J{UVr ^m {VH$s VWm {Magå V m {ÌH$s

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1 MSCPH-0 Jue - Examiatio 06 MSc (Previous) Physics Examiatio Mathematical Physics ad Classical Mechaics J{UVr ^m {VH$s VWm {Magå V m {ÌH$s Paper - MSCPH-0 Time : Hours ] [ Max. Marks :- 80 Note: The questio paper is divided ito three sectios A, B ad C. Write aswer as per the give istructios. Check Your paper code ad paper title before startig the paper. You are allowed to use o-programmable scietific calculator, however sharig of calculators is ot allowed. {ZX}e : h àíz Ì "A' "~' Am a "g' VrZ IÊS>m {d^m{ov h & àë oh$ IÊS> Ho$ {ZX}emZwgma àízm Ho$ CÎma Xr{OE& àíz Ì ewê$ H$aZo go yd àíz Ì H$moS> d àízì erf H$ Om±M bo& AmH$mo {~Zm àmojmq J dmbo gmbýg{w{ $H$ Ho$bHw$boQ>a Ho$ C moj H$s AZw {V h aývw Ho$bHw$boQ>a Ho$ hñvmývau H$s AZw {V Zht h & MSCPH-0 / 400 / () (P.T.O.)

2 Note: {ZX}e : 456 Sectio - A 8 6 Very Short Aswer Type Questios (Compulsory) Aswer all questios. As per the ature of the questio delimit your aswer i oe word, oe setece or maximum upto 0 words. Each questio carries marks. IÊS> - "A' A{V bkw CÎma dmbo àíz (A{Zdm ) g^r àízm Ho$ CÎma Xr{OE& AZo CÎma H$mo àízmzwgma EH$ eãx, EH$ dm³ m A{YH$V 0 eãxm [agr{ V H$s{OE& àë oh$ àíz Xmo A H$m H$m h & ) (i) Lagragia of a free particle i spherical polar coordiates is L m _ ro + r θo + r φo si θ i. The quatity that is coserved is (a) L ro (b) L θo MSCPH-0 / 400 / () (Cotd.) (c) L φo (d) EH$ w³v H$U H$m boja {O Z Jmobr {ZX}em H$ hmovm h, L m _ ro + r θo + r φo si θ hmojr& L + r θ φo o o i g a{jv am{e {ZåZ go L L L L (a) (b) (c) (d) + r θ ro θo φo φo o o (ii) For what value of the parameter α, the followig trasformatio is caoical? Q q cos α - p si α P q si α + p cos α am rq>a α Ho$ {H$g mz Ho$ {be {ZåZ ém VaU Ho$Zmo{ZH$b hmojr? Q q cos α - p si α P q si α + p cos α

3 (iii) Fid the Fourier trasform of f( t) k for O < t < a ad f( t) 0, otherwise. $bz f( t) k {X O < t < a AÝ Wm f( t ) 0. Bg $bz H$m w$[ae ê$mýva kmv H$amo& (iv) Fid the Laplace trasform of fuctio f( t) at + bt + c $bz f( t) at + bt + c H$m bmßbmg ê$mýva kmv H$amo& (v) Write the Bessel s differetial equatio. ~ {gb Ho$ AdH$bZ g rh$au H$mo {bi & d (vi) J ( x) dx _ 0 i is, where J ( x) is Bessel fuctio 0 (a) J ( x) (b) - J ( x) (c) ' J ( x ) d dx h? _ J ( x) i H$m mz ³ m hmojm, hm± J ( x) 0 (a) J ( x) (b) - J ( x) (c) ' J ( x ) - ' (d) J ( x) 0 a EH$ ~ gb $bz - ' (d) J ( x) (vii) How do the compoets of a cotravariat tesor of the secod rak, A ik, trasform uder coordiate trasformatio? Write the law. {ÛVr H$mo{Q> Ho$ H$m ÝQ >mdo[aa V Q> ga A ik Ho$ KQ>H$ H$mo{S> ZoQ> ê$mývau Ho$ AÝVJ V {H$g àh$ma émýv[av hmovo h & Bg {Z H$mo {bi & (viii)state trapezoid formula (trapezoid rule) for umerical itegratio. gm p» H$ BpÝQ>J«oeZ Ho$ {be Q >ooomobs>b gyì {bi & MSCPH-0 / 400 / () (P.T.O.)

