Quantum Mechanics Homework Solutions. 1.5 (page 16) Show that the z component of angular momentum for a point particle

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1 Qutu Mechcs Hoework Solutos.5 (pge 6) Show tht the z copoet of gulr oetu for pot prtcle Lz p p whe epressed sphercl coordtes becoes Lz p r s. rcoss rss r(cos s ) r( s ) s r cos cos r(s s ) r(cos ) s r s cos p p r cos s r(s s ) r(cos ) s r s cos r r s s r(cos s ) r( s ) s r cos cos (cos ) s (s ) s r s pr L.7 (p 8) Show tht the eerg of free prtcle be wrtte H r where L r p. Ht: Use the vector relto L r p r p r p defto p r p r. r together wth the Strt wth the Hlto for free prtcle sphercl coordtes: p ˆ r r p r p r p rpr r p H v L L r p r p r p r p r pr p p r r L p p r r pr L H r.8 (p 8) Show tht gulr oetu of free prtcle obes the relto p L L L Lz p. s of 3

2 Qutu Mechcs Hoework Solutos pr L H (fro lst proble) r pr p p H (fro book) r r s L p p p subtrct L p r r r s s. (p. 6) Use the epresso e cos s to derve the followg reltos: e cos s e e cos s cos s e cos s e e cos s cos s e cos s cos s ) cos cos cos s s (fro rel prt ) b) s cos s s cos (fro gr prt ) c) s s s cos (fro I +I ) d) cos cos cos cos (fro Re +Re ) e) cos cos s s (fro Re -Re ) f) cos cos (fro Re ) g) s cos (fro f wth dett s +cos =) h) e e s / / e e e cos s s cos s cos s cos s cos s e s cos s s e ) e e e e e e cos e e e e e e e e e e cos s s cos of 3

3 Qutu Mechcs Hoework Solutos j) Re z z z z e e e cos s cos s cos Re z k) I z z z e e cos s cos s s I z l) e z e z ep(cos s ) ep(cos s ) ep( e ) e ) ep z ep Re z z z z z zz Re z e e e e e e ) e, e, e,... e e cos 4 where,,,... s s s s s 5 9 cos cos cos cos 4 e e e debrogle p E 4.3 (pge 5) Show tht the de Brogle wvelegth of electro of ketc eerg E (ev) s e c d tht of proto s p c. E E h debrogle p p evs3 s h p h E h h c E E where rest ss ev evs3 cs evs3 cs E z 7.7 c e E 6.5 ev 9.86 c ev E E E 3 of 3

4 Qutu Mechcs Hoework Solutos Gve proof tht the Hlto d the ler oetu opertors for free prtcle hve coo egefuctos. Ths proof shows tht the cotrr ssupto leds to cotrdcto. Let be the egefucto of the Hlto opertor d be the egefucto of the ler oetu opertor. Opertor Egevlus for Free Prtcle Egevlue equto Soluto Egefucto Hlto Ler Moetu Hˆ pˆ k E p k Hˆ E pˆ p k k k k k k( k) k Ae k Be k Ce De k k Cobg the egefuctos both = -k : costt Show the costt s equl to, b substtutg the egefuctos d ther dervtves: kace k kbde k 4 of 3 kad kbc Itegrte to get l( ) l( ) costt. Ths s costt. Therefore d represet the se stte vector ccordg to the sttstcl terpretto of the wvefucto.

5 Qutu Mechcs Hoework Solutos 3.6 Estblsh the followg propertes of (): () () = (-) ()= whe d whe =, =- therefore () = (-) Note: Ths s lso see fro the setr the delt fucto plot. Ths s specl cse of prt (d), whch tur s specl cse of prt (). I prt (d), settg = -, () = (-) = - () = () (b) () = -'(-) () = -'() () = '() () = '() () = '() d d df f ( )( ) '( ) d ( f ) d ( f) d ( f ) d f ( ) d d d d therefore: () = '() d () = -'() ( c) () = whe =, () = whe, () =, so () = (d) ()= - () let = the d=d d d d Sce () = (-) f <, d 5 of 3 d

