Some Solutions to the Fractional and Relativistic Schrödinger Equations

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87- DOI:.593/j.ijtmp.555.3 Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions Yuchun Wei Deptment of Rdition Oncology, Wke Foest Univesity, NC Abstct Lskin intoduced the fctionl quntum mechnics nd sevel common poblems wee solved in piecewise fshion. Howeve, Jeng et l pointed out tht it ws meningless to solve nonlocl eqution in piecewise fshion nd tht ll the solutions in publiction wee wong except the solution fo the delt function potentil, which ws obtined in the momentum spce. Jeng s citique esults in cisis of fctionl quntum mechnics, tht is, in mthemtics it is quite difficult to find solutions to the fctionl Schödinge eqution nd in physics thee is no eliztion fo the fctionl quntum mechnics. In ode to eliminte this cisis, this ppe epots some nlytic solutions to the fctionl Schödinge eqution without using piecewise method, nd intoduces the eltivistic Schödinge eqution s eliztion of the fctionl quntum mechnics. These two siste equtions should be studied t the sme time. Keywods Fctionl quntum mechnics, Fctionl Schödinge eqution, Reltivistic quntum mechnics, Reltivistic Schödinge eqution. Intoduction In, Lskin intoduced the fctionl quntum mechnics [-3]. As the fist exmple he solved the infinite sque well poblem in piecewise fshion [3]. Since then, the piecewise method hs been widely used in this field. In, howeve, Jeng, et l [4] citicized tht it ws meningless to solve nonlocl eqution in piecewise fshion nd they demonstted thtit ws impossible fo the gound stte function to stisfy the fctionl Schödinge eqution ne the boundy inside the well. In seies of ppes [5-8], Byin insisted tht he explicitly completed the clcultion in Jeng s ppe nd the wve functions did stisfy the fctionl Schödinge eqution inside the well. Hwkins nd Schwz [9] climed tht Byin s clcultion contined seious mistkes. Luchko [] povided some evidence tht the solution did not stisfy the eqution outside the well. On the othe hnd, Dong [] e-obtined the Lskin s solution by solving the fctionl Schödinge eqution with the pth integl method. It is not esyfo edes to judge thei mthemticl gument [, 3], but wegee with Jeng tht the piecewise method to solve the eqution is wong, since ecently we explicitly nd ingubly showed tht the Lskin s functions did not stisfy the fctionl Schödinge eqution with = nywhee on the x-xis [4]. Accoding to Jeng, et l [4], only the solution fo the delt function potentil [5-7] ws cceptble nd they * Coesponding utho: yuchunwei@gmil.com (Yuchun Wei) Published online t http://jounl.spub.og/ijtmp Copyight 5 Scientific & Acdemic Publishing. All Rights Reseved themselves povided solution fo the one dimensionl hmonic oscillto potentil fo the cse = [4]. Redes hve been looking fowd to some othe solutions to the fctionl Schödinge eqution since the simple solutions fo the infinite sque well potentil wee dispoved. Jeng et l [4] lso showed thei concen tht thee ws no eliztion of the fctionl quntum mechnics. Jeng s citique esulted in cisis within fctionl quntum mechnics. In mthemtics it is not esy to find solution to the fctionl Schödinge eqution, nd in physics it is not esy to find eliztions fo the fctionl quntum mechnics. In ode to eliminte this cisis, this ppe epots some solutions to the fctionl Schödinge eqution without using the piecewise method, nd intoduces the eltivistic Schödinge eqution [8-], s eliztion of the fctionl quntum mechnics. Sevel solutions fo the eltivistic quntum mechnics e lso pesented.. The Reltivistic Schödinge Eqution: A Reliztion of the Fctionl Schödinge Eqution In this section we will list the stndd, fctionl, nd eltivistic Schödinge equtions in one- nd thee-dimensionl spces, nd explin why we clim the eltivistic Schödinge eqution is n ppoximte eliztion fo the fctionl Schödinge eqution... The Schödinge Eqution In the stndd quntum mechnics [, 3], the time-independent Schödinge eqution is

