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1 A Study of Frctonl Schrödnger Equton-composed v Jumre frctonl dervtve Joydp Bnerjee 1, Uttm Ghosh, Susmt Srkr b nd Shntnu Ds 3 Uttr Bunch Kjl Hr Prmry school, Ful, Nd, West Bengl, Ind eml- joydp1955bnerjee@gml.com Deprtment of Appled Mthemtcs, Unversty of Clcutt, Kolkt, Ind; eml : uttm_mth@yhoo.co.n b eml : susmt6@yhoo.co.n 3 Rector Control Dvson BARC Mumb Ind eml : shntnu@brc.gov.n Abstrct One of the motvtons for usng frctonl clculus n physcl systems s due to fct tht mny tmes, n the spce nd tme vrbles we re delng whch ehbt corse-grned phenomen, menng tht nfntesml qunttes cnnot be plced rbtrrly to zero-rther they re non-zero wth mnmum length. Especlly when we re delng n mcroscopc to mesoscopc level of systems. Menng f we denote the pont n spce ndt s pont n tme; then the dfferentls d (nd dt cnnot be tken to lmt zero, rther t hs spred. A wy to tke ths nto ccount s to use nfntesml qunttes s ( Δ (nd ( Δt wth 0< < 1, whch for very-very smll Δ (nd Δ t ; tht s trendng towrds zero, these frctonl dfferentls re greter tht Δ (nd Δ t. Tht s ( Δ >Δ. Ths wy defnng the dfferentls-or rther frctonl dfferentls mkes us to use frctonl dervtves n the study of dynmc systems. In frctonl clculus the frctonl order trgonometrc functons ply mportnt role. The Mttg-Leffler functon whch plys mportnt role n the feld of frctonl clculus; nd the frctonl order trgonometrc functons re defned usng ths Mttg-Leffler functon. In ths pper we estblshed the frctonl order Schrödnger equton-composed v Jumre frctonl dervtve; nd ts soluton n terms of Mttg- Leffler functon wth comple rguments nd derve some propertes of the frctonl Schrödnger equton tht re studed for the cse of prtcle n one dmensonl nfnte potentl well. Key-Words Jumre Frctonl Dervtve, Mttg-Leffler functon, frctonl Schrödnger equton, Frctonl wve-functon 1. Introducton Frctonl order dervtve re beng etensvely used by the uthors recently to study dfferent nturl process nd physcl phenomen [1-11, 18, 1, 5]. The mthemtcns of ths er re tryng to re construct generl form of clculus usng modfcton of the clsscl order dervtve-nd generlze the sme to rbtrry order. Remnn-Louvlle [11] defnton of frctonl dervtve dmts non-zero vlue for frctonl dfferentton of constnt. Ths ncreses the complety n clculton nd lso contrdcts the bsc propertes of clsscl clculus. To overcome ths problem Jumre [9, 19, 1] modfed Remnn-Louvlle defnton of frctonl dervtve to obtn zero for dervtve of constnt nd ths type of 1

2 dervtve s lso pplcble for contnuous but non-dfferentble functons. More of tht Mttg-Leffler [10] functon ws ntroduced n clsscl sense but t hs severl pplctons n frctonl clculus feld. The Mttg-Leffler functon wth comple rgument gves the frctonl sne nd cosne functons for frctonl order nvestgton [9]. Ghosh et l [11-1, 0] dscussed bout the solutons of vrous types of lner frctonl dfferentl equton (composed v Jumre frctonl dervtve n terms of Mttg-Leffler functons. All the prevous works leds to the generl formlsm for frctonl clculus whch meets clsscl clculus t certn boundry. On the other hnd reserchers re usng dfferent frctonl dfferentl equtons only ncorportng the frctonl order n plce of clsscl order but wht wll be the ctul equton n frctonl sense f we strt usng the bsc frctonl equtons? Ths s chllengng tsk to mthemtcns. If we hve pont n spce cll t, t orgntes from dfferentl d, tht s d =. Now wth dfferentl ( d, wth 0< < 1, we hve ( d > d, whle we hve d the nfntesml dfferentl pproches zero, gves spce trnsformed s ; tht s ( d [1, 3, 4]. So s the cse wth pont t tme cll tt.ths s corse grnng 0 n scle of observton, where we come cross the frctl spce tme, where the norml clsscl dfferentls d nd dt, cnnot be tken rbtrrly to zero, nd then the concept of clsscl dfferentblty s lost []. The frctonl order s relted to roughness chrcter of the spce-tme, the frctl dmenson [3-4]. Here n ths pper frctonl dfferentton of order s used to study the dynmc systems defned by functon f (, t.in [3-5] the demonstrton of the frctonl clculus on frctl subset of rel lne s Cntor Set, nd the order s tken ccordngly. In ths pper we hve studed the formulton on frctonl form of quntum mechncs usng frctonl clculus. We tred to fnd nternl behvour of quntum relm. For ths purpose we hve developed frctonl Schrödnger equton nd tred to understnd the nture of quntum mechncs for frctl regon. Ths formulton lso leds to norml quntum mechncs t lmtng condton. The nture of the soluton chnges dependng upon vrous frctonl orders of dfferentton, whch my led underlyng sgnfcnce of quntum mechncs. We hd to modfy De-Brogle s nd Plnk s hypothess n frctonl sense such tht they remn ntct f lmtng condtons re used. Ths pper s dvded nto seprte sectons nd sub-sectons. In secton-.0 some defntons of frctonl clculus re descrbed. In secton-3.0 we dscuss bout orgnl plne progressve wve. Secton 4.0 s bout dervton of frctonl Schrödnger equton. The secton-4.1 dels wth soluton of frctonl Schrödnger equton. The secton-4. s bout tme ndependent frctonl Schrödnger equton nd Hmltonn. In secton5.0 we developed equton of contnuty. In secton5.1 we dscuss bout propertes of frctonl wve functon. The secton5. dscusses on further study on frctonl wve functon, n secton-5.3 t s bout orthogonl nd norml condtons of wve functons. The secton- 6.0 s bout opertors nd epectton vlues. The secton-7.0 to secton7.1 we dscuss smple pplcton prtcles n one dmensonl nfnte potentl well. In secton7. we hve grphcl representton of frctonl wve functon. Net secton 7.3 s the grphcl representton of probblty densty nd then n secton 7.4 we dscussed energy clculton. In ppend-we hve defned frctonl qunttes used n the study. 0