6 Qutu Mechcs Hoework Solutos (e) ( - )= - [(-) + (+)] For the LHS, let = - = (-)(+). = t two plces d d = d Note: fctor of fro the two roots. d d d For the RHS: d d d d d d d (f ) b d b let f() = (-) the use f b bd f ( b b ) (g) f()(-) = f()(-) f d f f ( ) f ( ) d d therefore f f d f (b) 6 of 3

7 Qutu Mechcs Hoework Solutos (h) () = -() f d d d d d f d f d f d f d Ths estblshes df d 3.7 Show tht the followg re vld represettos of (): () e k d use the Fourer Trsfor defto of the drc fucto substtute k = f (b) l s d s d s d f ; f ; - f so s d for d 7 of 3

8 Qutu Mechcs Hoework Solutos 3. Clculte the ucertt Δp for prtcle the stte ψ gve b (3.37). Do ou fd our swer to be cosstet wth the ucertt prcple? (I ths proble oe ust clculte ˆp. The opertor: pˆ.) ˆ p p p ˆ p p p Usg equtos 3.37, 3.38, d 3.39 ( ) p (, t) Aep ep ep( t) 4 A troduce du vrbles d ( ) ( ) Doe book, equto (3.44) ˆ p p d d A p e d p A e d p A p Now fd <p > 8 of 3

9 Qutu Mechcs Hoework Solutos p pp ˆˆ d d ( ) p d ( ) p Aep ep ep( t) d 4 ( ) p ep( t) Aep ep 4 d ( ) p ( ) p Aep( t) ep ep d 4 d ( ) p d ( ) p ( ) p d ( ) p d ( ) p ( ) A ep d ( ) p ( ) p ( ) A ep d 3 p A 3 p A p 4 p 4 p p p p p p 4 p p p p p 4 p Ths s cosstet wth Heseberg s ucertt prcple. 9 of 3

10 Qutu Mechcs Hoework Solutos 4. Wht re the eerg egefuctos d egevlues for the oe-desol bo proble descrbed bove f the eds of the bo re t / d +/? [Check ou swer wth (6.).] ˆ for or H ˆ for - H boudr codtos fro regos outsde the well re: Hˆ E As k B cos k As k B cos k - - As k B cos k As k B cos k repetg As k B cos k - - As k B cos k As k B cos k where k E k k s k s k s k whe,4,6,... cos k cos k cos k whe,3,5,... k E E k whe,3,5,... of 3

11 Qutu Mechcs Hoework Solutos Fd the pltude of the eerg egefucto b orlzto / / / / / / B d B cos d B cos d B The soluto looks lke ths: / / / d B cos d B cos d / / / tegrl over cos s zero, tegrte B The eerg egefuctos re: / / d cos where,3,5,... slrl, b orlzg to get A, s wth (4.3) book: s where,4,6,... The eerg egevlues re: E k of 3

12 Qutu Mechcs Hoework Solutos ˆ ˆ 4.(b) Show tht AB B A Defto of hert djot O of O: ˆ O Oˆ ˆ ˆ ˆ AB ˆ ˆ AB ˆ ˆ ˆ Show tht AB ˆ ˆ Bˆ Aˆ to prove tht AB ˆ ˆ Bˆ Aˆ I Drc Notto: I Itegrl Notto: AB ˆ ˆ d ˆ A Bˆ ˆ ˆ BA AB ˆ ˆ d ˆ A Bˆ d ˆ ˆ BA d 4.4. If  s Hert, show tht A s rel; tht s, show tht A A. AB Hert opertors: If  s Hert, the A ˆ A ˆ. Ths proof uses, fro the defto of er products: A ˆ d ˆ d ( ˆ ) d ( ˆ A A ) d A A ˆ ˆ ˆ ( A) d sce A A A Aother proof: Strt wth the defto of Hert opertor: O ˆ ( ) ( ) ( ) ˆ d O d O( ) O ˆ ( ) ( ) ( ) ˆ d O d O ( ) B A d the propert of of 3

