Classification of Electromagnetic Fields in non-relativist Mechanics

Transcription

1 EJTP 5, No. 19 ( Elctronic Journal of Thortical Physics Classification of Elctromagntic Filds in non-rlativist Mchanics N. Sukhomlin 1 and M. Arias 2 1 Dpartmnt of Physics, Autonomous Univrsity of Santo Domingo, Santo Domingo, Dominican Rpublic 2 Dpartmnt of Physics-Mathmatics, Univrsity of Purto Rico at Cayy, Purto Rico Rcivd 18 July 2008, Accptd 16 August 2008, Publishd 10 Octobr 2008 Abstract: W study th classification of lctromagntic filds using th quivalnc rlation on th st of all 4-potntials of th Schrödingr quation. In th gnral cas w find th rlations among th quivalnt filds, currnts, and charg dnsitis. Particularly, w study th filds quivalnt to th null fild. W show that th non-stationary stat function for a particl in arbitrary uniform tim-dpndnt magntic fild is quivalnt to a plan wav. W prsnt that th known cohrnt stats of a fr particl ar quivalnt to th stationary stats of an isotropic oscillator. W rval that th only constant magntic fild is not quivalnt to th null fild (contrary to a constant lctrical fild and w find othr filds that ar quivalnt to th constant magntic fild. W stablish that on particular transformation of th fr Schrödingr quation puts a plan wav and Grn s function in a quivalnc rlation. c Elctronic Journal of Thortical Physics. All rights rsrvd. Kywords: Schrödingr Equation, Intrinsic Charactristics, Equivalnt Filds, th Shapovalov Group PACS (2008: w; z; D; p 1. Exposition of th Problm It is known that th Schrödingr quation for a particl in th lctromagntic fild in Cartsian coordinats ı A 2 (t, r] + ϕ (t, r. (1 ı Ψ t = ĤΨ with Ĥ = 1 2m must b compltd with th following known charactristics: a physical intrprtation of th wav function and th fact that th 4-potntial must b ral so that th Hamilton Corrsponding Author: ww @yahoo.fr

2 80 Elctronic Journal of Thortical Physics 5, No. 19 ( oprator is hrmitian (if not th normalization intgrals could not b constant. Ths charactristics ar intrinsic of th physics problm. Th Schrödingr quation compltd by thir intrinsic charactristics w call a physical problm in non-rlativist Mchanics. In this papr w raliz th classification on th st of all physical problms using th Shapovalov s approach 1]. It is vidnt that for diffrnt intrinsic charactristics w obtain diffrnt quivalnc rlations and, consquntly, diffrnt classifications. Traditional studis add to th list of known intrinsic charactristics th structur of th lctromagntic fild: E and B must b fixd. This imposition rducs th group of quivalnc to th Cartsian product of th gaug invarianc group and th Galilan transformations group (s, for xampl, 2]. On th contrary, Shapovalov s approach dos not includ th structur of th lctromagntic fild in th intrinsic charactristics. This allows us to find a broadr quivalnc group, and givs th nw xact solutions of diffrnt physical problms. Th following thorm dfins th quivalnc group of th Schrödingr quation (w call it Th Shapovalov group: Shapovalov s Thorm 1. Th Schrödingr quation in th Euclidian spac compltd by mntiond intrinsic charactristics admits following quivalnc group: G sh = R T V Γ Y. Whr Γ is th gaug invarianc group and R (θ 1 (t,θ 2 (t,θ 3 (t; T (c 1 (t,c 2 (t,c 3 (t and V (s (t ar th rotations, translations, and scal chang groups. Th Shapovalov group contains 7 arbitrary timdpndnt functions: s (t,θ i (t,c i (t and dt := s 2 > 0, and Y is th discrt dt invarianc group of th Schrödingr quation from Sction 4.5. Th Shapovalov group is vry ffctiv for finding nw solutions. W not that th traditional quivalnc group is a subgroup of th G sh : θ i,c i,s= const. For a spcification of th Shapovalov group, w compar two Schrödingr quations such as (1 with diffrnt 4-potntials: { A(t, r,ϕ(t, r } and { A (t, r, ϕ (t, r }. Additionally, w dsignat {t, r } to b th variabls of th scond quation, this allows us to distinguish quivalnt filds. Th quivalnc rlations must consrv th structur (1 of th Schrödingr quation. Whn using only this imposition w found 1] that th Shapovalov group of th Schrödingr quation is th following : ] dt t = t (t dt := s2 > 0 (2 r = s (t a (t r + c (t (3 Ψ (t, r =Ψ(t, r s 3/2. (4 Th svn arbitrary tim-dpndnt functions of th Shapovalov group man th following: s(t rprsnts a frdom of tim-scal and th coordinat-scal choic. Thr functions c i (t dfin thr indpndnt displacmnts with arbitrary tim-dpndnt linar vlocitis. Th orthogonal matrix a(t dscribs th rotations with thr arbitrary tim-dpndnt angular vlocitis. In Sction 4 w study th subclasss of th null fild quivalnc class gnratd by ach on of ths thr typs of quivalnc rlations.

