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PT-Rotations, PT-Spherical Harmonics and the PT-Hydrogen Atom Juan M. Romero,∗ R. Bernal-Jaquez,† and O. González-Gaxiola‡ Departamento de Matema´ticas Aplicadas y Sistemas Universidad Autońoma Metropolitana-Cuajimalpa, México 01120 DF, México (Dated: January 12, 2010) We have constructed a set of non-Hermitian operators that satisfy the commutation relations of theSO(3)-Liealgebra. Itisshownthatthisoperatorsgeneraterotationsintheconfiguration space andnotinthemomentumspacebutinamodifiednon-Hermitianmomentumspace. Thisgenerators arerelatedwithanewtypeofsphericalharmonicsthatresulttobePT-orthonormal. Additionally, wehaveshown that thisoperators representconserved quantitiesfor a non-Hermitian Hamiltonian with an additional complex term. As a particular case, the solutions of the corresponding PT- 0 Hydrogen atom that includes a complex term are obtained, and it is found that a non-Hermitian 1 Runge-Lenzvectorisaconservedquantity. Inthisway,weobtain asetofnon-Hermitianoperators 0 that satisfy theSO(4)-Lie algebra. 2 PACSnumbers: 11.30.Er,11.30.-j,03.65.-w. n a J 1 1 I. INTRODUCTION ] h Quantum mechanics is considered as one the most solid and well established theories in physics. Different p experiments have corroborated its predictions. However, at a theoretical level there are different facts that make us - h think that it could be necessary to modify or extend this theory. For example, it has not been possible to find a t consistent quantum-mechanical formulation of general relativity, then quantum theory maybe modified in order to a make it compatible with general relativity. m [ As it is well known, quantum mechanics has been formulated in terms of Hermitian operators in order to obtain 1 real spectra. However, it has become clear that Hermiticity is not a necessary condition to obtain real spectra. This v opens the possibility for quantum mechanics to be extended using non-Hermitian operators, this is the so called 9 PT-symmetry theory, see review [1] and its references. 7 5 The PT-version of quantum mechanics has strongly attracted attention because it gives a way to deal with some 1 problems that are out of the scope of conventional quantum mechanics. For example, we can solve certain kind of . 1 problems in which the potentials are given by complex-valued functions and whose spectra results to be real [2]. In 0 the same way, using this formulation it has been possible to achieve a consistent quantization of a system with high 0 order derivatives: the so called the Pais-Uhlenbeck oscillator model [3]. This opens the possibility to construct in a 1 consistent way high order derivatives field theories. This fact is important because different theories with high order : v derivativeshavebeenrecentlyproposed,forexample,in extensionsofthe standardmodel[4], inthe noncommutative i X spaces[5]andgravitytheories[6]. In this way,it becomes possible thatanextensionofPT-symmetry theoryapplied to field theory can give a consistent description of these systems. r a There is not a finished version of the theory, however, a growing number of themes are under study in the PT-framework, some of them can be found [7]. An