13 Qutu Mechcs Hoework Solutos 4.7. Cosder the opertor Ĉ, ˆ ( ) C ( ) ( ) Is Ĉ Hert? Ths proof shows tht the cotrr ssupto leds to cotrdcto. Strt wth defto of Hert opertor. A opertor s Hert f d ol f: ( ) ˆ ( ) d O d ( ) Oˆ ( ) ˆ C d ( ) ( ) d ( ) ( ) ˆ C d ( ) ( ) d ( ) ( ) d ( ) ( ) sce d ( ) ( ) d ( ) ( ) ˆ ( ) ( ) ( ) ˆ d C d C ( ) Cˆ s ot Hert ( b) Wht re the egefuctos of Ĉ? ( c) Wht re the egevlues of Ĉ? ˆ ( ) ( ) C Cˆ ( ) c( ) c( ) ( ) ˆ ˆ ˆ ˆ C ( ) CC( ) C ( ) ( ) ˆ ( ) c ( ) C c c sce c( ) ( ) f ( ) Re( ), the c f ( ) I( ), the c where cople fucto Hlbert Spce. 3 of 3

14 Qutu Mechcs Hoework Solutos (I Clss) Prove tht the egefuctos of Hert opertor re orthogol to ech other. Show tht the sclr product of two dfferet egefuctos s zero. Strt wth the defto of Hert opertor: ˆ ( ) ( ) ( ) ˆ d O d O( ) therefore Ad the egevlue equtos: ˆ ( ) ( ) ( ) ˆ d O d O( ) ˆ ( ) ( ) ( ) ˆ d O d O( ) ˆ ˆ O O d ( ) ( ) d ( ) ( ) d ( ) O ( ) d O ( ) ( ) d ( ) O ( ) [ O O ] d ( ) ( ) [ O O ] d ( ) ( ) ˆ ( ) ( ) ( ) ˆ d O d O( ) d O ( ) ( ) Oˆ ( ) O ( ) d Oˆ ( ) O ( ) 4 of 3

15 Qutu Mechcs Hoework Solutos 5.. If Aˆ, Bˆ, d Cˆ re three dstct opertors, show tht: () ˆ ˆ, ˆ ˆ, ˆ ˆ, ˆ A B C A C B C ˆ ˆ, ˆ ˆ ˆ ˆ ˆ ˆ ˆ A B C A B C C A B AC ˆ ˆ BC ˆ ˆ CA ˆ ˆ CB ˆ ˆ AC ˆˆCA ˆ ˆ BC ˆ ˆ CB ˆ ˆ ˆ, ˆ ˆ, ˆ A C B C ( b) ˆ ˆ, ˆ ˆ ˆ, ˆ ˆ, ˆ ˆ AB C A B C A C B Aˆ ˆ, ˆ ˆ, ˆ ˆ ˆ ( ˆ ˆ ˆ ˆ ) ( ˆ ˆ ˆ ˆ ) ˆ B C A C B A BC CB AC CA B AB ˆ ˆ Cˆ ACB ˆ ˆ ˆ ACB ˆ ˆ ˆ CAB ˆ ˆ ˆ ABC ˆ ˆ ˆ CAB ˆ ˆ ˆ ˆ ˆ, ˆ AB C 5.3. If  d ˆB re both Hert, show tht AB ˆ ˆ s Hert f ˆ, ˆ AB. Usg AB ˆ ˆ = ˆBA ˆ becuse the coute. ˆ ˆ ˆ ˆ If A A d B B : ˆ ˆ ˆ ABˆ Bˆ A BA ˆ If AB ˆ ˆ BA ˆ ˆ AB ˆ ˆ AB ˆ ˆ AB ˆ ˆ s Hert f AB ˆ ˆ BA ˆ ˆ whch s true f ˆ, ˆ A B. 5 of 3