3 . Some Defnton of frctonl clculus There re severl defntons of frctonl dervtve. The most ledng defntons re Remnn-Louvlle frctonl dervtve [5] Jumre modfed frctonl dervtve [9]. Remnn-Louvlle frctonl dervtve Remnn-Louvlle (R-L frctonl dervtve of functon f ( s defned s m+ 1 1 d m D f( = ( τ f( τ dτ Γ ( + m+ 1 d Where m < m+ 1, m s postve-nteger. Usng ths defnton, for non-zero constnt functon the frctonl dervtve s not-zero [6], ths s contrry to clsscl clculus. b Jumre modfed defnton of frctonl dervtve To get rd of the problem of R-L frctonl dervtve, Jumre modfed [9, 19, 1] the defnton of frctonl dervtve, for contnuous (but not necessrly dfferentble functon f (, n the rnge 0 such tht s 1 1 ( ξ f( ξ dξ, < 0 Γ ( 0 ( J 1 d 0 ( Γ (1 d 0 ( n f ( = D f( = ( ξ f( ξ f(0 dξ, 0< < 1, ( n ( f (, n < n+ 1, n 1 In Lebnz s clsscl sense the Jumre frctonl dervtve s defned v frctonl dfference. Let f :, f (, denote contnuous (but not necessry dfferentble functon, nd let h > 0 denote constnt nfntesml step. Defne forwrd opertor E f = f( + h ; then the frctonl dfference on the rght nd of order, 0< < 1of h [ f ( s ( ] ( + ( Δ f( = E 1 f( k= 0 h ( Ck f ( k h k = ( 1 + (! Where, = re the generlzed bnoml coeffcents. Then the Jumre frctonl C k k!( k! dervtve s followng ( ( Δ+ [ f( f(0 ] f+ ( = lmh 0 h d f( = d Smlrly one cn hve left Jumre dervtve by defnng bckwrd shft opertor. In ths Jumre defnton we subtrct the functon vlue t the strt pont, from the functon tself nd then the frctonl dervtve s tken (n Remnn-Louvell sense. Ths offsettng mkes the frctonl dervtve of constnt functon s zero, nd gves severl ese nd conjugton wth clsscl nteger order clculus, especlly regrdng chn rule for frctonl dervtves, frctonl dervtve of product of two functons etc [1]. 3

4 c Some technques of Jumrre dervtve Consder functon f [ u ( ] whch s not dfferentble but frctonlly dfferentble. Jumre suggested [1] three dfferent wys dependng upon the chrcterstcs of functon. ( D1 ( f[ u( ] = fu ( u( u 1 D ( f[ u( ] ( f / u ( f = u ( u u ( 1 ( D ( f[ u( ] = (1! u f ( u u ( 3 u d Mttg-Leffler functon nd Frctonl trgonometrc functons Mttg-Leffler functon [10] s defned s nfnte seres, s followng k z E ( z =, ( z,re( > 0 k= 0 Γ (1 + k The bove defnton s the one prmeter Mttg-Leffler functon. For = 1, t s smple eponentl functon.e. E ( 1 z = ez. The frctonl sne nd cosne functons [4] defned by Mttg-Leffler functon re k k E( t + E( t k t k t cos ( t = = ( 1 = ( 1 k= 1 ( k! k= 1 Γ (k + 1 (k+ 1 (k+ 1 E( t E( t k t k t sn ( t = = ( 1 ( 1 k= 1 ( k +! k= 1 Γ (k + +1 One of the most mportnt propertes of Mttg-Leffler functon [11] s J D E( = E(, menng tht Jumre frctonl dervtve of order of Mttg- Leffler functon of order n scled vrble s just returnng the functon tself. Ths s n conjugton to clsscl clculus smlr to eponentl functon nd very-useful n solvng frctonl dfferentl equton composed wth Jumre frctonl dervtve. 3. Orgnl plne progressve wve Plne wve s specl knd of wve whch does not chnge drecton wth the tme evoluton, nd progressve wve s not dsturbed by ny boundry condton [13]. Consder plne progressve wve whch s propgtng n the postve drecton wth constnt velocty v. The generl form s f ( t, = f( vt [13].We consders here the frctonl plne progressve wve n the followng form f (, t = f ( v t, 0 < 1 (1 Here v s frctonl velocty. When s tendng to one, ths plne progressve wve turn to one dmensonl plne-wve. Thus the wve wht we consdered n equton (1 s plne progressve wve n -th order frctonl wve, where the spce nd tme s re trnsformed to nd t respectvely, nd 0< 1, the vlue of s frctonl number. Thus the wve we consdered s frctonl plne wve movng n -drecton. Now the Jumrre type frctonl dervtve [9, 1] s used to fnd vrous physcl quntty nd physcl propertes of the correspondng wve. Let us defne the opertor J J D, D nd J J D t t, D t. t 4