aspect that has been scarcely treated in the PT-context is the study of symmetries and conserved quantities. In this work, we will study some aspects of this topic. We will obtain a non-Hermitian set of operators that satisfy the commutation relations of the Lie SO(3) rotation group. It will be shown that these operators generate rotations in the configuration space x , and not in the momentum space i p~ = −i∇ but in a modified non-Hermitian momentum space ~p = ~p+i∇~f, originally considered by Dirac in his f seminal book [8]. Also, we will show that the Casimir of the algebra has real spectra and that its eigenfunctions, under the PT-inner product, form a complete basis. This eigenfunctions will be called PT-spherical harmonics. ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] ‡Electronicaddress: [email protected] 2 Additionally we will study a centralpotential Hamiltonian with an additionalcomplex term. It will be shownthat the conventionalangularmomentumis nota conservedquantity anymoreandwe willhaveamodified non-Hermitian angular momentum operator. As a particular case, we obtain the solutions of the corresponding PT-Hydrogen atom that includes a complex term, and it will be found that a non-Hermitian Runge-Lenz vector is a conserved quantity. Then we will have the non-Hermitian generators of the SO(4)-Lie algebra. This work is organized as follows: section II make a brief review of PT-theory and of conventional spherical harmonics, in section III we study the PT-rotations, in section IV we study the completeness relation and some examples, section V is devoted to the study of symmetry transformations,in section VI we deal with the central po- tentialproblem,insectionVIIwewillstudytheHydrogenatomandatlastweconcludewithasummaryofourresults. II. PT-SYMMETRY AND SPHERICAL HARMONICS In this section, a review of some well known facts of PT-symmetry theory and spherical harmonics is made before consider the PT-transformed version of this functions A. PT-Inner product PT-theoryconsidersthe transformationsunder the parityoperatorP andthe time reversaloperatorT. Under the P-operator we have the transformation x,y,z →−x,−y,−z (1) and under T i→−i (2) In this way, any function f(~x) can be transform as PT (f(~x))=f∗(−~x). (3) Note that, in spherical coordinates P produces the transformation (r,θ,ϕ)→(r,π−θ,ϕ+π), (4) under P a f function transforms as P(f(r,θ,ϕ))=f(r,π−θ,ϕ+π), (5) therefore PT (f(r,θ,ϕ))=f∗(r,π−θ,ϕ+π). (6) Now, the PT-inner product is defined as hf|gi= d~x[PTf(x)]g(x). (7) Z This expressions will be used in sections bellow. An exhaustive study of the PT-theory can be found in [1]. B. Spherical Harmonics The angular momentum components are given by the Hermitian operators [9] ∂ ∂ ∂ ∂ ∂ ∂ L =−i y −z , L =−i z −x , L =−i x −y . x y z ∂z ∂y ∂x ∂z ∂y ∂x (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 3 Its algebra is given by [L ,L ]=iL , [L ,L ]=iL , [L ,L ]=iL . (8) x y z z x y y z x Considering L2 =L2 +L2+L2 and Eq. (8), we have x y z L2,L = L2,L = L2,L =0. (9) x y z An important equation in mathematica(cid:2)l-physic(cid:3)s is(cid:2)the eig(cid:3)enva(cid:2)lue equ(cid:3)ation L2Y = l(l+1)Y ,l = 0,1,2··· , that lm lm in spherical coordinates