16 Qutu Mechcs Hoework Solutos 6.6 ( ) Show tht ˆ tcoutes wth the oetu opertor pˆ. Ths s, show tht: ˆ, pˆ ˆ pˆ pˆ ˆ Fro the defto of the Prt opertor: ( ) ( ) Show tht: ˆ pˆ pˆ ˆ ˆ pˆ ˆ pˆ ˆ ˆ ˆ ˆ pˆ ( ) ( ) ˆ ( ) ˆ ( ) ( ) pˆ ˆ ( ) ˆ ( ) ˆ pˆ ( ) pˆ ˆ ( ) d ˆ pˆ pˆ ˆ ˆ ( b) Use our swer to prt ( ) to show tht ˆ coutes wth the ketc eerg ˆ opertor ˆ p T. The fctor / s ot portt, just show tht ˆ, pˆ ˆ ˆ ˆ, pˆ pp ˆ ˆ pp ˆ ˆ ˆ pp ˆ ˆ pˆ ( pˆ ˆ) ˆ pˆ pˆ pˆ ˆ pˆ ( pˆ ˆ) pˆ pˆ ˆ pˆ pˆ ˆ pˆ pˆ ˆ pˆ 6 of 3

17 Qutu Mechcs Hoework Solutos Prove the Cuch-Schwrtz Ieqult (fro Dr. She s otes ) vector legth d d d d d d d Here s the trck: = d = Substtute: Prove the Robertso-Schrödger relto: Aˆ d Bˆ re Heret opertors. ˆ ˆ ˆ A, B C Prove tht A B 4 C A A B B B A B A B A 4 4 B A B A B A B A 4 4 B A A B 4 AB BA 4 ( AB BA) 4 A, B 4 C 4 A B A, B A A B B C 7 of 3

18 Qutu Mechcs Hoework Solutos 5.8 (p 43) The TDSE perts the detfcto E t. Usg ths detfcto together wth the rule (5.95), gve forl dervto of the ucertt relto Et. Note tht the sttor stte (egestte H), E. The plcto for ths cse s tht sttor stte lst deftel. Fro the Robertso-Schrodger Equto: E t 4 E, t Et 4 E, t t t t t E, tg tg t g g t g t g g t t t t E, t E t 6... (pge 65) Fd ( t, ) d PE t t, relevt to prtcle oedesol bo wth wlls t (,) for ech of the followg sttes., A s 3 cos () Fd orlzto fctor: d A d A, s 3 cos (fro wolfr tegrtor) 4 A Now fd te depedece: Ht, t e, Ht, t e,,, t e c c Ht Ht Ht E t, t c e c e c e, c c c d, 8 of 3

19 Qutu Mechcs Hoework Solutos s 3 cos 4 4 c d 4 4 E Et, t e s 3 cos d 6. (p 7) Show tht dt A sttor stte, provded A t, usg couttor relto (6.68). d A dt H, A HA AH H A AH E A A 9 of 3

20 Qutu Mechcs Hoework Solutos 7.4 (pge 98) The dervto the tet of the egevlues of ˆN s bsed o the costrt tht there re o sttes correspodg to the egevlues of < -½. Ths costrt ws gurteed b settg ˆ. It would pper tht t c lso be gurteed b settg ˆ for the cse: ˆ Show tht for s defed s egefucto of ˆN wth the egevlue zero; hece s ore properl tered. ˆ ˆ ˆ ˆ Nˆ ˆ ˆ ˆ ˆ 7.5 (pge 98) Usg the fudetl couttor relto ˆ, pˆ ˆˆ,. of 3 show tht ˆ ˆ ˆ pˆ ˆ pˆ ˆ pˆ pˆ ˆ,,,, 7.8 (pge 7) Show drectl fro the for of gve b (7.57) A e tht ˆ ˆ ( ) e e A A A where ˆ s the prt opertor. e

21 Qutu Mechcs Hoework Solutos 7.9 (pge 7) () Show tht the orlzed th egestte s geerted fro the orlzed groud stte ˆ through ˆ ˆ ˆ ˆ. ˆ ˆ ˆ ˆ! ˆ! (b) Show tht prt () ples the followg reltos:!! ˆ ˆ ˆ! ˆ ˆ!ˆ ˆ! ˆ! ˆ ˆ of 3