88 Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions ( ) ( ) H, () whee ( ) is wve function defined in the 3 dimensionl Euclidenspce R 3, E is n enegy, nd is vecto in the 3 dimensionl spce. The Hmiltonin opeto is the summtion of the kinetic enegy opeto nd the potentil enegy opeto. The stndd kinetic enegy opeto is H = T + V ( ) () T = p m = m, (3) whee p = i is the momentum opeto. As usul, m is the mss of pticle nd ħ is the educed Plnk constnt. The one dimensionl time-independent Schödinge eqution is d + = ( x) V( x) ( x) E ( x) m dx whee the wve function ( x) nd the potentil V( x ) e defined on the x-xis... The Fctionl Schödinge Eqution nd Its Scling Popety In [-3], Lskin genelized the clssicl kinetic enegy nd momentum eltion (3) to p = = χ = ( ) /, (4) T D p mc D, (5) mc, whee is the fctionl pmete, the coefficient D χ mc /( mc) χ is positive numbe dependent on, nd c is the speed of the light. Oiginlly Lskin [-3] eve equied the fctionl pmete <, but in this ppe we llow < <, s in [4, 9], with n emphsis on the simplest nonlocl cse =. In the cse =, tking χ = /, the fctionl kinetic enegy is the sme s the clssicl kinetic enegy p p = p = = =. (6) T D mc mc m T In the cse =, tking χ =, the fctionl kinetic enegy is the ppoximte kinetic enegy in the extemely eltivistic cse, p T D mc c mc The definition of the fctionl kinetic enegy opeto is p 3 T( ) = D 3 ϕ( ) exp( i ) d 3 p p p R ( π ), (8) whee is clled the wvefunction in the momentum spce [, 3]. The fctionl Schödinge eqution is = p = = p. (7) p 3 ϕ( p) = 3( )exp( i ) d 3 R ( π ) (9)

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87-89 When fctionl Schödinge eqution is ( ) ( ) H () H = T + V(). () =, the fctionl Schödinge eqution ecoves the stndd Schödinge eqution; when = Fo =, the following scling popety is stightfowd. ( ) ( ), the H. () Scling popety. If wve function ( ) nd n enegy E is solution of the fctionl Schödinge eqution fo the potentil V ( ) then ( λ ) nd D p ( ) + V ( ) ( ) (), (3) λ E is solution of the fctionl Schödinge eqution fo the potentil λv ( λ ), i.e. D p ( λ) + λv ( λ) ( λ) = λe ( λ ). (4) In this ppe λ is n bity positive numbe. The poof is tivil. Fom (3) we hve D p ( λ) + V ( λ) ( λ) ( λ ). (5) λ Fo potentil stisfying λv ( λ ) = V ( ), such s () the coulomb V ( ) = Ze / electon nd Z the ode numbe of n tom, o () the dil delt function potentil V ( ) = V δ () V >, the scling popety cn be descibed simply s follows. with e the chge of n with the constnt Scling popety. Fo potentil V ( ) with popety V ( ) = λv ( λ ), if wve function ( ) E is solution of the fctionl Schödinge eqution H ( ) ( ), then the wve function ( ) enegy λ E is lso solution. The one dimensionl fctionl Schödinge eqution is When =, we hve ( ) ( ) H x E x nd n enegy λ nd the = (6) H = T + V( x). (7) ( ) ( ) H x E x = (8) d H T V( x) D p V( x) D V( x) dx = + = + = H +. (9) In this ppe, the bold H denotes the Hilbet tnsfom [4] while noml H denotes the Hmiltonin opeto. Fom the definition of the fctionl kinetic enegy opeto, we hve

9 Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions T ( x) = D p ( x) = D p ( x ') exp( ipx '/ ) dx ' exp( ipx / ) π D = k ( x ') exp( ikx ') dx ' exp( ikx) dk π D = ( i sgn( k))( ik) ( x ') exp( ikx ') dx ' exp( ikx) dk π D d = ( i sgn( k)) ( x ') exp( ikx ') π dx dx ' exp( ikx) dk d d = D H ( x) = DH ( x) dx dx d d = D ( x) * = D ( x) * dx πx dx πx = D ( x) * π x whee * denotes the convolution, nd / x nd (/ x)' = / x e genelized functions [5]. We point out tht the genelized function / ( π x ) is the well-known idel mp filte, which plys n impotnt ole in the theoy nd pplictions of Computed Tomogphy [6, 7]. We will discuss the eltionship between fctionl quntum mechnics nd the computed tomogphy in nothe ppe. The Dic delt potentil V ( x ) V δ ( x) V x = λv λx. Bsed on the scling = with V > stisfies ( ) ( ) popety, if wve function ( x) nd n enegy E is solution of the fctionl Schödinge eqution H E delt potentil well, then the wve function ( λ x) nd the enegy λ E is lso solution. See Poblems 3, 7, & 9..3. The Reltivistic Schödinge Eqution Accoding to the specil eltivity, the evised kinetic enegy is [8] () = with 4 p, () T = c + mc whee the subscipt mens specil eltivity. Fo the cse of low speed, the eltivistic kinetic enegy is ppoximtely the summtion of the est enegy nd the clssicl kinetic enegy ( = ) p T mc + = mc + T m, () nd fo the cse of extemely high speed, whee the est enegy cn be neglected, the eltivistic kinetic enegy is the fctionl kinetic enegy with = T p c= T. (3) Genelly speking, if the speed of pticle inceses fom low to high, the eltivistic kinetic enegy T will ppoximtely coespond to fctionl kinetic enegy T, whose pmete chnges fom to. Theefoe the eltivistic kinetic enegy is n ppoximte eliztion of the fctionl kinetic enegy. The definition of the eltivistic kinetic enegy opeto is 4 T() = c + mc() 4 p 3 = 3 ( ) exp( ) 3 ϕ p p c + mc i R ( π ),(4)