5 Consder f[ u(, t] = f( v t, 0< 1. Where ut (, = vt, 0 < 1 Now we choose the dfferentl trck tht s D f u u f u u. Here the 1 ( 3 ( [ ( ] = (1! u ( ( number 3 defnes the thrd trck nd fnlly 3 s not used n the dfferentl opertors. Now 1 ( D ( f[ u(, t] = (1! u f ( u u (, t We know from stndrd frctonl dervtve tht u ( u = Du(, t = D[ v t ] = D[ ] =! =Γ ( + 1 Clerly D f u t u f u 1 ( ( [ (, ] =!(1! u ( ( Smlrly From equtons ( nd( we get Opertng Now opertng D n both sde, 1 ( Dt ( f[ u(, t] = v!(1! u fu ( u ( D f[ u( ] = v D f[ u( ] t t [ ( ] [ ( ] [ ( ] D D f u = v D D f u = v D f u (b D t on both sdes D D f u = D f u = v D D f u (c t t [ ( ] t [ ( ] t [ ( ] Now usng the theorem n Append-8 nd combnng equtons (b nd (c we get J 1 D fu [ ( ] = D fu [ ( ] t v J ( D t f( v t = v D f( vt (3 Equton (4 represents the frctonl wve equton of order. If = 1 the equton turns to one dmensonl clsscl wve equton for the plne progressve wve. 4 Soluton of the wve equton Consder the soluton of equton (3 s of the type f ( t, g( = rt ( usng ths n equton (3 we get D g( = D r( t g ( v rt ( t Implyng 5

6 1 rt ( D g ( = g ( D rt ( t v Left hnd sde s spce dependent nd rght hnd sde s tme dependent. Clerly we cn equte ths equton wth constnt let The spce prt of the equton s now k 1 D g( = k or D g( = kg( g ( Soluton of the equton [11] s g ( = be ( ± k of the equton s D v t r( t = k r( t. The soluton s rt ( BE ( t ω ω = kv nd B s constnt. Thus the generl soluton s A s constnt., b s constnt. Smlrly the tme prt ( ( ω = ± where we put f ( t, = AE ± k E ± t (4 4. Dervton of frctonl Schrödnger equton Consder prtcle of mss m movng wth veloctyv. Accordng to de Brogle hypothess [14] there s wve ssocted wth every movng mterl prtcle. The mthemtcl form of de Brogle hypothess s p = k. Here momentum of the prtcle s denoted by p nd k s wve vector n one dmenson; s reduced plnk constnt. Now Plnk s hypothess [14] shows energy ε of prtcle n quntum level s proportonl to ngulr frequency tht s ε = ω. Wth ths contet t s ssumed tht De-Brogle hypothess nd Plnk s hypothess re lso vld n frctonl th order wth modfed form p ε = k (5 = ω (6 It s cler tht f =1 the equtons (5 nd (6 reduces to the orgnl form of de Brogle nd h Plnk hypothess. Here s reduced plnk constnt of order; = π, nd h s plnk constnt. Here we defned ω s frctonl order ngulr frequency nd k s frctonl order wve vector. The generl soluton for equton (4 s u = Af ( v t,0< 1 where A s constnt. To fnd the eplct form of the soluton Mttg-Leffler [10] functon s tken s trl soluton [11] s n clsscl dfferentl equton we consder the ep( s the trl soluton [15].Thus Here E ( k nd E ( t ( ( ω f (, t = AE k E t ω re Mttg-Leffler functons of one prmeter n comple vrble. Ths s trl soluton of the equton (4. Now frctonl velocty v cn be defned (7 6

7 s = ω / k. It s ssumed here tht the velocty v s constnt. Ths s the velocty of v prtcle s well s the group velocty of wve. Consderng the prtcle possess the constnt momentum p nd constnt energyε,.e. energy nd momentum does not vry wth the propgton of the wve n spce nd tme. Usng condtons of (5 nd (6 n the soluton of (7 t cn be wrtten f (, t AE p E t = ε (8 Ths prtcle hs some physcl propertes hdden nsde. To nvestgte them some opertons must be needed. It must be verfed tht how ths functon chnges wth the vrton of spce nd tme. Ths vrton my be mesured by n operton sy order frctonl dfferentton. Dfferenttng prtlly wth respect to of order of the equton (8, we get the followng J J p D f(, t A E p = E εt D f(, t = ( p f(, t (9 nd dong t once gn we get J 1 D f (, t p f(, t (10 = J We hve used 0 D E ( = E( [11]. Let s defne p = mε, where K m s mss (n frctonl frme, nd ε K s knetc energy of frctonl order. Then equton (10 cn be wrtten s J 1 D f(, t = ( m ε f(, t. Ths mples the followng K m J ( D f(, t =ε K f(, t (11 Now the vrton of the functon wth tme s studed. Thus repetng the bove steps.e. tkng Jumre frctonl dervtve of order w.r.t. tme, we hve followng J Dt f (, t = ε f (, t (1 Here ε s totl energy of the system. From the conservton of energy n frctl spce t cn be wrtten tht, Totl energy ( ε = (knetc energyε + (potentl energy V(, t.thus k K V( ε = ε +, t (13 7

