24 Concept of wave function. x 2. Ae is finite everywhere in space.

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1 4 Cocept of wve fucto Chpter Cocept of Wve Fucto. Itroucto : There s lwys qutty sscocte wth y type of wves, whch vres peroclly wth spce te. I wter wves, the qutty tht vres peroclly s the heght of the wter surfce lght wves, electrc fel getc fel vry wth spce te. For De-brogle wves or tter wves sscocte wth ovg prtcle, the qutty tht vry wth spce te, s clle wve fucto of the prtcle. The teporl sptl evoluto of qutu echcl prtcle s escrbe by wve fucto, t for -D oto r, t for 3-D oto.. It cots ll possble forto bout the stte of the syste. These re kow s cofgurto spce wve fucto. Coto for physclly ccepte, well behve, relstc wve fucto : (), t shoul be fte, sgle-vlue cotuous everywhere spce. () shoul be cotuous everywhere spce. But, y be scotuous soe cses s follows: () If the potetl uer whch the prtcle s ovg, hs fte out of scotuty t soe pots, (b) If the potetl uer whch the prtcle s ovg, s of rc elt ture. (), t shoul be squre tegrble.e. t () Asec (b) t, fte qutty Eple : Whch of the followg wve fucto s cceptble s the soluto of the Schroger equto for ll vlues of? A (c) Ae () Ae Asec t A s ot fte t Ae s ot fte t ; Correct opto s (). Ae s fte everywhere spce. Eple : A bll bouces bck off erth. You re ske to solve ths qutu echclly ssug the erth s ftely hr sphere. Coser surfce of erth s the org plyg V 0 ler potetl eleswhere (.e. V g for 0 ). Whch of the followg wve fucto s physclly ssble for ths proble (wth k > 0): () k e k (b) Ae k (c) Ae () Ae k

2 Cocept of wve fucto 5 Sce the erth hs bee tke s ftely hr sphere, therefore the wve fucto of the bll wll be zero for < 0, o-zero for > 0 zero t fty. Sce shoul be cotous everywhere, the wvefucto of the bll wll be zero t = 0. Correct opto s ().. Physcl sgfcce of wve fucto : Geerlly,, t s cople qutty. It c be ultple by y cople uber wthout ffectg ts physcl sgfcce. I geerl,, t hs o rect physcl sgfcce. But the qutty, t, t, t s rel, physclly sgfct s efe s posto probblty esty.e. probblty of fg the prtcle per ut legth t te t. Therefore, for -D oto the probblty of fg the prtcle betwee to + t te t s gve by, t, t, t for 3-D oto the probblty of fg the prtcle wth the volue eleet locte betwee r r r t te t s gve by r, t r, t r, t Now,, t or r, t spce shoul be equl to uty.e. shoul be chose such tht totl probblty of fg the prtcle the etre, t r, t, t (for -D oto) ll spce r, t r, t r, t (for 3-D oto) ll spce Ths s clle the orlzto coto of the wve fucto., t r, t s orlze t soe te t res orlze forever. Ths c be uerstoo s If or coservto of probblty or coservto of orlzto. Metho of Norlzto: Coser, t s uorlze wve fucto. We c costruct orlze wve fucto s, t N, t where N s the orlzto costt. Therefore,, t, t N, t, t N, t, Eple 3: Norlze the wve fucto gve by t Ne 0 Ne 0

3 6 Cocept of wve fucto Norlzto coto s 0 N e e 0 N N Orthogolty coto of wve fuctos : Two wvefuctos.e. f prtcle s the stte re s to be orthogol to ech other, f Orthoorlty coto of wve fuctos : Two wvefuctos.3 Hlbert Spce : 0, the the prtcle cot be the stte re s to be orthoorl to ech other, f 0 sulteously together. I vector spce, the set of ut vectors e, e, e 3... for the orthoorl bss.e. we c epress y vector ths vector spce s ler cobto of e, e, e 3... Slrly, spce c be efe whch set of fuctos 3,,... for the orthoorl bss of the coorte syste. The correspog fte esol ler vector spce s clle Hlbert spce. Propertes of Hlbert Spce: () The er prouct or sclr prouct of two fuctos efe s () Two fuctos b j j j j efe the tervl b s re s to be orthogol f ther er prouct s zero.e. b j j 0 Ths s kow s orthogolty coto of two wve fuctos. () The or of fucto s efe s b N (v) A fucto s s to be orlze f the or of the fucto s uty.e. b N Ths s kow s orlzto coto of prtculr wve fuctos. / /