3 Elctronic Journal of Thortical Physics 5, No. 19 ( If a rfrnc systm wr inrtial, th Shapovalov group puts it in quivalnc with th non-inrtial systms. This fact is usd frquntly, for xampl, to dscrib th circular movmnt of a classic particl with a constant linar vlocity. Aftr bing transfrrd to a rotating rfrnc systm, this physical problm is rducd to a fr particl (s, for instanc, 3], 4]. W mphasiz that in th traditional approach such obvious quivalnc dos not xist. At last, this fact shows a nd for xtnsion of th traditional quivalnc group. 2. Equivalnt Potntials and Filds Using th fact that th form (1 of th Schrödingr quation must b invariant and th quivalnc rlations of th Shapovalov group (2, (3 and (4, w obtain th quivalnt potntials: A k = s 1 a kj A j m s 2 x k, (5 j ϕ = 1 s 2 ϕ + k,j 1 s 3 ẋ ka kj A j k m 1 2 s 4 ẋ kẋ k, (6 whr ẋ k := x k. Rigorously, th Shapovalov group is spcifid through formulas (2, t (3, (4, (5, (6, and (39. Now, w can asily calculat th rlation among quivalnt lctromagntic filds: s 2 B i (t, r = k a ik B k (t, r+ m ε ilk ȧ lj (ta kj (t, (7 l,k,j s 2 E (t, r = 1 s a E(t, r r B + m ( r s 2. (8 For a compact prsntation, w introduc Larmor s notation: ω := B 2m and th magntic fild s anti-symmtrical 3 3 matrix: F ij := k ε ijkω k. Th rlation (7 btwn quivalnt magntic filds in matrix form appars as: s 2 F (t, r =af (t, ra T +ȧa T. (9 If, for xampl, w hav only tim-dpndnt magntic filds: B = B (t and B = B(t thy both ar quivalnt (a is an orthogonal matrix: ȧ(t =s 2 F (t a af (t. (10 Particularly, if both magntic filds ar constant, th following orthogonal matrix provids th quivalnt rlations: a(t = xp {F t (t Ft} ; t (0 = 0. (11