is written 1 ∂ ∂Y (θ,ϕ) 1 ∂2Y (θ,ϕ) L2Y (θ,ϕ) = − sinθ lm + lm lm sinθ∂θ ∂θ sinθ2 ∂ϕ2 (cid:20) (cid:18) (cid:19) (cid:21) = l(l+1)Y (θ,ϕ). (10) lm If ϕ∈(0,2π) and θ ∈(0,π), the solutions of this equation are given by the spherical harmonics [10] (2l+1)(l−m)! Y (θ,ϕ) = eimϕPm(cosθ), −l≤m≤l, (11) lm s 4π(l+m)! l with l =0,1,2,3··· , Plm(u) = (−)m(1−u2)m2 ddummPl(u), Pl(u)= 21ll!ddull u2−1 l (12) where Pm(u) denoted the associated Legendre polynomials and P (u) the Legendre polynom(cid:0)ials re(cid:1)spectively. The l l spherical harmonics satisfy the orthonormality relation <Yl′m′(θ,ϕ)|Ylm(θ,ϕ)>= dΩYl∗′m′(θ,ϕ)Ylm(θ,ϕ)=δmm′δl′l. (13) Z Given that the spherical harmonics constitute an orthonormal basis, we can write any function F(θ,ϕ) as a linear combination of them, that is l F(θ,ϕ)= C Y (θ,ϕ). (14) lm lm l≥0m=−l X X Using Eq. (13), we find C = dΩY∗ (θ,ϕ)F(θ,ϕ). (15) lm lm Z Substituting C in Eq. (14) and making the change of variables u′ =cosθ′,u=cosθ we obtain lm 2π 1 l F(θ,ϕ)= dϕ du′F(u′,ϕ′) Y∗ (u′,ϕ′)Y (u,ϕ) .  lm lm  Z0 Z−1 l≥0m=−l X X   Therefore, the expression inside the parenthesis must be equal to δ(ϕ−ϕ′)δ(u−u′), that is l Y∗ (θ′,φ′)Y (θ,φ)=δ(φ−φ′)δ(cosθ−cosθ′), (16) lm lm l≥0m=−l X X this expression is called completeness relation. Under the parity operator the spherical harmonics transform as P(Y (θ,ϕ))=Y (π−θ,ϕ+π)=(−)lY (θ,ϕ), (17) lm lm lm and under the PT-operator,we have PT (Y (θ,ϕ))=Y∗ (π−θ,ϕ+π)=(−)lY∗ (θ,ϕ). (18) lm lm lm This results will be used below. 4 III. PT AND ROTATIONS Given any f =f(r,θ,ϕ), we can define L =efL e−f. (19) fi i In general L is a non-Hermitian operator, however fi [L ,L ]=iL , [L ,L ]=iL , [L ,L ]=iL , (20) fx fy fz fz fx fy fy fz fx that we can identify as the SO(3)-Lie algebra commutation relations. Considering L2 = L2 +L2 +L2 and Eq. f fx fy fz (20), we have L2,L =0, (21) f fi therefore, we have the same algebra as the one sat(cid:2)isfied by(cid:3)L . Besides i L2Y (θ,ϕ)=l(l+1)Y (θ,ϕ), Y (θ,ϕ)=efY (θ,ϕ), (22) f flm flm flm lm that will be called PT-spherical harmonics. In this case, the PT-inner product is given by <Yfl′m′(θ,ϕ)|Yflm(θ,ϕ)>f = dΩPT (Yfl′m′(θ,ϕ))Yflm(θ,ϕ). (23) Z Under a PT-transformation,we have PT(Y ) = ef∗(r,π−θ,ϕ+π)(−)lY∗ (θ,ϕ). (24) flm lm therefore we can write <Yfl′m′(θ,ϕ)|Yflm(θ,ϕ)>f = (−)l dΩef∗(r,π−θ,ϕ+π)+f(r,θ,ϕ)Yl∗′m′(θ,ϕ)Ylm(θ,ϕ). Z It becomes clear that, under this inner product not any function f allows the set Y to be an orthogonal set. flm However, if the following condition is fulfilled ef∗(r,π−θ,ϕ+π)+f(r,θ,ϕ)=λ, λ=const (25) then we have <Yfl′m′(θ,ϕ)|Yflm(θ,ϕ)>f = (−)lλδl′lδm′m. (26) Inthis way,the sphericalharmonicsY (θ,ϕ) areorthogonalunder the PT-inner productonly if Eq. (25)is satisfy. flm It is worthy to mention that due to the parity of the wave functions, in some PT-symmetry systems the following orthogonality relations hφ |φ i=(−1)nδ , (27) m n m,n may be obtained [1]. IV. COMPLETENESS RELATION Using the PT-spherical harmonics Y (θ,ϕ), we can have the expansion flm l F(θ,ϕ)= a Y (θ,ϕ). (28) lm flm l≥0m=−l X X Appeling to the orthonormality relations Eq. (26), we find (−)l (−)l a = <Y (θ,ϕ)|F(θ,ϕ)> = dΩPT (Y (θ,ϕ))F(θ,ϕ). (29) lm flm f flm λ λ Z 5 substituting this result into Eq.