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87-9 whee ϕ( p ) is the wve function in the 3D momentum spce. The eltivistic Schödinge eqution is Accodingly, the D eltivistic Schödingee qutionis ( ) ( ) H (5) H = T + V. (6) H ( x) E ( x) = (7) 4 m c H = T + V = c + + V( x).(8) x As two sistes, the eltivistic nd fctionl Schödinge equtions should be studied t the sme time. 3. Solutions to the Fctionl Schödinge Eqution We will study the one dimensionl poblems fist nd then the 3D poblems. 3.. One Dimensionl Poblems Poblem. (Thefee pticle.) Fo V( x ) =, the fctionl Schödinge eqution H hs the solutions with < k <, o equivlently with k. ( x) exp( ikx) = (9) E = D ( k ) (3) ( x) sin( kx), o cos( kx) = (3) E = D ( k) (3) Poof. Accoding to the definition of the fctionl kinetic enegyopeto [-3], we obviously hve / H ( x) = DT ( x) = D( ) exp( ikx), = D ( k ) ( x) ( x) (33) with < k <. Futhe, fo k > we hve H sin ( kx) = H ( exp( ikx) exp( ikx) ) i = D ( k ) ( exp( ikx) exp( ikx) ) i, (34) = D ( k ) sin ( kx) = E sin ( kx), nd H cos( kx) = E sin ( kx), H =. (35) Genelly, if V( x) = V with V constnt, the eigen-functions do not chnge but the new eigen-enegies become ( ). (36) E = D k + V

9 Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions Poblem. (A peiodic potentil.) Fo the potentil whee k π / ( ) = ( ) V x D k b, (37) b+ cos( kx ) =, is positive el numbe (s thoughout this ppe), b is el numbe, nd < <, the fctionl Schödinge eqution H hs solution Poof. Since We hve D p ( ) x = b+ cos( kx) (38) E = D ( k ). (39) = p cos( kx) ( k) cos( kx) p = ( ) = D p b+ cos( kx) ( ) = + D( k) cos( kx) = D ( k ) b+ cos( k x) D ( k ) b b = D( k) b+ cos( kx) D( k) b+ cos( kx) V. ( ) ( ) b+ cos( kx ) This completes the poof. Futhe, we cn clculte the vege of the kinetic nd potentil enegies of the pticle. Since the nomlized function is The veges of the kinetic nd the potentil enegy e The vege of the totl enegy is Poblem 3. (The Delt potentil well.) * ( ) ( ) = ( + cos( )) x x dx b k x dx ( ) = b + / = (b + ) Ψ = + (b + ) ( x) ( b cos( kx) ) (4) (4) (4). (43) * ( ) ( ) ( ), (44) < T >= Ψ x D p Ψ x dx = D k b + * b ( ) ( ) ( ). (45) < V >= Ψ x V Ψ x dx = D k b + < H > = < V >+< T > = D k = E ( ). (46)

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87-93 Fo Dic delt function potentil V( x) = Vδ ( x) with V >, the fctionl Schödinge eqution H hs solution in the sense of cetin limit. Poof. The fctionl Schödinge eqution cn be ewitten in the momentum spce s [] ( x) δ ( x), E= - whee ϕ ( p) is the wvefunction in the momentum spce. = (47) ( ) ( ) ( ) D p x - V δ(x) x E x - V D p ϕ ϕ ϕ π ( p ) ( p) = E ( p) ( ) + V ( ) D p ϕ p E ϕ p = ϕ( p) π = (48) (49), (5) We fist chnge the integl limit in the bove eqution to finite positive numbe p, nd then let p. Thus we hve Tking n integl of the two sides, we hve If we hve ( ) + V + p D p ϕ p E ( p) = ϕ( p) π p ϕ V π D p + E + p p ( p) = ϕ( p) V + p + p + p ϕ p π p D p + E p ( p) = ϕ( p) ϕ ( p) = + p p ( p) (5) (5) (53) ϕ (54) V + p π p D p + E = V ln / π D + ( pd E) Dp E = exp( π D / V ) (55) (56) (57) Dp + E Dp / E +. (58) Hee mens tht the wve functions t both sides e equl except fo constnt coefficient. Obviously, constnt coefficient is not impotnt fo n eigen-wvefunction. Theefoe we hve