4 82 Elctronic Journal of Thortical Physics 5, No. 19 ( Thn w obsrv that naturally th quivalnt filds (7 and (8 simultanously vrify th first pair of th Maxwll quations. It is asy to find that: s 3 B = B, s 5 E + B ] = sa E t + ] B ( divb t r. If on rfrntial systm dos not hav a magntic charg, than no othr quivalnt systm will hav it. Th scond pair of th Maxwll quations givs th rlations btwn th quivalnt dnsitis of charg and currnt: s 4 ( ρ (t, r ɛ 0 2m ω 2 = J (t, r +ɛ 0 E t ρ(t, r ɛ 0 = 1 s 3 a 2m ω2 + s 2 rrot B 3m f(t, (12 ( J(t, r+ɛ E 0. (13 t Whr ω(t is th Larmor frquncy, E and E vrify (8 and ( 1 f(t :=s. (14 s 3. Filds Equivalnt to th Null Fild Hr w study th null fild quivalnc class. If E = B =0, thn rlations (7, (8, and (10 allow us to dscrib all filds of this quivalnc class: B = B(t dfind by (7 ȧ = af (t; (15 E(t, r = m b(t r + d(t ], (16 whr w dnot th matrix: b(t := sa T (sa with f(t from (14, I unity matrix, and th vctor s 2 ] d(t := sa T c s 2 ] = f(ti + F F 2, (17. (18 W conclud from (15 and (16 that th widst fild quivalnt to th null fild can contain an arbitrary tim-dpndnt magntic fild, whras th lctric fild is linar rlatd to r (with th matrix of tim-dpndnt cofficints : B = B(t (19

5 Elctronic Journal of Thortical Physics 5, No. 19 ( ( m E(t, r = r ω ] +( ω r ω + f(t r + d(t, (20 whr th scond trm of E has th form of th cntrifugal forc. W obsrv that th class has non-orthogonal filds E B 0, but all ar quivalnt to on orthogonal. From (12 and (13 w find th null quivalnt dnsitis of charg and currnt: 4. Particular Cass ρ(t =2mɛ 0 ω 2 (t+3mɛ 0 f(t, (21 E J(t, r = ɛ ] 0 t = ɛ 0m ḃ r + d. ( Exclusivly Rotation Equivalnc (s = 1, c = 0 Now w trat th subclass of th filds which ar quivalnt to th null fild using only th rotating quivalnt rfrnc systms. From (19 and (20 w hav: B = B(t, (23 ( m E(t, r = r ω ( m + ( ω r ω. (24 Th following 4-potntial crats such a fild: ( m ( m A = ( ω r, ϕ = ( ω r 2 2, whr th scalar potntial rprsnts th cntrifugal nrgy. W obsrv that th lctric ( and magntic filds ar not orthogonal in th gnral cas: E B = 1 ( B 2 B r, but thy ar if, for xampl, B B. Hamilton s oprator of this systm has th form: Ĥ = 2 2m Δ ω(t ˆ L, (25 whr Δ is th Laplacian and ˆ L := r ˆ p is th oprator of th angular momntum. Th formula (25 corrsponds xactly to th classic mchanics thorm of transformation of nrgy aftr passing from an inrtial rfrnc systm to a systm in rotation: E rotation = E inrtial ω L (s for instanc 4]. If th fild (23 and (24 is quivalnt to th null fild, thn th systm with Hamilton s oprator (25 has a symmtry that corrsponds to on of a fr particl. Th first intgrals of motion of (25 ar: ˆ X 1 = a(tˆ p, ˆ X 2 = a(t t ] m ˆ p + r, ˆ X 1 = a(tˆ L. (26 Ths first ordr oprators of symmtry ar uniqu for th systm and constitut th bas of all suprior-ordr symmtry oprators. Th corrsponding intgrals of motion of