(28), we obtain l (−)l F(θ,ϕ) = dΩ′PT (Y (θ′,ϕ′))F(θ′,ϕ′)Y (θ,ϕ) flm flm λ l≥0m=−l Z X X l (−)l = dΩ′F(θ′,ϕ′) PT (Y (θ′,ϕ′))Y (θ,ϕ) flm flm  λ  Z l≥0m=−l X X   l ef∗(r,π−θ′,ϕ′+π)+f(r,θ,ϕ) = dΩ′F(θ′,ϕ′) Y∗ (θ′,ϕ′)Y (θ,ϕ) ,  λ lm lm  Z l≥0m=−l X X   therefore l ef∗(r,π−θ′,ϕ′+π)+f(r,θ,ϕ) δ(φ−φ′)δ(cosθ−cosθ′)= Y∗ (θ′,ϕ′)Y (θ,ϕ). (30) λ lm lm l≥0m=−l X X ThisisthecompletenessrelationforthePT-sphericalharmonics. Asimilarcompletenessrelationisfoundfordifferent systems in PT-quantum mechanics [1]. A. Examples on Completeness In this subsection we will see some examples of functions that satisfy Eq. (25). Let us suppose that a is a real constant, then we can define f(r,θ,ϕ)=aθ (31) therefore ef∗(r,π−θ,ϕ+π)+f(r,θ,ϕ)=ea(π−θ)+aθ =eaπ, (32) where λ=eaπ. (33) In this case, the orthonormality relations are given by <Yfl′m′(θ,ϕ)|Yflm(θ,ϕ)>f = (−)leaπδl′lδm′m (34) and the completeness relation is given by l δ(φ−φ′)δ(cosθ−cosθ′)= ea(θ−θ′)Y∗ (θ′,ϕ′)Y (θ,ϕ). (35) lm lm l≥0m=−l X X Now consider f(r,θ,ϕ)=aisinθ, (36) then we obtain ef∗(r,π−θ,ϕ+π)+f(r,θ,ϕ)=e−aisinθ+iasinθ =1=λ (37) with the orthonormality relations <Yfl′m′(θ,ϕ)|Yflm(θ,ϕ)>f = (−)lδl′lδm′m. (38) In this case the completeness relation is given by l δ(φ−φ′)δ(cosθ−cosθ′)= eai(sinθ−sinθ′)Y∗ (θ′,ϕ′)Y (θ,ϕ). (39) lm lm l≥0m=−l X X 6 Using f(r,θ,ϕ)=acosθ, (40) we have ef∗(r,π−θ,ϕ+π)+f(r,θ,ϕ)=e−acosθ+acosθ =λ=1 (41) in this case the orthogonality relations are given by <Yfl′m′(θ,ϕ)|Yflm(θ,ϕ)>f = (−)lδl′lδm′m, (42) and the completeness relation is given by l δ(φ−φ′)δ(cosθ−cosθ′)= ea(cosθ−cosθ′)Y∗ (θ′,ϕ′)Y (θ,ϕ). (43) lm lm l≥0m=−l X X With the function f(r,θ,ϕ)=aiϕ (44) we obtain ef∗(r,π−θ,ϕ+π)+f(r,θ,ϕ)=e−ai(ϕ+π)+iaϕ =e−iaπ =λ, (45) with the orthogonality relation given by <Yfl′m′(θ,ϕ)|Yflm(θ,ϕ)>f = (−)le−iaπδl′lδm′m, (46) and the completeness relation l δ(φ−φ′)δ(cosθ−cosθ′)= eai(ϕ−ϕ′)Y∗ (θ′.ϕ′)Y (θ,ϕ). (47) lm lm l≥0m=−l X X Considering the above examples, it is clear that many functions satisfy Eq. (25). V. SYMMETRY TRANSFORMATIONS Consider A,B,C that satisfy the following commutation relations [A,B]=C. (48) Transforming the A,B,C operators, we obtain A =efAe−f,B =efBe−f y C =efCe−f, then f f f [A ,B ]=C . (49) f f f We know that [L ,x ]=iǫ x . If we consider the transformation x =efx e−f =x , we arrive to i j ijk k fi i i [L ,x ]=iǫ x , (50) fi j ijk k therefore the operators L generate infinitesimal rotations in the space x . However, as p =−i ∂ , then fi i i ∂xi [L ,p ]6=iǫ p , (51) fi j ijk k and we say that the operators L does not generate infinitesimal rotations in the space p . Now, if p is given by fi i fi p =efp e−f, then we have fi i [L ,p ]=iǫ p . (52) fi fj ijk fk 7 Note that p~ =efp~e−f =p~+i∇~f. (53) f this operator was studied by Dirac in his seminal book [8]. If the Hamiltonian operator is given by ~p2 H = +V(r), (54) 2m then [L ,H]=0. (55) i Defining H by f H =efHe−f, (56) f we have that, in general [L ,H ]6=0, (57) i f therefore the angular momentum L is not a conserved quantity for Hamiltonians of the form H . However i f [L ,H ]=0, (58) fi f then the modified angular momentum L is conserved. Note that the Hamiltonian H is given by fi f ~p2 m 2 H = f +V(r)= ~p+i∇~f +V(r) (59) f 2m 2 (cid:16) (cid:17) In