sme EF s Cke queston: Wht s spe bout the -s? We bt hoose nd to hve the sme EF beuse

25 s the most smpe opeto. OTHERWISE THERE IS NOTHING SPECIAL ABOUT THE Z AXIS!!! 55

26 56 Souton of the EV/EF Pobem: Sphe Hmons: Let Y be the EF s of nd by Y Y Y Septon of vbes: Y = Q The es one fst: Q Q Q Y Y Dvde both sdes b Y / Ae d d A bt onstnt In ode fo to hve peodt of.e. = + = mħ m ntege EV of = A e m

27 Phs ptue: the ngu momentum ong the -s s qunted nto ntege mutpes of ħ. It s onvenent to nome eh funton septe. uns fom to so d nd theefoe A = -/ nd m e EF of End Letue /9/

28 58 Lst hou: Angu momentum Angu momentum opetos: sn ot os os ot sn sn ot Wthout poof: A ngu momentum opetos e Hemtn. Commutto etons: ] [ nd pemuttons ] [ nd o n hve ommon EFs but the nnot hve the sme EF s. We bt hoose nd to hve the sme EF beuse s the most smpe opeto.

29 To sove the EV/EF pobem use septon of vbes: Y = Q Fo : m e EF of wth EV mħ 59

30 6 Now we sove the seond equton: m m e bq e Q Y n sphe oodntes Q b Q m Q Q e bq e Q m m sn ot sn ot An ppoh sm to tht of the H.O. bngs us to soutons nvovng the so-ed ssoted Legende ponoms P m os. These ponoms e defned s m m m m d d P os os os! os / = 3... ; m In tot the egenfuntons of nd e: m m m e P m m Y Y os!! 4 / The e ed sphe hmons.

31 The egenvues e mħ fo nd ħ + fo. =: Y s-obt 4 =: Y 3 sn e 8 Y 3 os SKIPPED =: 5 Y sn 3 e Y 5 8 sn os e Show sde 6

32 End Skpped Note:. In the bsene of eten feds eh vue of hs + degenete subeves oespondng to dffeent m vues. Cke queston: How do the Y m ete to the obts ou know?. Lne ombntons of these degenete eves ed known foms of pdf... obts. 3. These ne ombntons e EF s of wth EV s ħ + but not of. Chemsts ke e obts Phssts ke EF s of. 4. The obt ngu momentum devbe fom p hs EV s ħ + =3... In moe gene ptue of ngu momentum EV s e ħ jj+ whee j n be hf-nteges. 6

33 63 Ldde Opetos Apped to Angu Momentum Sm to H.O. but ths tme sng nd oweng m the pojeton quntum numbe of the ngu momentum onto the s. Shows how m = -... n steps of Stt wth ommutto etons: ] [ ] [ nd pemuttons Ĵ s n bt ngu momentum n nude e.g. spn et. Ths s suffent to deve tht we got fom pevous tetment the ommutto etons defne ngu momentum Defne: sm to  ± wth espet to ^p^ Tsk: Look fo ommon EF s of nd Ĵ usng on opeto geb! Fst get some ommutto etons: ] [

34 64 Sm ] [ Aso: ] [ ] [ ] [ Lkewse ] [ ] [ o Fn: ] [ ] [ ] [ Consde Y Y nd Y b Y b dmensoness Wht s the effet of Ĵ on the EF s of nd Ĵ? Y Y Y Y

35 Y e EFs of wth the sme EV: ħ. Y Y b Y b Y Y e EFs of Ĵ wth EVs b±ħ Contnue ths poess: Ldde of EV s b+h b+h bh b-h b-h Cke Queston: Wht woud be menngfu esut of - jm>? END Letue --- //

36 Lst hou: EFs of nd e sphe hmons Y m wth EVs mħ fo nd ħ + fo. In the bsene of eten feds eh vue of hs + degenete subeves oespondng to dffeent m vues. Lne ombntons of these degenete eves ed known foms of pdf... obts. In gene ngu momentum quntum numbes n be hf-ntege Ldde Opetos Apped to Angu Momentum [ ] ] [ [ ] 66

37 Note:. Thee e mn EF s of Ĵ wth EV s septed b nħ n ntege quntton of Ĵ. The Ĵ se/owe the EF s of stte w.. to Ĵ but the don t hnge the EV wth espet to. The gve se to set of sttes wth the sme mgntude of the ngu momentum but dde of dffeent -pojetons. Rnge of EV s of Ĵ : Y Y b Y s so EF of wth EV -b ħ -b Y beuse -pojeton nnot be gete thn ength of veto b.e. b s bounded bove nd beow. 67

38 Consde EF of Ĵ wth EV b m : Ym Ym we nnot se Y m! ħ b m ħ ħ b m Y m = = b m + b m = b m b m + Sm: = b mn b mn - b m b m + = = b mn b mn - b m = -b mn = j = h[jj+] Summ: mh Note:. EV s of e j j+ ħ. EV s of Ĵ e m ħ m = -j... j n steps of 3. j n be nteges 3... o hf-nteges / 3/ 5/...: e.g. : j = 3/ m = -3/ -½ ½ 3/ ows fo the estene of hf-ntege ngu moment e.g. spn! 68