6 84 Elctronic Journal of Thortical Physics 5, No. 19 ( a fr particl ar: ˆ X i = a T ˆ Xi (i =1, 2, 3. It is known that th ignvctors of th linar momntum and th initial coordinat oprators ˆ X 1 = ˆ p, ( t ˆ X 2 = ˆ p + ˆ r, (27 m ar rspctivly a plan wav and Grn s function (w rmov th prims on top of wav functions and variabls: Ψ plan wav = C xp ı E + ı p ] r ; Ψ Grn = C xp ı m( r ] λ 2. (28 t3/2 2 t In Sction 4.5 w show that an quivalnc xists among ths functions and among th oprators (27. Now, using th known solutions (28, w can construct nw solutions to th Schrödingr quation for th fild (23 and (24 for an arbitrary B(t. Th first oprator of (26 ˆ X 1 has th following ignfunction: Ψ(t, r =C xp ı λ T a(t r ] ı λ2 2m t. (29 This solution dscribs a non-stationary stat, it is quivalnt to a plan wav for a λ fr particl (stationary stat function. Th constant 2 is quasi-nrgy, th xprssion p 0 := a T (tλ is undrstood to b a quasi-momntum. W did not nd to solv th 2m Schrödingr quation for a particl in an arbitrary tim-dpndnt fild (23, (24. But w dirctly wrot its solution (29 without solving this quation, only using an quivalnc rlation of th Shapovalov group. W obsrv that contrary to th null fild, for a particl in th fild (23 and (24 th vctor momntum is not a first intgral of motion. Th Hamilton s oprator (25 and th vctor oprator ˆ X 2 of (26 provid th solution of th Schrödingr quation that corrsponds to th Grn s function: Ψ(t, r = C xp t3/2 ı m( r r 0(t 2 2 t ], (30 whr r 0 (t :=a T (t λ, λ is th initial coordinat in th inrtial rfrnc systm; r 0 (t is th sam initial coordinat in a rotating rfrnc systm. Th vctor λ is an arbitrary constant vctor (ignvalu of ˆ X 2. An orthogonal matrix a(t vrifis (15 with th arbitrary vctor function ω(t. It is known that th solution (30 can b orthonormalizd to th Dirac dlta function, but its intrprtation is difficult. It is a cas of quivalnc of a non-stationary solution to anothr non-stationary on. Th third vctor oprator of symmtry (26 ˆ X 3 dos not allow a dirct construction of a solution bcaus as its componnts do not commut, it is ncssary to us th st quivalnt to: Ĥ, ˆ L 2, ˆL z.

7 Elctronic Journal of Thortical Physics 5, No. 19 ( Exclusivly Euclidan Translational Equivalnc (th Unity Matrix a = I and s =1 Th subclass of th Euclidan translational displacmnts ( c 0 rlats to th quivalnc btwn rfrnc systms moving with arbitrary linar vlocitis. For xampl, th only tim-dpndnt uniform lctrical fild (16 dfind by vctor d(t from (18, with th uniform corrsponding currnt (22 is quivalnt to th null fild without a currnt. Naturally, Galilo s quivalnc blongs to th Shapovalov group (s Sction Isotropic Oscillator (Exclusivly Tim Scal Equivalnc: th Unity Matrix a = I and c =0 Lt b B =0 = b(t is a constant matrix. Using formulas (19 and 20, for c = 0 and ω =0, w find th fild of th isotropic oscillator: E = ( m f(tr. In cas of f(t =const, it corrsponds to th potntial nrgy U = mω2 0 r2, with ω 2 0 = const. W not that constant ω 0 dos not hav any rlation to th Larmor frquncy ω (which is null hr. From this form of potntial nrgy and dfinition (14 w obtain th wll known rlation: ( 1 s + ω 2 0 ( 1 =0. s Solving it, w find th tim-scal quivalnc (scal chang group of Shapovalov s thorm: dt dt := s2 = sc 2 (ω 0 t = ω 0 t = tan(ω 0 t, r = s(t r. (31 This quivalnc rlation provids an quivalnc btwn th fr Schrödingr quation ı Ψ t and th quation for th isotropic oscillator: ı ψ nlm(t, r t = Ĥψ nlm(t, r := = Ĥ Ψ = 2 2m Δ Ψ, (32 ] 2 0r 2 2m Δ+mω2 ψ nlm (t, r. 2 Th solutions of th last quation ar known (for instanc, 5], 7]. W obsrv that ths ˆ stats ar dfind by th following st of intgrals of movmnt: Ĥ, L, 2 ˆL z. In addition, in th quivalnc rlation (31, it is ncssary to us gaug invarianc: Ψ nlm(t, r = ψ nlm(t, r xp ı m ] ṡ s 3/2 2 s r2 ; (33 with th wav function: ψ nlm (t, r =ϕ nlm ( r xp ı E ] nl t ; (34