22 Qutu Mechcs Hoework Solutos 7. (pge 7) Show tht the th egestte of the hroc osclltor, the verge ketc eerg <T> s equl to the verge potetl eerg <V> - the vrl theore. Tht s, V T p E k Usg: d Ad: of 3 k For the Potetl Eerg: V Vˆ k k 4 k 4 k 4 k ( ) k ( ) Tˆ p 4 4 ( ) For the Ketc Eerg: T V 4 ( ) ( ) ( ) T p

23 Qutu Mechcs Hoework Solutos 7.34 (pge ) Show tht the curret dest J be wrtte J ˆ ˆ p p where ˆp s the oetu opertor. Show tht t s equvlet to equto 7.7: J Here s the th: pˆ pˆ pˆ pˆ pˆ pˆ J ths s equto (pge ) Show tht for oe-desol wvefucto of the for [where t, s rel], t Aep, t J A., Strt wth equto 7.7: J J J, t Aep, t, t, t, t, t A e Ae Ae A e A e e e e A e e A A e e 3 of 3

24 Qutu Mechcs Hoework Solutos 8.34 (pge 33) Costruct the egesttes d egeeerges of prtcle twodesol rectgulr bo of edge legths d. Tke the org to be t the corer of the rectgle. Accout geoetrcll for the reovl of ost of the degeerc preset the cse of the squre, two-desol bo descrbed prevousl. The degeerc preset for ths cofgurto (e.g., the eerg 5E s doubl degeerte) s soetes cll ccdetl degeerc, tht t s ether echge- or setrdegeerte. E E, s s E E E s E E E Degeerc = +(/) (/) For =squre_root( ) - Ol survvor: (/) Double degeerc t 5E wth,=(4,3) d (3,4). Degeerc occurs t Pthgore trples (pge 36) Wht s the order of degeerc of the egestte E s the two-desol hroc osclltor? S of The degeerc equls the uber of ws of wrtg teger s s the ordered su of two whole teger ubers (strtg t ). There re (s+) ws to do ths. 4 of 3

25 Qutu Mechcs Hoework Solutos 8.36 (pge 36) () Wrte dow the Hltos, egeeerges, d egesttes for two-desol hroc osclltor wth dstct sprg costts K d K. V()= K /= / where = K / d V()= K /= / where = K / The Hlto for ths -D hroc osclltor: H H H p K p K p p The Schroedger Equto s: Seprto of vrbles c be used to epress s: (,, z) X ( ) Y ( ) E X ( ) Y ( ) X ( ) Y ( ) Y ( ) X ( ) X ( ) Y ( ) X ( ) Y ( ) X ( ) Y ( ) ( ) ( ) X( ) X ( ) Y( ) Y ( ) X ( ) Y ( ) EX ( ) Y ( ) X Y Ech vrble s -D Hroc osclltor Defe:, E E E E E d E The egesttes d egeeerges of the Hltos H d H re: k X A e E ( ) H ( ),,,3, k Y A e E ( ) ( ),,,3, H th Where H re the order Herte polols d A re the orlzto costts. The totl egestte s d egeeerg s: (, ) A H ( ) H ( ) e E E E where,,,,3, ( ) 5 of 3

26 Qutu Mechcs Hoework Solutos (b) If K=4K show tht the egeeerges be 3 correspods to oto d wrtte: E where oto. The egeeerges of the Hltos H d H re: to k k 4k k E,,,3,,,,3, 3 E E E E where +,,,,3, ( c) For prt (b), wht s the order of degeerc of E,3? Lst the correspodg egesttes. Accout for the bsece of setr degeerc og these sttes. +,,,,3, =+6+=9 How ws c we get =9? There re 5 dfferet ws to get =9. The bsece of setr degeerc rses becuse of the fctor of o. 6 of 3