8 Usng ths condton of equton (13 n equton (1 nd combnng equton (11 nd (1 we get the followng m J ( ( ε D f(, t = ( V(, t f(, t by rerrngng bove we obtn the followng (14 J J ( D f(, t + V(, t f(, t = ( Dt f(, t (14 m Where f (, t AE p E t = ε Ths s the frctonl Schrödnger equton of -th order. At the lmt = 1 the equton reduces to the Schrödnger equton n one dmenson spce nd tme. Ths equton hs the soluton whch wll led to certn nterestng physcl propertes. 4.1 Soluton of frctonl Schrödnger equton The bsc method of the soluton of equton (14 s the method of seprton of vrbles. Method of seprton of vrbles s by ssumng tht the soluton s dentfed s the product of two dfferent functons Φ ( nd Tt (, where Φ( depends on the spce vrble nd Tt ( depends on tme vrble. The functon s f (, t =Φ( T( t. Here Φ ( s the sptl functon.e. solely dependent on trnsformed-spce nd Tt ( s nother functon whch s only functon of trnsformed-tmet. Substtute f (, t =Φ( T( t n equton (14, we obtn the followng ( ( + Φ = ( ( ( 1 d d T t Φ V( ( m Φ d T t dt (15 Left hnd sde of the equton s spce dependent nd rght hnd s tme dependent. Thus, to stsfy the equton (15 both sdes must be equl to some constnt. Now on the left sde of the equton hs frctonl potentl term. Ths hs the dmenson of frctonl energy, tht s[ ML T ] [ M L = T ]. Clerly the constnt must hve the dmenson of frctonl energy due to homogenety of dmenson. From the rght hnd sde of the equton, the dmenson nlyss llows us to choose the unt of the constnt ε s (Joule for frctonl vlues of. Also t s supported by equton (1 tht ths constnt sε, ths s frctonl energy. Now equton (15 cn be wrtten s two dfferent equtons, one s solely tme dependent nd nother s only dependent on spce. d Tt ( T( t = dt ε (16 8

9 d ( ( 1 m Φ d Φ( + V( = ε (17 Soluton of equton of type (16 ws found by Ghosh et l [11] usng the Mttg-Leffler functons n the followng form Tt ( E ( εt s (omttng the ntegrl constnt s. Thus the soluton of the equton (15 ( f (, t =Ψ =Φ( E ε t (18 For = 1,.e. n lmtng cse the soluton (18 turns to the soluton of one dmensonl clsscl Schrödnger wve equton. 4. Tme ndependent frctonl Schrödnger equton nd frctonl Hmltonn The equton (17 hs no tme dependent soluton s well s the equton hs no effect wth the vrton of tme. Thus the equton (17 cn be rerrnged s d Φ( ( ε V( Φ ( = 0 ( m d (19 Ths s the tme ndependent Schrödnger equton. Ths equton s potentl dependent. So t s not possble to solve the equton wthout knowng the chrcter of potentl functon. But t cn be confrmed tht the soluton hs only spce dependency. So ths equton sys bout only the chrcterstc of the prtcle wth the vrton of spce. Ths equton s energy equton. Thus Hmltonn cn be constructed wth the nlogy of Schrödnger s one dmensonl quntum wve equton. The Hmltonn n terms of non-nteger order dervtve s defned s ^ H d = V( ( m d (0 Therefore the equton (19 cn be wrtten n terms of Hmltonn s ^ H Φ = Φ (1 ε Ths equton s nothng but n Egen equton wth the Egen vlueε. The Egen functon of the equton s Φ. The Egen functon s the nformton centre of prtcle. One cn operte t n vrous wys to fnd the correspondng physcl property. From equton (1 t s cler tht the Hmltonn s such n opertor, dong the sme job. The Hmltonn gves the correct nformton bout the energy of the prtcle. 9

10 5. Equton of contnuty Consder the Schrödnger equton prevously derved n equton (14 J J ( D f(, t + V(, t f(, t = ( D t f(, t m Multply the equton wth the comple conjugte of the soluton sy * f (, t of the equton nd rewrtng the equton s followed m J ( * * f (, t D f(, t + V(, t f (, t f(, t J ( t * = f (, t D f(, t n rght sde ( Let s tke comple conjugte of the equton (14 nd multply wth the functon f n rght sde of the equton nd the new equton s m J ( f t D f t + V t f t f t * * (, (, (, (, (, J * ( t = f(, t D f (, t Subtrctng equton ( from equton (3 we hve followng (by droppng, t (3 m ( ( [ ] ( [ ] ( ( ( t f D f f D f = f D f + f D f * J J * * J J * t (4 Now the equton cn be rewrte n the followng form J * J J * D f D f f D f m ( [ ] ( * J J * ( ( t [ ] ( t = f D f + f D f (5 * J J * Let s defne ( f D f f D f = j tht s probblty current densty of -th order m nd * f f * f f = ρ s probblty densty of -th order. For = 1 the frctl probblty densty = ρ turns to the one dmensonl probblty densty. Thus the equton (5 reduces to J J D [ j ] = D [ ρ ] (6 t Ths s the equton of contnuty of -th order n one dmenson. If probblty densty * f f = ρ s ndependent of tme, rght hnd sde s zero. Thus the left hnd sde s lso equl to zero. Ths mples tht the one dmensonl vrton of current densty wth spce s zero. Physcl sgnfcnce of the fct s tht there s no source or snk of probblty current 10

