Wter 3 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs... 3 Chpter Itro to Mth Techques for Qutum Mechcs Itro to dfferetl equtos Fucto y y( ) s to stsfy dfferetl equto d y dy 5 6y () For ths type of Dfferetl equto (more lter), try soluto dy The e d y d y e e y e Substtute to dfferetl equto () Or e 5e 6e 5 6 ( 3)( ) Solutos: 3, 3 e : e : 3 3 3 9e 5e 6e 3 3 3 4e e 6e But: y ler combto of solutos s lso soluto
Wter 3 Chem 356: Itroductory Qutum Mechcs y( ) c e c e 3 9c e 4c e 3 5c e c e 3 6c e 6c e 3 Let us try other oe d y y e e Geerl Soluto: ce ce Altertve wy to wrte: e cos s e cos s( ) cos s y( ) ( c c )cos ( c c )s d cos d s defe d c c, d ( c c ) choose d, d rel Verfy: d d d cos ( s ) cos d s (cos ) s d y y, s epected Type of solutos e, e, cos, s, rel, > Chpter Itro to Mth Techques for Qutum Mechcs 3
Wter 3 Chem 356: Itroductory Qutum Mechcs Whe does ths work? 3 dy d y d y 3 3 cb y c c c... () Costt coeffcets frot of y d ts dervtves d y y ot: () Ler fucto y d y dy ot: y (3) Homogeeous equto ot: y y c c c y 3 For homogeeous dfferetl equto: Fd prtculr soluto y P( ) dd to ths the geerl soluto of homogeeous equto. For more complcted dfferetl equtos (e. Not homogeeous DE wth costt coeffcets) solutos re ofte hrd to fd My trcks of the trde Use symbolc mth progrm (t kows my of the trcks) Numercl pproches (ofte work very esly pcture of soluto) Chpter Itro to Mth Techques for Qutum Mechcs 4
Wter 3 Chem 356: Itroductory Qutum Mechcs Boudry Codtos Let us cosder our orgl dfferetl equto. d y dy 5 y y( ) c e c e 3 Now mpose further codtos. Eg: y() c c dy y( ) e e 3c c c c c 3c c c 3 sttstcs DE d boudry codtos Soluto s completely specfed f oe supples s my codtos s oe hs free coeffcets c, c,. the soluto lwys ler set of equtos So recpe s very smple Try y( ) e d work t out! Prtl dfferetl equtos d seprto of vrbles Cosder problem of vbrtg strg (eg. gutr, vol) We wt to descrbe the mpltude u(, t ) Chpter Itro to Mth Techques for Qutum Mechcs 5
Wter 3 Chem 356: Itroductory Qutum Mechcs strg) u u DE (, t) (, t) v t v : Velocty of wve propgto strg, relted to sprg costt, (s soud of the Boudry Codto: u(, t) u(, t) t u(, t ) : fucto of vrbles use prtl dervtves I mth we typclly do ot wrte s kept costt u (compre thermodymcs) t How to solve PDE (prtl dfferetl equto)? Try soluto u(, t) X ( ) T( t) Smple product of fucto of d fucto of t Boudry Codto: X () X ( ) Substtute trl fucto to PDE d X ( ) d T T( t) X ( ) v dt Dvde both sdes by X ( ) T( t ) : d X ( ) d T X ( ) v T( t) dt oly depeds o Lke f ( ) g( t) should be true for ll, t oly depeds o t f ( ) g( t ) f( ) s costt gt ( ) f( ) s costt, should be sme g( t) f ( ) gt () s costt f( ) f ( ) g( t) t, C oly be true f both fuctos re costt! Ie. The sme costt Let us cll ths costt, the seprto costt k (for lter smplcty) Chpter Itro to Mth Techques for Qutum Mechcs 6
Wter 3 Chem 356: Itroductory Qutum Mechcs d X k X ( ) v dt dt X () X ( ) k T() t o boudry codtos Now we c use techques dscussed before ( k s costt) Try X ( ) e e k e k, k X ( ) c e c e k Note k could be mgry m k ( m) m k However we kow tht the strg wll oscllte, d hece c tcpte Usg wht we dd before c e ( c c )cos k ( c c )s k k k ce d cos k d s k Ths s geerl soluto. Now cosder boudry codtos. : dcos k ds k dd d : d s( k) d (flt strg possblty) or s( k) Whe s s( k)?,,, 3... k k k e, wth k rel Geerl soluto: X ( ) d s Ths s soluto for t prtculr vlue for k k Now cosder correspodg soluto for Tt () Chpter Itro to Mth Techques for Qutum Mechcs 7