8 86 Elctronic Journal of Thortical Physics 5, No. 19 ( E nl = ω 0 2n + l + 3 ] ; n, l =0, 2,...; m =0, ±1,..., ±l; 2 ϕ nlm (r, θ, φ = 1 ξ R mω0 nl(ξy lm (θ, φ; ξ := r. Hr th functions Y lm (θ, φ ar th sphrical harmonics and R nl (ξ ar th radial functions xprssd by conflunt hyprgomtric functions. Th non-stationary solutions Ψ (33 of th fr Schrödingr quation (32 ar known as cohrnt stats and ar quivalnt to th stationary stats of th isotropic oscillator. For xampl, th function of th fundamntal stat that vrifis th quation (32 can b writtn (w abandon th prims on top of th variabls and th wav function: Ψ 000 (t, r = C 4π (1 + ω 2 0 t 2 xp mω 0 1 3/4 2 1+ıω 0 t r2 ı 3 ] 2 arctan(ω 0t. (35 W obsrv that to stablish th quivalnc of th fr particl s cohrnt stats to th stationary stats of an isotropic oscillator, th tim scal quivalnc rlation (31 is indispnsabl. For anothr quivalnt solution to (35 of th fr Schrödingr quation s (40. Th quivalnc btwn a fr particl and an isotropic oscillator of on dimnsion is discussd in 8]. If f(t from (14 is not a constant, w hav an arbitrary tim-dpndnt radial dilation. In this cas, nw solutions of othr physical problms can b constructd. 4.4 Constant Magntic Fild In th null fild quivalnt class (15 and (16, w find a fild with a constant magntic componnt: B =(0, 0, B=const; E(t, r = m f(t r + ω 2 (x 1,x 2, 0 ] + ε(t. (36 Hr f(t is dfind by (14, ε(t is an arbitrary tim-dpndnt vctor function. Th first trm of th lctrical componnt corrsponds to an arbitrary tim-dpndnt dilation; particularly to an isotropic oscillator from Sction 4.3 with an arbitrary timdpndnt frquncy. Th scond trm corrsponds to th constant cntrifugal forc fild and th third trm rsults from Euclidan translational frdom (c(t 0. Hr ar two particular cass of intrst: if f =0 (s =1 = B =(0, 0, B; E(t, mω 2 r = (x 1,x 2, 0 ; (37 ( 1 if f := s = ω0 2 = B s =(0, 0, B; E(t, mω 2 r = (0, 0, x 3. W can s that only constant magntic fild dos not blong to th null fild quivalnt class, but always som lctrical componnt of th fild is prsnt. By using formulas (7

9 Elctronic Journal of Thortical Physics 5, No. 19 ( and (8, w find that only constant magntic fild is quivalnt to th constant cntriptal forc fild: B =(0, 0, B =const; E =0 B =0; E(t, mω 2 r = (x 1,x 2, 0. (38 This rsult xplains th formula (37. If f = const, ε = const, th fild (36 dfins th widst stationary fild quivalnt to th null fild. 4.5 Fr Schrödingr Equation s Nuclus It is intrsting to find th st of quivalnc rlations that put th null fild in quivalnc to itslf (s also 6]. By choosing E = B = E = B =0, w asily find that such a nuclus is th Galilan group: t = α 2 t + β; r = αa r + ν 0 t + r 0 ; Ψ = α 3/2 Ψ with arbitrary constants α, β, ν 0, r 0, and a constant orthogonal matrix a. In addition, on isolatd quivalnc opration xists and lavs th fr Schrödingr quation invariant: t = 1 t ; r = 1 t r; Ψ (t, r =Ψ(t, rt 3/2 xp ı m ] r 2, (39 2 t ] t 2 i + Ψ t 2m Δ Ψ= t i ] Ψ t + 2 2m Δ Ψ. Lt us dnot this transformation. It is asy to show that {, 2, 3, 4 } constituts th discrt invarianc group of th Schrödingr quation Y from th Shapovalov s thorm. This transformation has a vry intrsting proprty: it transforms Grn s function (28 to th plan wav propagating in th dirction λ : Ψ (t, r =C xp ıω t ı ] k r, ω := mλ2 2 ; k := m λ. And invrsly, th quivalnc rlation (39 transforms a plan wav into Grn s function. Of cours, th rlation (39 also puts in quivalnc th oprator of linar momntum ˆ p and th oprator of initial coordinat: ( t ˆ p + r (s formula (27. This m opration is similar to passing to momntum spac. This transformation dos not mak sns if λ =0. W asily find that rlation (39 dos not chang th angular momntum: L = ˆ L. Consquntly, th wll-known stats of a fr particl with a constant angular momntum ar invariant in rspct to rlation (39. Anothr xampl, rlation (39 puts th cohrnt stat (35 in quivalnc with th following function: Ψ 000(t, r = C 4π (ω t 2 xp m 3/4 2 1 ı 3 ω 0 + ıt r 2 2 arctan(ω 0 ] t. (40