11 densty. Ths s the condton of sttonry stte. To stsfy the bove condton of ρ the ( soluton must of type f(, t =Ψ ( E ( t =Φ ε /. Ths s the sttonry stte of -th order. For = 1 the stte s sme s of the one dmensonl sttonry stte. 5.1 Propertes of frctonl wve functon For further nvestgton t s needed to chrcterze the bsc propertes of the soluton of frctonl Schrödnger equton. The frctonl wve functon must be contnuous nd should be sngle vlued. As the prtcle hs physcl estence, the frctonl wve functon of the prtcle must be contnuous t every poston of spce nd tme. If the frctonl wve functon s not contnuous for some poston or tme then the prtcle wll vnsh n the mddle of ts trjectory whch s not possble t ll. Frctonl wve functon must be sngle vlued.e. for every poston of spce tme the property of the prtcle s unque. b The frctonl wve functon must be squre ntegrble n frctonl sense.e. b Ψ Ψ d < n the regon * b c Lner Combnton of solutons of the frctonl Schrödnger wve equton tself s soluton of the system. Thus lner combnton of the wve functon s nother wve functon. d The frctonl wve functon must vnsh t the boundry. If t s does not then, the boundry tself loses ts sgnfcnces. The boundres seze the moton of prtcle to go further. As result the prtcle hs to stop t the boundry nd consequently the frctonl wve functon vnshes. Here boundry mens perfectly rgd boundry. If we hve n nlogy wth vbrtng strng bounded by two certn ponts, then we cn get no mpltude on the two end ponts. Mthemtclly the condton my be descrbed s Ψ ( =Ψ ( b = 0 f the wve s n the regon b e The frctonl Schrödnger equton suggests tht the -order frctonl dervtve J of wve functon s contnuous nd sngle vlued. D f The order frctonl dervtve of the wve functon must vnsh t the boundry. If not, condton of sttonry stte wll volte s suggested n the equton of contnuty. g The wve functon must be normlzed; tht sgnfes the estence of the prtcle s certnly mesured wthn boundry. 11

12 5.Further study on frctonl wve functon The generl soluton of frctonl wve equton s f(, t ( E ( εt / Its comple conjugte s * * ( E ( t / * Ψ =Φ ε. Multplyng Ψ wth we Ψ * * Ψ = Φ( Φ (. =Ψ =Φ. Ψ get Ths quntty s ndependent of tme. We defne ths quntty s estence ntensty nd Ψ or s estence mpltude. In certn boundry the prtcle ests certnly. So t cn be wrtten n mthemtcl form. Let Ψ s defned n the boundry + +. Then Ψ Ψ = constnt. Note tht the notton f ( d mples frctonl ntegrton tht s * d 1 1 ( = Γ( ( ( f d ξ f ξ dξ wth > 0. If the prtcle does not est n the boundry the ntegrton vnshes. Now we cn defne f the prtcle ests certnly nd + Ψ Ψ * d = + * Ψ d Ψ = 0 f the prtcle does not est nywhere. Clerly esten * * ce prmeter ΨΨ d s such tht the condton 0 Ψ d Ψ 1, gets stsfed. Consder we hve to fnd the nformton bout estence over certn regon nsde the boundry. Then the quntty + + b * Ψ d l Ψ = should be less thn 1. It defnes tht prtcle s not loclsed nd t s convenent becuse the prtcle behves lke wve nd wve s not loclzed. If ll ths estence prmeter or probblty dds, the whole probblty s unty. From the equtons (9, (10, (1, (17 we found tht wve functon s Egen functon of vrous opertors Orthogonl nd norml condtons of wve functons Two functons F( nd G ( defned n the regon b re orthogonl f ther nner product s zero [15]. From the nlogy of ths orthogonl condton n {} spce we cn defne the orthogonl condton for { } spce wth followng frctonl ntegrton operton such tht, b * ( ( F G = F G d = 0 (7 Here b * ( ( F G = F G d s defned s nner product of order where comple conjugte. F * ( s In the sme wy norml condton cn be defned by the followng frctonl ntegrton b * = ( ( = 1 F G F G d (8 1