Wter 3 Chem 356: Itroductory Qutum Mechcs v d T dt dt w () T t dt w Tt () v Smlr equto s before: T( t) c s w ( t) c cos( w t) If we combe ths wth X we get vt vt u(, t) As s Bs cos Ths s soluto for y vlue of A, B d y vlue of,,, 3 Most geerl soluto: u(, t) s A s( wt) B cos( wt),,,3... You c verfy tht ths deed stsfes PDE How c we terpret ths? s : Chpter Itro to Mth Techques for Qutum Mechcs 8
Wter 3 Chem 356: Itroductory Qutum Mechcs s s : Sme soluto, restrct,,,3 (, othg etr) So strg vbrtes s ler combto of modes, ech of the modes osclltes tme t dfferet frequecy. s v w w v All multples of fudmetl frequecy w Ths defes the ptch of the soud w The other modes re clled over toes Meg of coeffcets A, B? v t B cos s The tl shpe of fucto du dt v A cos the tl velocty of the strg. Dfferet strumets, gutrs, vols, cello Chpter Itro to Mth Techques for Qutum Mechcs 9
Wter 3 Chem 356: Itroductory Qutum Mechcs dfferet A, B How you ttck the strg determes the tl shpe/velocty compre Chese zther: ht the sre dfferet spots or twg the strg Noe of ths ffects the ptch w the geerl hrmoc Perod s( wt) s w( t T) s( wt ) s( wt) wt T ; w w T Itroducto to Sttstcs We wll see tht qutum mechcs s essetlly sttstcl theory. We c predct the results d ther dstrbuto from lrge umber of repeted epermets oly. We cot predct (eve prcple) the outcome of dvdul epermet. Let us therefore tlk bout sttstcs usg smple emple: the dce If you throw the dce, ech throw wll yeld the result,, 3, 4, 5 or 6. If you throw the dce my tmes, sy 6 tmes, we mght get For fr dce, ech umber hs equl chce, d so we sy the probblty to throw for emple 3 s. Ths s reflected by the ctul umbers we got the emple. 6 P 6 6 6 98 3 995 4 5 5 3 6 96 Totl 6 I the lmt tht we throw very lrge (fte) tmes, we get closer d closer to 6 Chpter Itro to Mth Techques for Qutum Mechcs 3
Wter 3 Chem 356: Itroductory Qutum Mechcs P oly hs meg for my repeted epermets N totl P We mght cll the ctul outcome of epermet, here =,, 3, 4, 5, 6 The verge s gve by P N tot For dce: ( 3 4 5 6) 3 6 6 The verge vlue does ot eed to be possble outcome of dvdul throw. We re lso terested the vrce of the results. We cll the verge A or A. The the vrce s gve by: (both re used) P ( A) A Let us tke dce wth 5 sdes to mke the umbers eser,, 3, 4, 5, A 3 P 5 [( 3) ( 3) (3 3) (4 3) (5 3) A ] 5 [4 4] 5 Stdrd Devto A I c wrte the vrce dfferetly s P A P A A ( ) ( ) P A P A P A AA AA Chpter Itro to Mth Techques for Qutum Mechcs 3
Wter 3 Chem 356: Itroductory Qutum Mechcs A A A Let us check for the 5 fce dce: ( 3 4 5 A ) 5 ( 4 9 6 5) 5 (55) 5 A 33 9 A A A (s before) Of course: We proved ths s true mthemtclly! Ths cocludes (for ow) dscusso of dscrete sttstcs. Now cosder the cse of cotuous dstrbuto. For emple desty dstrbuto. ( ) dm the mss betwee d ( dmeso) ( ) M totl mss b Also ( ) mss betwee pots d b If we ormlze P( ) ( ) M probblty to fd frcto of the totl mss betwee d We c defe verge posto P( ) P( ) Chpter Itro to Mth Techques for Qutum Mechcs 3
Wter 3 Chem 356: Itroductory Qutum Mechcs Emple: tke bo betwee d Uform P ( ) elsewhere ) ) 3) 4) P( ) ormlzed P( ) 3 P( ) 3 3 3 Alwys More complcted dstrbutos re possble of course Fmous s the Guss dstrbuto The P( ) ce prmeter (wll be ) P ( ) c e c : ormlzto costt e Odd fucto f ( ) f ( ) s dvertsed (ths ws the reso to defe the Guss lke ths) See book for dscusso tegrls Chpter Itro to Mth Techques for Qutum Mechcs 33