4 Cocept of wve fucto (v) Fuctos whch re orthogol orlze re clle orthoorl fuctos they wll stsfy the coto: (v) A set of fuctos 3 b j j j,,... s lerly epeet f there est relto lke c c c , where ll c, c, c 3... re zero. Otherwse, they re s to be lerly epeet. A set of lerly epeet fuctos s coplete..4 Opertor forls : A opertor s thetcl rule (or proceure) whch opertg o oe fucto trsfors t to other fucto.e.  opertor.. Every ycl vrble Qutu echcs represete by Opertos Sybol Result of the operto Tkg the squre root / Dfferetto w.r.t. Posto Opertor :, y, z Moetu Opertor :, p py, p z p y z Potetl eergy Opertor : V Ketc eergy Opertor : Couttor Brcket: K p The couttor brcket of two opertors A B s efe s A, B AB BA. Eple:,, () p p p p (),, Slrly, y p, y Ler Opertor: If opertor z, p z  s s to be ler f () A c ca () A A A All Qutu echcl Opertors re ler ture. Propertes of the couttor brcket:. A, B B, A.,...,, A B C D A B A C... 7

5 8 Cocept of wve fucto 3. A, B B, A 4. A, BC A, B C B A, C 5.,,,,, 0 A BC B C A C A B 6. A, f A 0 7. f A G A f A, G B 0 oly f, A B 0, A B B A, B Eple 4: F the couttor brcket A, B, B, A Sol: A, B, B, A A, B B, A B, A A, B = AB BA BA AB BA AB AB BA, 0 0.,, A B A A B = ABBA ABAB BABA BAAB BAAB ABAB BABA ABBA 0 Eple 5: F the couttor brcket, p. Sol:, p, pp, p p p, p, pp p, p p p Eple 6: If p H V, the clculte,, H. p H V p p p p Sol:,,,, 0,,, H, p, p Egevlues Egefuctos: If opertor  opertg o fucto gves A, the s clle the egefucto of  correspog to egevlue the bove equto s kow s egevlue equto of opertor Â. Eple: A e A e 4e 4 s egefucto of  correspog to egevlue 4. If the couttor brcket of two opertors A B s equl to zero.e. A, B 0, the the physcl observbles correspog to these opertors re sulteously ccurtely esurble they hve coplete set of sulteous egefuctos. Eple 7: The opertor hs the egevlue. Detere the correspog wvefucto. Sol: Egevlue equto:

6 Cocept of wve fucto l l ep Eple 8: F the costt B whch kes Wht s the correspog egevlue? B e B e Sol: 4 For e to be egefucto of the opertor e egefucto of the opertor B ust be epeet of.e. 4 B, the the egevlue 4 B 0 B 4 Therefore, B e e ; Thus the egevlue of the opertor s (-). Hert opertor: A opertor  s s to be hert f A A ts shoul stsfy the followg relto: A A A Drc represetto: Schroger represetto:, A A, A A B. Propertes: () Egevlues of hert opertors re rel. () Egefuctos correspog to fferet egevlues of hert opertor re orthogol. (3) All physcl observble qutu echcs re represete by hert opertors., p Eple: Hece, the oetu opertor p s hert ture. p Slrly, we c see tht Projecto opertor: s t-hert.e. A opertor s s to be projecto opertor f A A A A Propertes: () Egevlues of projecto opertor s 0. () Prouct of two projecto opertors s lso projecto opertor f p p (3) Su of two projecto opertor s projecto opertor f p p, 0, 0

7 30 Utry Opertor: A opertor  s s to be utry f A A Prty Opertor: Prty opertor correspos to spce reflecto bout the org.e P Cocept of wve fucto I geerl, P r r ; where r yj zk r yj zk Whe prty opertor cts o wvefucto, the followg chges tke plce vrous co-orte syste: () Crtes co-orte syste:, y y, z z () Sphercl polr co-orte syste : r r,, () Cyllrcl co-orte syste:,, z z Propertes: () Egevlues of the prty opertor s, -. P eve prty. P o prty () Prty opertor s hert ture (3) Prty opertor coutes wth hlto opertor f the potetl uer whch prtcle s ovg.e. V() s syetrc ture. Eple 9: Let, N b b where the opertor b stsfes the relto the egevlues of N re () +ve ve tegers (b) ll +ve tegers (c) 0 oly Sol: N b b b b b bb b b b b b b b b b bb b b 0, b b 0 ; bb b b () 0 oly Eple 0: Let the wve fucto of prtcle of ss t gve stt be Wht wll be the fucto K ketc eergy? The ketc eergy opertor s K / Ae. where K s the ketc eergy? Is ths wve fucto egefucto of. Thus, / A / K Ae e A / A / / e e e A / e Ths s ot egefucto of ketc eergy. Eple : A opertor A s efe s A For the gve opertor, A. Is ths ler opertor?