39 4. EFs e beed wth the quntum numbes: nd m> o jm> fo nd Ĵ. 5. In sstem desbed b jm> mesuement of o n on ed jj+ħ nd mħ espetve. 6. m < j j m m = j; m mn = -j 7. -s pojeton ws < mgntude of 8. n neve pont et ong -s 9. thee w ws be unetnt n the obt pne of moton but fo ge j ss mt m m j j so n get ose to -s. 69

40 Lenng gos Chpte 5 6. The Hdogen Atom the on tom wth nt souton foms phs bss fo undestndng ge toms nd moeues fu QM tetment to ompe wth smpe Boh mode Potent: V = Couomb potent = e /4 Hmtonn: In Ctesn oodntes: H t 4 Ze ese n sphe oodntes: H L Ze sph 4 A ngu dependene n s n L obt ngu momentum. Septon of vbes: We ook fo soutons of the fom = R Y m 7

41 7 We ssume tht the ngu dependene n be desbed wth Y m beuse must be EF of L In TISE: 4 m m m m Y R E Y R Ze Y R L Y R Re: L Y m = ħ +Y m Use EV s of L nd dvde b Y m : Rd S.E. The d Shödnge equton tkes the fom 4 ER R Ze R New effetve potent Ze V eff 4 Cke queston: Wht w be the effet of noneo ngu momentum: A eeton w hve ge pobbt t the nueus

42 B eeton hs eteme ow pobbt to ppoh nueus V eff hb +/ -e / Effets of ngu momentum:. mkes t hd to ppoh : ngu momentum be. Beuse V eff hs be fo > ntpte R to be qute dffeent fo = nd > End Letue /8/

43 Lst hou: EV s of e j j+ ħ EV s of Ĵ e m ħ m = -j... j n steps of j n be nteges 3... o hf-nteges / 3/ 5/... m < j j n neve pont et ong -s H L Ze sph 4 Soutons hve the fom = R Y m Rd Shödnge eq.: R Ze R 4 ER New effetve potent V eff Ze 4 73

44 74 Soutons of the d Shödnge equton:!] [! 4 / n n n e n L n n n n R n L n = ssoted Lguee ponoms n k n n k n k L!!!! At sm : R n fo At ge : e - R n fo Ostng n between fo n> Some d funtons: We hve 3 quntum numbes: n m n = 3... pnp quntum #

45 =...n- obt ngu momentum quntum # m = pojeton quntum # But wthout eten feds on n detemnes the eneg: E nm e 4 En ndependent of m sme s n Boh s 4 n mode Degene of eh eneg stte: g n n n n n n n n n n n m Fo eh n thee e n soutons n fod degene How mn bound sttes? E n -/n nfnte numbe of bound sttes! Due to popet of the Couomb potent: t goes ve sow to eo! Note: 75

46 Athough these wve funtons e on et fo H-ke toms we w use them n ge toms nd moeues b owng Z to v wth she Htee-Fok Fo hgh eted sttes e - s f w fom oe ooks ke H tom n hgh n stte Rdbeg stte E n Rdbeg -/n- = quntum defet depends on tom nd on SKIPPED see Homewok! Wh s estted to n-? Knet eneg n ngu moton: T ng T ng e 4 n 4 4 = <T tot > fo m T ng T tot H 4 e n 76

47 e 4 n e n n n = 3...n End Skpped Pobbt denst n voume eement dv: Css: = = m fnte pobbt t = no pobbt t = P d d d = d d d dv = d d d = d sn d d P dv = R n Y m d sn d d Cke Queston How does the pobbt denst P n d R n d ook ke t the nueus? 77

48 78 A Thee s mmum t the nueus fo s-eetons but not fo > eetons pdf et. B Thee s mmum t the nueus fo obts C The pobbt denst s eo t the nueus. Rd pobbt denst ntegted ove nges : d R d P d R Y d d d R d Y R d d n n n m n m n sn sn sne Y nomed to

49 Obts e epesentton of pobbt denst: thee s etn pobbt to fnd the pte nsde the ontous of obts END Letue //

50 Lst hou: H-tom Soutons of the d Shödnge equton: R n n 4 4 n 3 n! 3 [ n!] / L n n e n L n n = ssoted Lguee ponoms L k n n! k k n!! n! At sm : R n fo nd > At ge : e - R n fo Ostng n between fo n>; numbe of d nodes: n-- Wthout eten feds on n detemnes the eneg: E nm e 4 En 4 n ndependent of m Degene of eh eneg stte n nfnte mn bound sttes Fo hgh eted sttes Rdbeg sttes E n Rdbeg -/n- = quntum defet depends on tom nd on 8

51 Fo toms wth hghe Z nd mutpe eetons nd fo moeues: no moe nt sovbe ppomte methods needed vton theo nd petubton theo 8