13 The generl soluton of wve functon s Ψ = cψ, here c s some constnt. Here s * c * * j jψ j dummy nde. The comple conjugte of the soluton s Ψ = the nner product usng Drc s Brcket notton * cj c j j Ψ Ψ = ψ ψ (9 From orthogonl nd norml condton we hve the followng * Ψ Ψ = cj c ψ jψ = 0 f j nd j j.in the smlr wy = j * * cj c ψ jψ 1 j Ψ Ψ = = ψ ψ = 1 f = j.clerly * Ψ 1 Ψ = cc =.More c precsely t cn be wrtten Ψ Ψ =. Now we cn defne c s estence coeffcent or probblty coeffcent. = 1 f 6. Opertors nd epectton vlues In quntum mechncs ll the mesureble qunttes tht cnnot be mesured drectly re mesured by epectton vlues [15]. So n the cse of order quntum mechncs t needs to defne opertors for every mesurble quntty. For ths purpose there must be some rules of choosng opertors. Every opertor must be Egen opertor of the wve functon. Egen vlue of the opertor defnes mesurble quntty. Epectton vlue of n opertor s the mesure of the correspondng opertor. Consder n opertor ^ A opertes on certn functon ^ λ A Ψ such tht Ψ = Ψ (30 From the generl form of Ψ, the equton turns to A ^ [ ψ ] = c ^ A ψ = λcψ. Thus λ cnnot be determned drectly. For the correct nformton of the system we hve to fnd the men vlue or epectton vlue of the system. Epectton vlue of n opertor s defned s A = + ^ * ψ A ψ d + * ψ ψd (31 13

14 For every physcl mesurble quntty there s correspondng epectton vlue. 7. Smple pplcton-prtcles n one dmensonl nfnte potentl well Consder prtcle s bounded by one dmensonl nfnte potentl well wth length = 0 to = for 0. The potentl s defned here s of the type of V = 0 f 0 nd V = otherwse. Thus the prtcle s strctly bounded by the potentl well n the trnsformed scle too. So wve functon s lso zero outsde the well. For contnuty, the wve functon must vnsh t the boundres lso.e. Φ (0 =Φ ( = 0. The frctonl Schrödnger equton s suggested n equton (19 s d Φ( ( ε V( Φ( = 0 ( d m In ths we tke V( = 0. Thus the equton s of the form s follows Rerrngng we get the followng Let s tke d Φ( ε ( 0 Φ = ( d d m Φ( ( m ε + Φ ( = 0 d ( m ε = k (31 nd the equton s now s followng Ths equton hs soluton s suggested by Ghosh et l [11] d Φ( + k ( 0 Φ = (3 d ( ( Φ ( = AE k + BE k Usng boundry condton Φ (0 =Φ ( = 0, we get A+ B= 0. Thus the soluton (3 s Φ ( ( = B E ( k E ( k Usng the defnton of frctonl sne functon [9] we wrte the followng Usng boundry condton on equton (34 we get gn (33 ( Φ = Csn ( k (34 ( Φ = Csn ( k = Φ (0 = 0 (34 14

15 As defned by Jumre [11] sn ( = sn (( + M here we defned M s frst order zero or frst zero crossng [16] for -th ordered sn functon. Sncesn (0 = 0, therefore sn (( M = 0 (34b =, we get k = ( M Comprng equtons (34 nd (34b sn (k sn (( M mplyng k M = ( M Usng the vlue of k = ( n equton (34 the soluton s M Φ ( = sn ( C (34c 7.1 Normlzton of wve functon The normlzton condton for wve functon of -th order s * Φ d Φ = 1 (35 0 * Now Φ ( = Csn ( k s rel so ΦΦ = Φ ; then Φ d = 1. Here * ΦΦ =Φ s ( Φ = C sn ( k. Thus the ntegrton s sn ( C k d 1 0 =. To ntegrte the equton n dentty must be developed. By defnton we hve followng cos Now we hve followng denttes By defnton we hve ( cos 1 = ( cos 1 = ( E ( + E ( = E( + E( cos ( 1 = 1 E( + E( cos ( 1 = ( ( E ( + E ( E ( E ( ( E( E( sn ( E ( E ( = (36 b 0 15

16 So we get the followng dentty ( ( Usng the dentty of equton (37, we hve followng Also C sn ( k 1 cos = sn (37 = 0 C sn ( k d 1 1 C ( (1 cos k d = ( ( ( C d C cos k d = = C sn ( k = 1 Γ (1 + k = 0 C sn ( k = 1 Γ (1 + C s zero s suggested by boundry condton. Thus ( ( Γ (1 + M =. Now the soluton s ( sn ( Γ (1 + ( (1 1 Γ + = or Φ =. For = 1 the soluton s converted to one dmensonl soluton for one dmensonl Schrödnger equton of nfnte potentl well. 7. Grphcl representton of wve functon Grphcl presentton of order for =10 Γ (1 + ( sn ( k Φ = for dfferent vlues of the frctonl unt s shown n the fgure-1. Before the plot we need to know the vlues of M for vrous. We found usng Wolfrm Mthemtc-9 the vrous ppromte vlues of ( M for terms of Mttg-Leffler sn functon nd numerclly t cn be shown tht sn ( losses perodcty for < 1. They re lsted below n Tble-1 16

17 Tble-1: Frst zeros of functonsn ( fter = 0 for dfferent. ( M

18 =0.736 = = 0.80 = = 0.9 = = 1.0 Fgure-1: Grphcl presentton of Γ (1 + ( sn ( k Φ = for = 0.736,0.75,0.8,0.85,0.9,0.95nd