8 Cocept of wve fucto 3 A A Thus, A A hece A s ot ler opertor. Eple. Prove tht Xp, H p X p, V opertor.,,, where p Xp H X p H X H p =, p X p, V X V p X p,,, V p X p X p p p X p, V p p p X p, V p H V, V s potetl eergy Eple 3. be two ortoorl stte vectors. Let A. Is projecto opertor? A A Hece, A s hert. Now, A Sce, re orthoorl, A A Therefore, A s ot projecto opertor. Eple 4. Prove tht Xp p X t t p X, H X, V X p p X p X p p,,, p X X p p X Xp Xp p X Eple 5. Prove tht the opertors ( / ) / re Hert. [ ] Therefore, / s Hert.

9 3 Cocept of wve fucto Thus, / s Hert. Eple 6: F the followg coutto reltos: (), (), F( ) () 3 3, (), F( ) ( F ) F = F F F F F Thus,, F( ) Eple 7: F the equvlece of the followg opertors: () () () ( ) = = Therefore, () Slrly, Eple 8. By wht fctors o the opertors ( p p ) / ( p ) p ffer? f ( p p ) f ( f ) f ( f ) f f f f f f f f 4 f f ( f ) ( p p ) f ( p p)

10 Cocept of wve fucto 33 f ( p p ) f f f f ( f ) f f f f f f 4 f f f 8 f f 4 The two opertors ffer by ter (3 / ) Eple 9. F the egevlues egefuctos of the opertor /. Egevlue equto: ( ) k ( ) (where k s the egevlue ( ) s the egefucto) k Ce k. Cse : k s rel postve qutty, s ot cceptble fucto sce t tes to or s or. Cse : k s purely gry (sy ), the Ce whch wll be fte for ll vlues of. Hece, y k ce s the egefucto of the opertor / wth egevlues k, where s rel. Epectto vlue of yc vrbles: It s efe s the verge of the result of lrge uber of epeet esureets of physcl observble o the se syste. Note:    () If the stte of the prtcle s egefucto of the opertor Â, the the epectto vlue of the physcl observble correspog to  wll be equl to the egevlue of  correspog to the stte of the prtcle. () The stte of the prtcle s gve s: C C C C where,... re the egefucto of the opertor Â, the epectto vlue of the the physcl observble correspog to  wll be A C where re the egevlues of opertor  correspog to.

11 34 Cocept of wve fucto Eple 0: A prtcle of ss s cofe -D bo fro L to L. The wve fucto of the prtcle ths stte s cos 0 4L () F the orlzto fctor. () The epectto vlue of P ths stte. () L L 0 0 L L L cos cos 4 4L L 0 4L 0 s 4L L L 0 L L P L L = L cos L L 4L 4L. L cos L cos 3 L 6L L L 3L 0 = L 6L Eple : F the epecte vlue of posto oetu of prtcle whose wve fucto s / k e ll spce Epecte posto of the prtcle Epecte oetu of the prtcle / e. 0 / e p / k / k / e e e k = / / e e k e e / / k /4 Eple. The wvefucto of prtcle stte s N ep( / ), where N (/ ). Evlute ( )( p). Sce s syetrcl bout 0, 0 N ep Sce, s rel wve fucto, p 0 p ( ) N ep ep N N ep ep

12 Cocept of wve fucto 35 Therefore, p p p Eple 3. The Hlto opertor of syste s H ( / ). Show tht N ep( / ) s egefucto of H etere the egevlue. Also evlute N by orlzto of the fucto. Nep( / ), N beg costt H N ep 3 ep ep N ep 3N ep 3 Thus, the egevlue of H s 3. The orlzto coto gves N e N N / The orlze fucto / ep Eple 4. Coser the wve fucto ( ) Aep ep( k), where A s rel costt. () F the vlue of A, () clculte p for ths wve fucto. () Norlzto coto: A ep A / / A () p k k ( ) A ep e k ep e ( ) ep ( )( k) A ep I the frst ter, the tegr s o the tegrl s fro to. Hece the tegrl vshes. Therefore, p k, sce A ep