10 88 Elctronic Journal of Thortical Physics 5, No. 19 ( It is a nw solution of th fr Schrödingr quation (32. dt Also, w can not that formula (33 with th rlations dt := s 2 (t and r = s(t r contains th opration of quivalnc (39 if w just impos th transformation of th Grn s function to a plan wav or invrs. 4.6 Classic Particl Th Shapovalov group was found by Shapovalov and Sukhomlin (1974 1] who also numratd all cass of sparation of variabls in a parabolic quation and provd that all of ths cass rcur in quantum mchanic and in classical Hamilton Jacobi approach. In 1980, S. Bnnti and M. Francaviglia 9] applid th Shapovalov group to th Hamilton-Jacobi quation and G. Rid 10] xtndd it in 1986 to th spac of n-dimnsions. A study of th Hamilton-Jacobi quation givs th sam quivalnc group that th Schrödingr quation (2, (3, (4 dos. In fact, th quivalnt potntials and th filds ar th sam as (5, (6 and (7, (8. In particular, w rfr to th rsults to th quivalnc study in som cass. Using (31 w stablish th rlation btwn quivalnt actions as in Sction 4.3: W (t, r =W (t, r+ m 2 ṡ s r2, (41 whr W (t, r corrsponds to th isotropic oscillator and W (t, r to cohrnt stats of a fr particl. Finally: ( W (t,r = α cos 1 1 ω 0 2α + mω 0 2 mω ω0t 2 r 2 + mω 0 2 r 1 1+ω 2 0t 2 ( 2α mω 2 0 r 2 1+ω0t 2 2 t 1+ω 2 0t 2 r 2 α ω 0 arctan(ω 0 t. (42 Hr α is nrgy. As in Sction 4.5 w can vrify that th nuclus of th Hamilton-Jacobi quation is spcifid by th sam rlation (39 as gaug invarianc: W (t, r =W (t, r m 2 r 2 t, (43 whr W (t, r is (42 without th prims. Th solution of Hamilton-Jacobi that corrsponds to (40, prsnts: ( W (t,r = α cos 1 1 ω 0 2α mω 2 0 ω0 2 r + t 2 + mω 0 2 r 1 ω t 2 ( 2α mω 2 0 r 2 ω0 2 + t 2 + mω 0 r 2 2t ω0 2 + t α arctan( ω 0. (44 2 ω 0 t