19 Γ (1 + The plot s drwn Φ ( = sn ( k gnst. Here the bo wdth s tken s 10 unt tht s =10. From numercl nlyss we found tht the quntum boundry condtons re stsfed up to from = 1. The plot suggests tht the mm of the wve functon shfts to the rght wth the ncrese of vlue. More thn tht the nture of wve functon chnges wth vlue. But t = 1 the plot s sme s suggested by one dmensonl Schrödnger potentl bo problem. At 0.736, the curve s not symmetrcl nd more re covers n the left sde thn other. Less the vlue mens more symmetrcl s the plot. These plots hve one zero crossng [16]. Ths mens the quntum number of the system s 1. The system s n ground stte. To compre the wve functons for vrous vlues we hve nother plot whch s gven below-(fgure Fgure-: Grphcl presentton of Φ ( for dfferent vlues of for =10. If we choose the bo length s 5.7 unts, the plot wll be s below (Fgure-3 19

20 Fgure-3: Grphcl presentton of Φ ( for dfferent vlues of for = Probblty densty As we got wve functons for vrous vlues of, we cn lso get probblty densty * ρ = ψψ. Probblty densty plot for vrous s gven below * Fgure-4: Grphcl presentton of ψ ψ for dfferent vlues of for =10. 0

21 7.4 Energy clculton Now we cn clculte energy of the prtcle. Usng equtons (31 nd (34c tht s ( M mplyng ( / ( / ( m ε / = / 1 ( ( ε = m M. For = 1the energy s ε = / m π /. Ths s the energy for frst quntum stte s descrbed n quntum mechncs. 8.0 Conclusons Usng frctonl dervtve of Jumre type we found tht quntum mechncs n frctonl regon.e < 1 regon quntum behvour of the prtcle chnges drmtclly. In ths regon equton of contnuty s successfully mntned nd the sttonry condton lso holds. For = 1ll the equtons re norml clsscl the Schrödnger equton n norml spce. We studed prtcle n bo problem nd found tht the wve equton n frctonl sense lso meets the condton. Further the wve functon s not symmetrc tll = 1. For < < 1the pek of wve functon s left sded.e. the pek s on the left sde of the mddle pont of the bo. The estence mpltude.e. wve functon hs hgher mpltude for hgher nd t s mmum when = 1. Thus the wve functon or estence mpltude s dependent. We need further study on the frctonl Quntum mechncs of < to understnd the nternl behvour of the quntum sttes. 9.0 Reference [1] S. Ds. Functonl Frctonl Clculus nd Edton, Sprnger-Verlg 011. [] S. Zhng nd H. Q. Zhng, Frctonl sub-equton method nd ts pplctons to nonlner frctonl PDEs, Phys. Lett. A, [3] J. F. Alzdy. The frctonl sub equton method nd ect nlytcl solutons for some non-lner frctonl PDEs. Amercn Jounl of Mthemtcl Anlyss ( [4] H. Jfr nd S. Momn, Solvng frctonl dffuson nd wve equtons by modfed homotopy perturbton method, Phys. Lett. 007 A [5] K.S Mller nd B Ross. An Introducton to the Frctonl Clculus nd frctonl Dfferentl Equtons.John Wley & Sons, New York, NY, USA; 1993 [6] I. Podlubny. Frctonl Dfferentl Equtons, Mthemtcs n Scence nd Engneerng, Acdemc Press, Sn Dego, Clf, USA. 1999;198. [7] K. Dethelm. The nlyss of Frctonl Dfferentl equtons. Sprnger-Verlg, 010. [8] A. Klbs, H. M. Srvstv, J.J. Trujllo. Theory nd Applctons of Frctonl Dfferentl Equtons. North-Hollnd Mthemtcs Studes, Elsever Scence, Amsterdm, the Netherlnds, [9] G. Jumre. Modfed Remnn-Louvlle dervtve nd frctonl Tylor seres of non-dfferentble functons Further results, Computers nd Mthemtcs wth Applctons, 006. (51, [10] G. M. Mttg-Leffler. Sur l nouvelle functon E (, C. R. Acd. Sc. Prs, (Ser. II 137, (