4 Note: 456 Sectio - B 4 8 (Short Aswer Questios) Aswer ay four questios. Each aswer should be give i 00 words. Each questio carries 8 marks. (IÊS> - ~) (bkwîmamë H$ àíz) {ZX}e : {H$Ýht Mma àízm Ho$ CÎma Xr{OE& AmH$mo AZo CÎma H$mo A{YH$V 00 eãxm [agr{ V H$s{OE& àë oh$ àíz 8 A H$m H$m h & ) Derive the Rodrigues formula for the Legeder polyomials. d ( x) ( x - )! dx P boo S >r mo{bzmo{ ëg Ho$ {be amos >r½g Ho$ gyì ( x) H$mo ñwm{v H$a & d ( x - )! dx P ) Prove the relatio related to Bessel fuctios. J ( x) - J ( x) J ' ( x) ad hece prove the relatio - + ' ad hece prove the relatio J ( x) - J ( x) 0 ~ {gb $³eZ go g ~ {YV {ZåZ g ~ Y {gõ H$ao& J ( x) - J ( x) J ' ( x) - + AV> {ZåZ g ~ Y àmßv H$a ' J ( x) - J ( x) 0 MSCPH-0 / 400 / (4) (Cotd.)

5 4) Fid the Fourier trasformer of the fuctio F(t) shower i the figure give below: {ZåZ {MÌ {XE hþe $³eZ F(t) H$m y$[ae ê$mývau kmv H$a & 5) The Lagragia of a particle of mass m movig i oe dimesio αt mx kx is, L e < o - Fwhere α ad k are positive costats. Show that the equatio of motio of the particle is xp + α xo + m k x 0 EH$ H$U {OgH$m Ðì mz h VWm EH$ {d^r J{V mz h & H$U H$m boja {OAZ {ZåZ h, αt mx kx L o e - < F hm± a α VWm k YZmË H$ {Z Vm H$ h & {gõ H$amo {H$ H$U H$s J{V H$m g rh$au xp + α xo + m k x 0 h & MSCPH-0 / 400 / (5) (P.T.O.)

6 α 6) Show that the trasformatio Q q e cos p P - α q e si p, is caoical. {gõ H$s{OE {H$ {ZåZ ém VaU P α q e si p, Q - H $Zmo{ZH$b h & α q e cos p 7) Solve harmoic oscillator problem by Hamilto-Jacobi method. ho{ ëq>z - OoH$mo~r {d{y Ûmam gab AmdV Xmo{bÌ Ho$ {be hb àmßv H$ao& 8) Fid a real root of the equatio x e x - 0 usig Newto-Raphso method, where e g rh$au x e x - 0 H$m dmñv{dh$ yb Ý yq>z aoâgz {d{y Ûmam kmv H$amo, Ohm± e ) Obtai the Lagragia of a free particle i spherical polar coordiates. EH$ wº$ H$m boja {O Z H$s Jmo{b {ZX}em H$m àmßv H$a {bi & Sectio - C 6 (Log Aswer Questios) Note: Aswer ay two questios. You have to delimit your each aswer maximum 500 words. Each questio carries 6 marks. (IÊS> - g) (XrK CÎmar àíz) {ZX}e : {H$Ýht Xmo àízm Ho$ CÎma Xr{OE& Am AZo CÎma H$mo A{YH$V 500 eãxm [agr{ V H$s{OE& àë oh$ àíz 6 A H$m H$m h & MSCPH-0 / 400 / (6) (Cotd.)

7 0) (i) Write the law of trasformatio of a cotravariat tesor of secod rak uder coordiate trasformatio. Also write the law of trasformatio of covariat tesor of secod rak. (ii) Fid iverse Laplace trasform of s + s + s (iii) Show that the Fourier sie ad cosie trasforms of e -at are g ( w) w s w + a g ( w) w a + a c (i) {ÛVr H$mo{Q> Ho$ H$m Q >mdo[ae Q> Q> ga H$m H$mo{S> ZoQ> Q >m±g $mo }gz Ho$ AÝVJ V {Z {bio& "H$modo[aE Q> Q>oÝga ({ÛVr H$mo{Q>) Ho$ {b o ^r H$mo{S> ZoQ> Q >m±g $mo }gz Ho$ AÝVJ V ê$mývau {Z {bio& (ii) $bz s s s + + Ho$ {b o ì wëh«$ bmßbmg ê$mýva kmv H$amo& (iii) {gõ H$amo {H$ $bz e -at H$m y$[a o Á m VWm y$[a o H$moÁ m ê$mýva {ZåZ h g ( w) w w + a s g ( w) w a + a c MSCPH-0 / 400 / (7) (P.T.O.)