94 Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions Accodingly, in el spce we hve We cn simply wite the solution s lim E =, (59) p p ( p) lim ϕ =. (6) p ( x) lim δ( x). (6) ( x) δ ( x), E= - =, (6) which completes the poof. Let us compe this solution with the solution in the stndd quntum mechnics. This solution indictes tht the pticle flls inside the potentil well completely, while the solution to the stndd Schödinge eqution with delt potentil well indictes tht the pticle ppes outside the well minly. Theefoe we see tht Lskin s pticles e esie to be tpped thn Schödinge s pticles. The second diffeence is tht the delt potentil well poblem hs unique bound stte in the stndd quntum mechnics but hs moe thn one bound sttes in the fctionl quntum mechnics ( = ), s we will see soon. Poblem 4. (The line potentil.) Fo line potentil V = Fx with F >, the solutions to the fctionl Schödinge eqution H e whee the functions C() nd S() e Fesnel integls, nd π ξ π ξ (x) = C( ξ /) cos + S( ξ /) sin 8 4 8 4 (63) with E R. F E ξ = x D F (64) Poof. The fctionl Hmilton is In the momentum epesenttion [] The Fctionl Schödinge eqution is Its solution is with A n bity el numbe. In the el spce, the wvefunction is H H =D p =D p + Fx (65) d + i F (66) d D p ϕ( p) + i F ϕ( p) = Eϕ( p) (67) D d E ϕ( p) = i p i ϕ( p) F F. (68) D E ϕ ( p) = Aexp( i p p i p) F F (69)

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87-95 whee the coefficient This completes the poof. Futhemoe, since (x) = ϕ( p)exp( ipx / ) π A id exp( ie ix = p p p+ p) π F F A D E x = cos( p p+ p) π F F A D E = cos( p + ( x ) p) π F F A F F E = cos( u + ( x ) u) du π D D F F = A cos( u ξu) du D π + = C the limit behvio of the wvefunction is Poblem 5. (A peiodic potentil.) The peiodic function X(x) defined by cos( u + ξu + ξ /4 ξ /4) du = C cos ( u + ξ /) ξ /4 du = C cos( u + ξ /) cos ξ /4 + sin( u + ξ /) sin ξ /4du ξ cos( + ξ + + ξ ξ = C co s u / ) du C sin sin( u / ) du 4 4 ξ cos + sin sin ξ/ ξ/ ξ = C cos u du C u du 4 4 π ξ π ξ = C C( ξ / ) cos + C S( ξ / ) sin 8 4 8 4 F C A D π (7) =. (7) C( + ) = S(+ )= π / 8, C( ) = S(- )=- π / 8, (7) (x) = C x + ξ π π sin( + ) x 4 4 (73) X( x) = x + / x [, ] (74) X( x+ ) = X( x) x (, ) (75) is clled tingul wve, whee > is el numbe, whose popeties hve been studied cefully in electonics [8-3].

96 Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions Fo the potentil D π x = ln tg π X( x) ( ) V x the fctionl Schödinge eqution H hs the solution Poof. We hve Theefoe we hve which completes the poof. In the bove poof, we used fomul ( x) X ( x) (76) = (77) E =. (78) H ( x) = D p X( x) d = D H ( X( x) ) dx π = D H( sgn(sin x)) π x = D ln tg π = VxXx ( ) ( ) = V( x) ( x). (79) H ( x) + V( x) ( x) =, (8) π π x (sgn(sin x)) = ln tg π which cn be seen in book [3] (Eqution 6.4, pge 9, vol. ). Poblem 5. (The Dic comb.) The Schödinge eqution H with the Dic comb potentil hs solution nd Poof. Since H (8) V ( x) = δ ( x n) m n= ( x) X ( x) (8) = (83) E =. (84) d d π n X ( x) = sgn(sin x) = ( ) δ ( x n) m dx m dx m n=, (85)

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87-97 we hve V( x) X ( x) = X ( x) δ ( x n) m n= n = δ( x n) X ( n) = ( ) δ( x n), m m n= n= (86) d X ( x) + VX ( x) =, (87) m dx which completes the poof. We include this well-known esult in stndd quntum mechnics hee so tht the ede cn compe the stndd nd the fctionl Schödinge equtions conveniently. Poblem 6. Fo the potentil V ( x) = D, (88) x + with >, the fctionl Schödinge eqution H hs two nd only two bound sttes Poof. We need well-known Hilbet tnsfom pi [4] Tking the deivtive of the bove two equtions, we hve Multiplying by the constnt D, we hve Theefoe we hve ( ) = x, = + / x E D, (89) x ( x) =, E = x +. (9) x H = (9) x + x + x = x + x + H. (9) H = + (93) d dx x + x + x + ( ) d x x dx x + ( x + ) H = (94) d D D D dx x = H x + (95) + + x + x + d x x D H = D (96) dx x + x + x + D p ( x) ( x) V ( x) (97) D p ( x) = E V ( x) (98)