27 Qutu Mechcs Hoework Solutos Proble to solve. Wht s the groud stte eerg for ech of the followg -Prtcle sstes?.) H, deutero d electro.) He +, sgle ozed Helu to 3.) Postrou, boud postro d electro 4.) Ecto, wth ε= (delectrc costt) gore CM sste ' k E Z,,, M N C 9.9 kg.6 C e ev N s.6 J ev e =9.9-3 kg p = kg = kg Z ' Z 4 e 4 kg ev ' E ev Z + p e 9.65e p e 9.65e e e e e e e of 3

28 Qutu Mechcs Hoework Solutos 9.5 p 365 Show tht the frequeces of photos due to eerg decs betwee successve levels of rottor wth oet of ert I re gve b l or l ˆ ˆ L H I ˆ ˆ L,, ( ), l, H l l l l l E l I I ˆ ˆ L,, ( ), l, H l l l l l E l I I ˆ ˆ L,, ( )( ), l,. H l l l l l E l I I E El El l( l ) l( l ) l l I I I I E El El ( l )( l ) l( l ) l l I I I I 9.6 p 365 A HCl olecule rotte s well s vbrte. Dscuss the dfferece esso frequeces ssocted wth these two odes of ectto. Assue tht ol lltrstos betwee rottol sttes re llowed. Assue the se for vbrtol levels. For rottol levels ssue l 5. Sprg costt d oet of ert be ferred fro the equvlet teperture vlues for HCl: kb 45 K; IkB 5.K Hˆ rottol l l5 ˆ L I Erottol El El l ~ kb (5. K) l ()(5.)( K) ev/k=.6 ev I Erottol El El l ~ kb (5. K) (5)(5.)( K) ev/k=.3 ev I E vbrtol E E E 45K eV/K=.36eV vbrtol l l The vbrtol eerg s orders of gtude lrger th the lowest rottol sttes, but roud rottol sttes t l=5 t s ol 3 eerg. I I 8 of 3

29 Qutu Mechcs Hoework Solutos 9.3 (pge 385) Assue tht prtcle hs orbtl gulr oetu wth z ll. copoet d squre gtude () Show tht ths stte L L. L l, Lˆ l, l, Lˆ l, l, Lˆ l, l l l, l, l l l, l, L l, Lˆ l, l, Lˆ l, l, Lˆ l, l l l, l, l l l, l, (b) Show tht L L. Lˆ L L L L L L L L L L 4 Lˆ L L L L L L L L L L L L L L L L 4 4 Lˆ Lˆ L L Lˆ Lˆ Lˆ Lˆ Lˆ Lˆ Lˆ Lˆ z z L l, Lˆ l, l, Lˆ Lˆ l, l l l, l, z 9.4 (p 385) The se codtos hold s Proble 9.3. Wht s the epectto of the opertor LL LL the Y l? LL LL L L L L L L L L L L L L L L L L L L L L L L L L L L LL 4 L L 4 L L Yl Y L L L L Y Y l l l Y L c Y Y L d Y l 4 l, l l 4 l, l Y c c Y Y d d Y l 4 l, l, l l 4 l, l, l c c Y Y d 4 l, l, l l 4 l, dl, Yl Y l 9 of 3

30 Qutu Mechcs Hoework Solutos.6 (p 43) The curret vector J ssocted wth wvefucto rt, 7.7: J. J. p correspodg to the egevlue of k? The wvefucto be tered source-free f () Wht s the egefucto of ˆ r pˆ r r r r pˆ k r k k rk kk r r rk k r r k e e A k r r r kr kr Et / krt A k A e e (b) Clculte J for ths egefucto. s gve b A A A A J e e e e r r r r A krt A krt A krt e r e kr e r r r r r r r A krt krt Ak krt k e kr ke e r r A krt A krt A krt e r e kr e r r r r r r r A krt krt Ak krt k e kr ke e r r A krt Ak krt A krt Ak J e e e e r r r r k A r kr t kr t kr t kr t krt 3 of 3

31 Qutu Mechcs Hoework Solutos.45 p 55 () For sp correspodg to s=/, show tht the egevectors S d S re: Obes the egevlues equtos for s=/. S z S z (b) Wht re the egevlues correspodg to these egevectors? S S S S ( c) Show tht the egevectors coprse two sets of orthoorl vectors. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 of 3