11 Elctronic Journal of Thortical Physics 5, No. 19 ( Conclusions (1 Th null fild is quivalnt to th uniform magntic fild with arbitrary tim dpndnc. Th corrsponding lctric componnt (linar rlatd to r is dfind by (16 or (20. Th quivalnc rlation btwn such a magntic fild and th null fild is similar to th passing from an initial rfrnc systm to on which is in rotation dfind by an orthogonal matrix from (15. (2 Th null fild quivalnc class has svral non-orthogonal filds. (3 Th arbitrary-uniform charg distribution (tim-dpndnt or not is quivalnt to th null charg dnsity according to (21. Its tim dpndnc coms from two typs of quivalnc: on from rotation of a rfrnc systm and th othr from tim scal quivalnc. It is possibl to hav a null charg distribution simultanously for both filds: on of typ (15, (16 and th null fild. Its currnt dnsitis can b null also. Th widst currnt dnsity quivalnt to th null currnt is linar rlatd to r (22. (4 Th tim scal quivalnc (ṡ 0 has an important rol in th quivalnc rlations on all physical problm sts (s th isotropic oscillator, Sction 4.3. Particularly it is known that th non-stationary cohrnt stats of a fr particl (33 ar quivalnt to th stationary stats of a harmonic oscillator (34. Anothr xampl is whn concrt rlations stablish th quivalnc btwn a non-dissipativ wav packt and on with dissipation 5]. (5 Only Euclidan translational quivalnc ( c(t 0 stablishs a corrspondnc btwn arbitrary tim-dpndnt uniform lctrical filds and th null fild (s Sction 4.2. (6 Sinc th fr Schrödingr quation has both stationary and non-stationary solutions, in th null fild quivalnc class w find four typs of quivalnc rlations: th stationary wav functions quivalnt to stationary or non-stationary stats. Also non-stationary stats could b quivalnt to on othr stationary or non-stationary wav functions. For xampl, th non-stationary wav function (29 is quivalnt to a stationary on (plan wav (28; th non-stationary function (30 is quivalnt to anothr non-stationary on (Grn s function of th fr Schrödingr quation (28. (7 Th widst stationary fild quivalnt to th null fild is dfind by formula (36 with conditions f = const, ε = const. (8 Th formulas (36 and (38 show that only a constant magntic fild dos not blong to th null fild quivalnt class (contrary to a constant lctrical fild. (9 Th known isolatd quivalnc rlation (39 puts th fr Schrödingr quation in quivalnc to itslf. Th plan wav is quivalnt by mans of this rlation to Grn s function. It is similar to th rotation in a phas spac.

12 90 Elctronic Journal of Thortical Physics 5, No. 19 ( Acknowldgmnts Th authors ar gratful to Prof. Franklin Garcia Frmin and to th UASD Dpartmnt of Physics for thir coopration. Also, th authors xprss thir gratitud to th Editor in Chif of th EJTP and to anonymous rviwrs. Rfrncs 1] V. Shapovalov, N. Sukhomlin, Sparation of th variabls in th non-stationary Schrödingr quation, Izv. Vyss. Uchb. Zavd., Physika, 12, 100 (1974 (Sov. Phys. Journ., 17, 1718 ( ] R. Zhdanov, A. Zhalij, On sparabl Schrödingr quation, Journal of Mathmatical Physics,40, 6319 ( ] H. Goldstin, C. Pool, J. Safko, Classical Mchanics, Addison-Wsly, 3rd d., 134 ( ] L. D. Landau, E. M. Lifshitz, Cours of Thortical Physics, Mchanics, 1, 3rd d., Oxford: Prgamon prss, 126 ( ] M. Moshinsky, D. Schuch, A. Suarz-Morno, Motion of wav packts with dissipation, Rvista Mxicana d Fisica, 47, 7 ( ] C. Boyr, Th maximal kinmatical invarianc group for an arbitrary potntial, Hlv. Phys. Acta, 47, 589 ( ] S. Flügg, Practical Quantum Mchanics, vol. I, Springr-Vrlag, Brlin, Hidlbrg, Nw York (1971; problm 65. 8] W. Millr Jr, Symmtry and Sparation of variabls, Addison-Wsly, London (1977; Chaptr 2. 9] S. Bnnti, M. Francaviglia, Th Thory of Sparability of th Hamilton-Jacobi Equation and its Applications to Gnral Rlativity, A Gnral Rlativity and Gravitation, Brn, Switz, 393 ( ] G. Rid, R-Sparation for Hat and Schrödingr Equations I, SIAM, 17 Issu 3,646 (1986.