98 Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions whee E = D /, E =. (99) Fom the shpes of the wve functions [4], we know tht ( ) is the gound stte, nd ( ) x x is the fist excited stte. Since the excited enegy E =, we know tht thee e no moe excited sttes. The pticle hs only two bound sttes, the gound stte nd the excited stte. This completes the poof. Let us futhe clculte the veges of the kinetic nd potentil enegies in this elegnt poblem. The nomlized functions e ( x) Ψ = π x ( x) Ψ = π x + x Since the wvefunction Ψ is even nd Ψ is odd, the two sttes obviously e othogonl, i.e. + * * ( x) ( x) dx ( x) ( x) dx In the gound stte, the veges of the kinetic nd potentil enegies e In the excited stte, the veges of the kinetic nd potentil enegies e Poblem 7. Fo the delt function potentil the fctionl Schödinge eqution hs two solutions () () Ψ Ψ = Ψ Ψ =. () 3 D D < V >=, < D p > =. (3) D D, < V >= < D p > =. (4) ( ) V x = D πδ ( x) (5) ( ) ( ) H x E x ( x) Poof. In the bove exmple, let, nd notice tht This completes the poof. The two solutions cn lso be witten s = (6) = πδ ( x), E = (7) ( x ) =, E = (8) x lim = πδ ( x), x + x lim =. x + x ( x) ( x ) (9) = δ( x), E = () =, E π x = ()

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87-99 This esult is consistent with the solution fo Poblem 3 nd the scling popety discussed in Sec. II. Poblem 8. Fo the potentil with >, the fctionl Schödinge eqution H ( x) E ( x) 4 V ( x) = D () x + = hs bound stte x D ( x) =, E= ( x + ) Poof. Tking the deivtive of the two sides of Eqution (95), we hve ( x + ) (3) d D 4 D H D dx = + (4) x + x + x + D p E - V( x) = (5) x + x + x + This completes the poof. Notice tht the potentil in this poblem is just times the potentil in Poblem5, but thei solutions e completely diffeent. Poblem 9. Fo the potentil the fctionl Schödinge eqution hs bound stte ( ) V x = 4 D πδ ( x) (6) ( ) ( ) H x E x ( x) πδ '( x), E Poof. By letting in Poblem8, this sttement follows immeditely. 3.. Thee Dimensionl Poblems Poblem. (The Fee pticle.) Fo V () =, the solutions fo the fctionl Schödinge eqution H e = (7) = =. (8) ( ) exp( ik ) = (9) E = D ( k). () with k ny thee dimensionl vecto. An ltentive fom of the eigen-functions is m ( ) = jl( k) Yl ( θϕ, ) () m whee j l is the spheicl Bessel function of ode l, l (, ) [], (, θϕ, ) is the spheicl coodinte system, nd k k is the length of the vecto k. Poof. Fo V () =, the stndd Schödinge eqution Y θϕ is the spheicl hmonic function of degee l nd ode m

Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions hs the solutions H m ( ) = ( ) = EV ( ) ( ) exp( i ) () = k (3) Fom the eltionship between the fctionl Hmiltonin nd the stndd Hmiltonin we see tht the fctionl Schödinge eqution hs the solutions k E =. (4) m / / / = = = ( ) = ( ) = ( ) H T D p D p D mt D mh, (5) H () = E () (6) ( ) exp( ik ) In the spheicl coodinte system, the clssicl Schödinge eqution hs solutions Since H = (7) E = D ( k). (8) m ( ) = ( ) = EV ( ) m ( ) = j ( k) Y ( θϕ, ) (9) l l. (3) m / m l( ) l ( θϕ, ) = ( ) l( ) l ( θϕ, ) / m m D k jl k Yl θϕ D k jl k Yl H j k Y D mh j k Y = ( ) ( ) (, ) = ( ) ( ) ( θϕ, ) the solutions to the fctionl Schödinge eqution hs n ltentive fom Poblem. The function m ( ) = j ( k) Y ( θϕ, ) (3) l l, (3) E = D ( k). (33) Y 3 ( ) 3 3 exp( ) ( ) R i p = π D d p p nd E= is solution of the fctionl Schödinge eqution H with the potentil V () = δ ()/ Y (), (35) whee δ() = δ() x δ() y δ() z = δ()/( π ) is the Dic s delt function in the 3D spce. Poof. It is esy to veify tht (34) D p Y () + VY () () = δ() δ() =. (36) This completes the poof. Thee e two specil cses, whee the wve functions nd the potentil enegy cn be given explicitly:

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87- () When = we hve nd Theefoe we sy tht fo the centl potentil solution m () = nd E=. π () When =, we hve nd m p 3 Y ( ) = 3 3 exp( i ) d p ( π ) R p m 3 m m = 3 3 exp( ik ) d k = = 4 ( π ) R k π π (37) V () = δ ()/ Y ( ) = δ ) (. (38) m V( ) = δ ) (, the Schödinge eqution H hs m p 3 3 Y ( ) = 3 3 exp( i ) d p = 3 3 exp( i ) d ( ) R D k k π p ( π) R D k, (39) 4π = = D D 3 3 3 exp( ik ) d k ( π) R 4 π k π V () = δ ()/ Y () = D δ (). (4) Theefoe, fo the centl potentil V ( ) = D δ ( ), the fctionl Schödinge eqution H hs solution () = nd E=. D π Poblem. (The hmonic oscillto potentil.) Fo hmonic oscillto potentil V() = k, (4) the fctionl Schödinge eqution H hs solutions(in the momentum epesenttion) ϕ( p ) = ϕ( p) = Ai( κp n ) (4) p En /3 ( k D ) n whee Ai(x) is the Aiy function, n ( D / ( k /3 )) Poof. The fctionl Hmiltonin is =. (43) κ. H = D p + k. (44) In the momentum epesenttion, the Hmiltonin opeto nd its eigen-eqution e p p (45) H = k + D k p ϕ( p) + D p ϕ( p) = Eϕ( p) (46)

Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions In the spheicl coodinte system, the Lplce opeto p is expessed s p p (sin θ ). (47) = + + p p p sinθ θ θ p sin θ ϕ Fo s-stte wve function, ( p ) = ( p), the eigen-eqution becomes Let u( p) = p ( p). Then we hve ( ) ( ) ( ). (48) k p p + D p p = E p p p d ( ) ( ) ( ). (49) k u p + D pu p = Eu p The solution of this eqution unde the condition u() = u( ) = is whee Ai is the Aiy function, n Poblem 3. (The Coulomb potentil.) Fo the Coulomb potentil eqution hs solution ( ) = / u( p) = Ai( κ p n ) (5) /3 E = ( k D ), (5) n is its n-th zeo point, nd /3 ( D / ( k )) n κ [4]. This completes the poof. V () = Ze / with Z>, nd e is the chge of the electon, the fctionl Schödinge with E= when Ze = D / π. nd Poof. In fct we hve Theefoe we hve This completes the poof. 3 D p = D p 3 3 exp( i ) d 4π k k ( π ) R k D 3 D 3 = 3 3 k exp( ik ) d k = 3 3 exp( i ) d ( ) R k k π k ( π) R k (5) D4π 3 D4π = 3 3 exp( ik ) d k = 4 3 ( π ) R π k ( π ) D = π D. (53). Ze V() = = 4 π 4 π π D p + V = 4 π 4π D p () + V () =. (54)

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87-3 4. Solutions to the Reltivistic Schödinge Eqution Agin, let us study the one dimensionl poblems fist nd then the 3D poblems. 4.. One Dimensionl Poblems is Poblem i. (The fee pticle.) Fo V(x)=, the solution fo the eltivistic Schödinge eqution < k <. H ( x) exp( ikx) = (55) 4. (56) E = ( kc) + m c with Poof. Accoding to the definition of the sque oot opeto [8-] the bove sttement is obvious. The solution cn lso be witten s 4 4 + exp( ) = + exp( ), (57) p c m c ikx k c m c ikx k R ( x) sin( kx) o cos(kx) = (58) 4. (59) E = ( kc) + m c with k. Futhe, when V(x)=V with V constnt, the wvefunctions do not chnge but the new enegy levels become Poblem ii. (A peiodic potentil.) Fo the potentil 4 (6) E = ( kc) + m c + V 4 ( ) = ( ( ) + ) V x k c m c mc b (6) b+ cos( kx ) whee k = π /, is length, nd b is el numbe, the eltivistic Schödinge eqution H Poof. Fom the definition of the sque oot opeto [8-], ( ) hs solution x = b+ cos( kx) (6) 4. (63) E= ( kc) + mc We get tht 4 4 + = +, (64) p c m c exp( ikx) k c m c exp( ikx) k R 4 4 + cos = + cos( ) (65) pc mc kx kc mc kx Theefoe we hve 4 p c + m c = mc. (66)

4 Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions T 4 ( cos( )) = pc + mc b+ kx ( ) 4 = ( k c) + m c cos( k x) + mc b ( ) ( ) 4 4 = ( kc) + mc b+ cos( kx) ( kc) + mc mc b b = ( kc) + mc b+ cos( kx) ( kc) + mc mc b+ cos( kx) V. ( ) ( ) ( ) 4 4 b+ cos( kx ) This completes the poof. Futhe, we cn clculte the vege of the kinetic nd potentil enegies. The nomlized function is Ψ = + (b + ) The veges of the kinetic nd the potentil enegies e The totl enegy is ( x) ( b cos( kx) ) (67). (68) * b 4 ( ) ( ) mc k c m c. (69) < T >= Ψ x T Ψ x dx = + b b ( ) + + + 4 ( ( kc) m c mc ) * b Ψ ( ) Ψ ( ) =. (7) < V >= x V x dx + b + < H>=< T > + < V>= ( kc) + mc = E 4. (7) Poblem iii. (The delt potentil well.) Fo Dic delt function potentil V = V δ ( x) with V >, the eltivistic Schödinge eqution H hs solution in the sense of cetin limit. Poof. The poof is simil to Poblem 3. The eltivistic Schödinge eqution 4 cn be ewitten in the momentum spce s [] We fist chnge the integl limit to positive numbe p ( x) δ ( x), E=- = (7) ( ) ( ) ( ) - (x) pc + mc x Vδ x x (73) 4 V p c + mcϕ p ϕ p = Eϕ p π ( ) - ( ) ( ) 4 V p c + mcϕ( p) + E ϕ( p) = ϕ( p) π > mc, nd then let p +. Thus we hve (74) (75) 4 V + p pc + mcϕ( p) + E ( p) = ϕ( p) π p (76)