8 ) (i) Fid the Lagragia for a system i which bob of simple pedulum of mass m, with a mass m at the poit of support which ca move o a horizotal lie i the plae i which m moves (see figure). The system is placed i a uiform gravitatioal field (acceleratio g ), legth of massless strig is l as show i figure. Show that the Lagragia is _ m + m ixo + m _ l φo + lxφ cos φ + m gl cos φ o o i L (ii) Let f( p, q, t ) be some fuctio of coordiates, mometa ad time. Show that its total time derivative is df dt f + 7 H, f A t where [ H, f H f H f ] // f - p q q p p is R k k k k the Poisso bracket of the quatities H ad f. MSCPH-0 / 400 / (8) (Cotd.)

9 (i) EH$ gab bmobh$ {OgHo$ Jw Q>o H$s g h{v m h VWm {OgHo$ Ambå~Z {~ÝXþ a m Ðì mz H$m H$U h, Vmo Bg gab bmobh$, {OgH$s bå~mb l h, H$m boja {O Z kmv H$a & m H$U {MÌmZwgma {XImB j {VO aofm Ho$ AZwgma J{V H$a gh$vm h & {ZH$m EH$ Jwê$Ëd joì (ËdaU g ) J{V mz h & {gõ H$a {H$ boja {O Z {ZåZ h _ m + m ixo + m _ l φo + lxo φo cos φ i + m gl cos φ L (ii) mzm {H$ H$moB $bz f( p, q, t ) g doj p, H$mo{S> ZoQ> q, VWm g t a {Z^ a H$aVm h & {gõ H$a {H$ Bg $bz H$m g Ho$ gmw yu AdH$bZ {ZåZ g rh$au go {X m OmVm h ; df dt f + 7 H, f A t Ohm± a [ H, f H f H f ] // f - p q q p p R k k k k am{e H VWm f H$m mobe±m ~«oho$q> àx{e V H$aVm h & MSCPH-0 / 400 / (9) (P.T.O.)

10 ) The Hermite equatio is y'' - xy' + y 0 (i) Show that a Hermite polyomial ( x) (- d ) e e dx x -x H satisfies this equatio, (ii) Show that H ( x) ca be expaded as ( - ) - ( - )( - )( - ) ( x) ( x) - ( x) ( x) +... H (iii) Write H ( x), H ( x), H ( x), H ( x) 0 (iv) Show that the ormalizatio itegral is w - -x e H ( x) dx! h m BQ> g rh$au {ZåZ h y'' - xy' + y 0 (i) {gõ H$a {H$ h m BQ> mobrzmo{ Ab x H e -x ( x) (- d ) e dx h m BQ> g rh$au H$mo g VwîQ> H$aVm h & MSCPH-0 / 400 / (0) (Cotd.)

11 (ii) {gõ H$amo {H$ H ( x) H$m {dñvma {ZåZ h ; ( - ) - ( - )( - )( - ) ( x) ( x) - ( x) ( x) +... H (iii) H ( x), H ( x), H ( x), H ( x) 0 H$m mz {bi & (iv) {gõ H$a {H$ Zmoa obmbooez BpÝQ>J«b {ZåZ h : w - -x e H ( x) dx! ) Fid the Laplace trasform of the followig fuctio for (A), (B) ad (C) ad iverse Laplace trasform for part (D). (A) f( t) si at cosh bt αt β you ca use property L # e si βt - ( s - α) + β - t (B) f( t) e cos t αt s you ca use property L # e cos βt - - α ( s - α) + β Z ] si at - k ; t (C) f( t) H [ ] 0 ; t < \ (D) Fid Iverse Laplace trasform of s + s + s MSCPH-0 / 400 / () (P.T.O.)

12 ^mj (A), (B), (C), Ho$ {b o bmßbmg ê$m Va VWm ^mj (D) Ho$ {b o ì wëh«$ bmßbmg émýva kmv H$amo& (A) $bz f( t) si at cash bt Am αt β L # e si βt - H$m C moj H$a gh$vo h & ( s - α) + β - t (B) $bz f( t) e cos t αt s Am L # e cos βt - - α ( s - α) + β Z ] si at - k ; t (C) f( t) H [ ] 0 ; t < \ s s (D) $bz s + + Ho$ {be ì wëh«$ bmßbmg ê$mýva kmv H$amo& MSCPH-0 / 400 / ()