22 [11] U. Ghosh, S. Sengupt, S. Srkr nd S. Ds. Anlytc soluton of lner frctonl dfferentl equton wth Jumre dervtve n term of Mttg-Leffler functon. Amercn Journl of Mthemtcl Anlyss. 015; 3( [1] U. Ghosh, S. Srkr nd S. Ds. Solutons of Lner Frctonl non-homogeneous Dfferentl Equtons wth Jumre Frctonl Dervtve nd Evluton of Prtculr Integrls. Amercn Journl of Mthemtcl Anlyss. 015; Amercn Journl of Mthemtcl Anlyss, 015, Vol.3, No.3, pp [13] D.P. Ry-Chudhur. Advnced Acoustcs. The new Book Stll [14] J. L. Powell nd B. Crsemnn, Quntum Mechncs, Addson-Wesley, [15] G. B. Arfken, H. J. Weber nd F. E. Hrrs, Mthemtcl Methods for Physcst. Acdemc press. Seventh edton. 01. [16] D. J. Grffths, Introducton to Quntum Mechncs, Person Educton,Inc., second edton,nnth mpresson, 011 [17]G. Jumre. An pproch to dfferentl geometry of frctonl order v modfed Remnn-Louvlle dervtve. Act Mthemtc SINICA ( [18] S Ds. Mechnsm of wve dsspton v Memory Integrl vs--vs Frctonl dervtve, Interntonl Journl of Mthemtcs & Computton Vol-19, Issue-, 013, pp [19] U Ghosh, S Srkr, S Ds. Frctonl Weerstrss Functon by Applcton of Jumre Frctonl Trgonometrc Functons nd ts Anlyss, Advnces n Pure Mthemtcs, 015, 5, pp [0] U Ghosh, S Srkr, S Ds. Soluton of System of Lner Frctonl Dfferentl Equtons wth Modfed Dervtve of Jumre Type, Amercn Journl of Mthemtcl Anlyss, 015, Vol. 3, No.3, pp7-84. [1] G. Jumre. On the dervtve chn-rules n frctonl clculus v frctonl dfference nd ther pplcton to systems modellng. Cent. Eur. J. Phys ( [] L. Nottle, Frctl Spce Tme n Mcrophyscs, World Scentfc, Sngpore, [3] Abhy Prvte, A D Gngl, Clculus on Frctl Subset of Rel-Lne-I: Formulton, Frctls, Vol 17, No. 1, (009, [4] Abhy Prvte, A D Gngl, Frctl Dfferentl Equtons nd Frctl-tme dynmcl systems, Prmn- Journl of Physcs, Vol. 64, No. 3, pp , 005. [5] G. Jumre. Modfed Remnn-Louvlle dervtve nd frctonl Tylor seres of non-dfferentble functons Further results, Computers nd Mthemtcs wth Applctons, 006. (51,

23 Append 1. Frctonl mss Frctonl mss m my be defned s m = ρ d, where ρ s frctonl lner mss densty n one dmenson. We hve consdered tht the densty of mss s sme s t s n the cse = 1..Frctonl velocty The chnge of frctonl dsplcement d per unt chnge n frctonl tme dt s the frctonl velocty.e. = v d dt 3. Frctonl wve length Frctonl wve length cn be demonstrted by plot (Fgure: A-1 of frctonl wve of the order = 0.8 Fgure A1: Showng frctonl wve-length The wve length s the dstnce AB. Tht s the dstnce covered by frctonl wve n full frctonl cycle. Frctonl wve length s not fed quntty. It chnges wth the evoluton of frctonl tme. 3

24 4. Frctonl Tme perod The tme tken N to wve to cover the dstnce AB s the frctonl tme perod. We should tke cre tht ths s frst order tme perod. As wvelength chnges, the tme perod lso chnges wth the wve propgton. But we ssume tht λ = vn 5. Frctonl ngulr frequency Fgure A: Showng concept of frctonl ngulr frequency The bove plot s the Polr plot of the frctonl wve of the order = 0.8. In ths polr plot we cn esly see tht the wve s returned to the sme pont fter completng frctonl cycle.e. n ts orgn. By polr plot we cn sy tht the ngle trversed n full frctonl cycle s π. Thus frctonl ngulr frequency cn be ssgned s ω = π N From ths we cn see tht the product of frctonl ngulr momentum nd frctonl tme perod N s lwys π though both re vryng. 6. Frctonl wve constnt (or vector n 3Dmenson From the nlyss of frctonl wve of equton (8 f k ω t = 0 tht s phse prt of the wve s zero, we cn get k ω ( t Now = ( ω / = π /( k = / = ω / v s / t = v s frctonl velocty. k v v N, usng ppend (5. Now usng ppend (4 we hve = π / λ 7. Frctonl reduced Plnk constnt In ths pper we hve ntroduced frctonl Plnk constnt s bsc constnt for ordered frctonl system. For the lmtng condton of ths constnt s of the form of reduced Plnk constnt. 4

25 8. Theorem If f ( y, be functon whch s frctonlly -th order dfferentble wth respect to both the vrble nd t then D y D f(, y D D y (, y = f or (, f y = f ( y, equvlently where 0 1. Proof: Consder functon φ ( = f( y, + k f( y,, k> 0. Now frctonl men vlue theorem sttes tht [5] h φ( + h φ( = φ ( + θh, where 0< θ < 1,0 1 Γ (1 + h φ( + h φ( = [ f ( + θh, y+ k f ( + θh, y] Γ (1 + Let F( y = f ( +θh, y. Then usng frctonl Men vlue theorem we hve h h k φ( + h φ( = [ F( y+ k F( y] = [ Fy ( y+ θ1k Fy ( y] Γ (1 + Γ (1 + Γ (1 + y h k = [ fy ( + θh, y+ θ1k]where 0 < θ, θ1 < 1,0 1 Γ (1 + Γ (1 + On the other hnd φ ( + h = f ( + h, y+ k f ( + h, y. Therefore φ( + h φ( = f( + hy, + k f( + hy, f( y, + k + f( y, f(, y+ k f(, y fy (, y =Γ (1 + lm k 0 k or f (, (, y y+ h fy y fy (, y =Γ (1 + lm h 0 h f ( + h, y+ k f( + h, y f(, y+ k + f(, y = ( Γ (1 + lm lm h 0 k 0 k h φ( + h φ( = ( Γ (1 + lm lm h 0 k 0 k h = Γ (1 + lm lm f ( + θh, y+ θ k ( h 0 k 0 or fy (, y = fy (, y Hence the theorem s proved. Whch we hve used n our detled dervton. y 1 y 5