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87-5 ϕ Tking integl of the two sides, we hve + p If ϕ ( p) p, we hve V ( p) = ϕ( p) π 4 pc + mc + E V + p p + p + p + p ϕ p π p 4 p pc + mc + E ( p) = ϕ( p) (77) (78) V V = =. (79) π pc + mc + E pc + mc + E p p p 4 π 4 [ mc, p ] [, p ], we hve If the bove integtion is clculted only on subintevl Since we hve Futhe we hve We see nd hence the bound stte enegy The wve function ϕ ( p) V π p mc <. (8) 4 pc + mc + E 4 p> mc p > mc pc > mc 4 pc > pc + mc 4 pc > p c + m c 4 pc+ E > p c + m c + E < pc+ E pc + mc + E 4 V π p mc < pc+ E V p + E /( ) ln c < π c mc+ E /( c) π p c mc exp( c / V )) E > exp( π c/ V ), (8) (8) (83). (84) E, s p, (85) E =. (86) (87) 4 4 pc + mc + E pc + mc / E +

6 Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions Fom Eqution (85), we hve Accodingly, in the el spce we hve ( p) s p ϕ. (88) ( x) δ( x) s p. (89) Agin, pticle with eltivistic kinetic enegy is esie to be tpped thn pticle with Newtonin kinetic enegy. Poblem iv. (The line potentil.) Fo line potentil V = Fx with F >, the solution to the eltivistic Schödinge eqution H is with E R. bu u b u u uξ du (9) (x) = cos( + + ln( + + ) ) 3 E mc m c ξ = ( x ), b= F F (9) Poof. The eltivistic Hmilton is In the momentum epesenttion, The eltivistic Schödinge eqution is 4 H = pc + mc + Fx. (9) 4 d H = pc + mc + if. (93) 4 d pc + mcϕ( p) + i F ϕ( p) = Eϕ( p) (94) This eqution cn be solved esily ϕ d i 4 ie ϕ( p) = pc + mc ϕ( p) F F. (95) ln ( p) i i pc mc E F F ic ln ϕ ( p) = F ( p + m c E / c) ic p+ p + mc ie = p p + mc + mc ln p F mc F 4. (96) 3 ( p) Aexp ln p + p p m c p + mc (97) ic im c ie = + + F F mc F p, (98) whee A is n bity constnt. Vi Fouie tnsfom, we cn get the wvefunction in the el spce

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87-7 (x) = ϕ( p)exp( ipx / ) π 3 3 3 3 A ic im c p+ p + mc ie ipx = exp( p p + m c + ln p + ) π F F mc F A c mc p + p + mc E p = cos( p p + m c + ln + ( x ) ) π F F mc F Amc mc mc E umc = cos( u u + + ln( u+ u + ) + ( x ) ) du π F F F Futhe, we point out tht = C cos( bu u + + b ln( u + u + ) + uξ ) du Amc E mc C =, p = umc, ξ = ( x ), π F (99) () (x) s x +. () It is becuse the vlue of (x) is equl to the sum of n ltenting seies nd the bsolute vlues of the tems in the seies become smlle when x +. Specificlly, let us conside the integl with Fo fixed ξ >, suppose tht un with n=,,,3 stisfy tht We hve With Since the tems I( ξ ) = cos G( u, ξ ) du () G(u, ξ) = ( bu u + + b ln( u + u + ) + uξ. (3) Gu (, ξ) = nπ + π / (4) n cos Gu (, ξ ) =. (5) n I( ξ ) = I + I + I + + In + (6) u u u I ( ξ ) = cos G( u, ξ ) du I ( ξ ) = cos G( u, ξ ) du I n ltentely chnge thei sign, nd un un I ( ξ ) = cos G( u, ξ ) du n (7) I < I < < I n <. (8) The seies of I( ξ ) conveges fo ny given ξ >. As ξ, the intevl between ny two djcent points, un+ un, becomes close to ech othe, evey tem

8 Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions In ( ξ ), nd hence thei ltenting summtion I( ξ ). In othe wods, fo ny fixed ξ, we hve (x) s x +. (9) It is eltively complicted to discuss the limit behvio of the wvefunction s x, so we omit it tempoily. Inteested edes cn obseve its behvio intuitively on gph. 4.. Thee Dimensionl Poblems Poblem v. (The fee pticle.) Fo V () =, the solutions to the eltivistic Schödinge eqution H() () e ( ) exp( ik ) = () 4 k () E = c + mc with k thee dimensionl vecto. An ltentive fom of the eigen-functions is Poof. Fo V () =, the stndd Schödinge eqution hs the solutions H m ( ) = j ( k) Y ( θϕ, ) l l. () m ( ) = ( ) = EV ( ) ( ) exp( i ) (3) = k (4) whee k = k is the length of the vecto k. Fom the eltionship between the eltivistic Hmiltonin nd the stndd Hmiltonin we know tht the eltivistic Schödinge eqution hs the solutions k E = (5) m 4 4 4 p = = (6) H = c + mc mct+ mc mch+ mc H () () (7) ( ) exp( ik ) = (8) 4 k. (9) E = c + mc Obviously, the wvefunction cn lso be expessed in the spheicl coodinte system. Poblem vi. The function p 3 Y ( ) = 3 3 exp( i ) d p ( π ) R 4 p c + mc nd E= is solution of the eltivistic Schödinge eqution H () () with the potentil Poof. It is esy to veify tht () V () = δ ()/ Y (). ()

Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87-9 This completes the poof. p c + mcy 4 () + V() Y() = δ() δ() =. () Poblem vii. (The hmonic oscillto potentil) Fo hmonic oscillto potentil V() = k stisfy /3 ( k c ) ( ) n < En < n ω + mc whee n =,, 3,, ω k / m Poof. In the momentum spce, we hve, the s-stte enegies fo the eltivistic Schödinge eqution, n E, (3), nd n is the n-th zeo point of the Aiy function Ai(x). The Schödinge eqution is 4 4 H = p c + mc + k = p c + mc k p (4) 4. (5) k pϕ( p) + p c + mcϕ( p) = E ϕ( p) Up to some constnts, this eqution is mthemticlly the sme s the Schödinge eqution in the el spce with sque oot potentil () k () E () m + + =, (6) with k nd e positive numbes. We lso know tht in stndd quntum mechnics the enegy levels become highe if the potentil becomes highe. Now let us etun to the cuent poblem. Since 4 p c < p c + m c < mc + p, (7) m we see tht the eltivistic kinetic enegy is smlle thn the clssicl kinetic enegy plus the est enegy mc, but gete thn the enegy levels of the fctionl enegy with = nd D = c. Specificlly, fo s-stte, we hve /3 ( k c ) ( ) n < En < mc + n ω (8) n =,, 3,. Hee we used the esult of Poblem nd the enegy fomul fo the clssicl hmonic oscillto []. Poblem viii. (The Coulomb potentil.) Fo the Coulomb potentil V = e /, wheee is the chge of the electon, the enegy eigen vlue of the eltivistic Schödinge eqution H() () is n 3 64 Enl = mc + + O n n l + /. 4 5π n ( ) 4 5 6 4 δ 3 l whee =e / c is the fine stuctue coefficient, n is the pinciple quntum numbe, l is the ngul momentum quntum numbe. Only in this poblem nd its solution, is not fctionl pmete. Poof. The eltivistic Hmiltonin is (9)

Yuchun Wei: Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions The Schödinge eqution 4 e p - (3) H= c + mc 4 e H() = p c + mc() - () () (3) hs no nlytic solutions by now. Since the eltivistic effect is vey smll, we cn use the petubtion method bsed on the clssicl Hmiltonin The clssicl Schödinge eqution hs the well-known wve function ( ) H e mc - = + p m (3) p e H () = mc () + ()- () E () m = (33) nlm nd enegy levels En mc = n. (34) Accoding to the petubtion theoy [], the fist ode ppoximtion of the enegy is whee 4 e E nl = < H> = < p c + mc > < > * 4 3 = 3nlm( p) p c + mcnlm( p) mc R n n 3 4 64 5 6 = mc δ 4 + 3 l + O( ) n n l + / 4 5π n φ p p ( ) ( π ) 3 nlm( ) = 3nlm exp( i ) d 3 R (35) (36) is the wvefunction in the momentum spce[3]. The detils of the clcultion cn be found in [9]. 5 The new enegy levels contin vluble tem, which is 4% of the obseved Lmb shift [9]. We e tying to find the exct solutions fo the eltivistic Schödinge eqution with Coulomb potentil to see whethe we cn explin the Lmb shift bette in the fmewok of quntum mechnics. 5. Conclusions Jeng s citique esulted in cisis of fctionl quntum mechnics, tht is, the fctionl Schödinge eqution ws difficult to solve in mthemtics nd hd no eliztion in the el wold. To eliminte this cisis, we pesent vious solutions to the fctionl Schödinge eqution, nd intoduce the eltivistic Schödinge eqution s eliztion of the fctionl Schödinge eqution. Sevel solutions to the eltivistic Schödinge eqution e lso pesented. The stndd, fctionl nd eltivistic Schödinge eqution should be studied togethe. We wish tht the winte of the fctionl quntum mechnics could go wy nd its sping could come soon. ACKNOWLEDGEMENTS The esech on the eltivistic Schödinge eqution ws suppoted by Gnsu Industy Univesity (cuently clled Lnzhou Univesity of Technology) duing 989-99, with poject title On the solvbility of the sque oot eqution in the eltivistic quntum mechnics. Coopetive esech, joint gnt pplictions nd semins on the new quntum mechnics e welcome.

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