the next section we will consider one important example. VI. THE CENTRAL PROBLEM Consider the Hamiltonian m H = p~2+V(x,y,z), (60) 2 then m 2 H = efHe−f = ~p+i∇~f +V(x,y,z) f 2 m (cid:16) (cid:17) 2 = ~p2+2i∇~f ·~p+(∇2f)− ∇~f +V(x,y,z), (61) 2 (cid:18) (cid:16) (cid:17) (cid:19) that is a non-Hermitian Hamiltonian. If the potential is given by m 2 V(x,y,z)=− ∇2f − ∇~f +U(x,y,z), (62) 2 (cid:18) (cid:16) (cid:17) (cid:19) then we can write m H = ~p2+2i∇~f ·~p +U(x,y,z). (63) f 2 (cid:16) (cid:17) This kind of Hamiltonians naturally arise in some statistical models [11]. Note that if ψ is an eigenfunction in the equation Hψ =Eψ (64) 8 then we can define the f-states ψ =efψ that satisfy f H ψ =Eψ . (65) f f f It follows that although H is a non-Hermitian operator it does has a real spectrum. f As an example, let us consider the central potential problem V(r) whose Schrodinger equation is given by [9] m Hψ = ~p2+V(r) ψ =Eψ (66) 2 (cid:16) (cid:17) and whose solutions ψ (r,θ,ϕ)=φ (r)Y (θ,ϕ) (67) E E lm satisfy the orthogonality relations <ψE′(r,θ,ϕ)|ψE(r,θ,ϕ)>= drr2dΩψE∗′(r,θ,ϕ)ψE(r,θ,ϕ)=δEE′. (68) Z Then the solutions of the equation m 2 H ψ = p~2+2i∇~f ·p~+∇2f − ∇~f +V(r) ψ f f f 2 (cid:20) (cid:18) (cid:16) (cid:17) (cid:19) (cid:21) = Eψ (69) f are given by ψ (r,θ,ϕ)=ef(r,θ,ϕ)φ (r)Y (θ,ϕ). (70) Ef E lm The PT-inner product for the ψ -states is given by f <ψE′f(r,θ,ϕ)|ψEf(r,θ,ϕ)>f= drdΩPT (ψE′f(r,θ,ϕ))ψEf(r,θ,ϕ) Z = drdΩ(−)lef∗(r,π−θ,ϕ+π)+f(r,θ,ϕ)φ∗E′(r)φE(r)Yl∗′m′(θ,ϕ)Ylm(θ,ϕ). Z Besides, if f(r,θ,ϕ) satisfies Eq. (25) and considering Eq. (68), we have <ψE′f(r,θ,ϕ)|ψEf(r,θ,ϕ)>f = (−)lλ drdΩφ∗E′(r)φE(r)Yl∗′m′(θ,ϕ)Ylm(θ,ϕ) Z = λ(−)lδEE′. (71) Given that [L ,H ]6=0, it follows that L is not a conserved quantity. However, as i f i [L ,H ]=0, (72) fi f then L is conserved. fi VII. THE HYDROGEN ATOM In the case of the Hydrogen atom, we have the potential Ze2 V(r)=− , (73) r where the solutions are given by ψNlm(ρ,θ,ϕ) = N22saZ3R3B (N(N−+l−l)1!)!ρlL2Nl+−1(l+1)(ρ)e−ρ2Ylm(θ,ϕ), Ze2 E = − , N =n+l+1, n=0,1,2··· , N a N2 RB 9 where a is the Bohr radius and RB −2mE ρ=αr, α=2 . (74) ~2 r Taking into account Eq. (69), we have the equation m 2 Ze2 H ψ = ~p2+2i∇~f ·p~+∇2f − ∇~f − ψ =Eψ (75) f f f f 2 r (cid:20) (cid:18) (cid:16) (cid:17) (cid:19) (cid:21) whose solutions are given by ψ (ρ,θ,ϕ) = ef(r,θ,ϕ)ψ (ρ,θ,ϕ) (76) fNlm Nlm this are orthogonal functions if equation (25) is satisfied. A remarkable fact is that L is a conserved quantity. In fi the conventional Hydrogen atom, the Runge-Lenz vector is also conserved[9] 1 Ze2~r R = L~ ×p~−p~×L~ + . (77) i 2 r (cid:18) (cid:19)i In the case of the Hamiltonian H , we have f [R ,H ]=0, (78) fi f and we can say that the transformed non-Hermitian Runge-Lenz vector is conserved. Note that in this case, we have obtained a set of conserved quantities L ,L2,R , that are the non-Hermitian fi f fi generators of the SO(4) algebra. VIII. SUMMARY Inthisworkwehaveconstructedasetofnon-HermitianoperatorsL thatsatisfythecommutationrelationsofthe fi SO(3)-Lie algebra. We have shown that this operators generate rotations in the configuration space and not in the conventionalmomentumspacebutinamodifiednon-Hermitianmomentumspacep~ =~p+i∇~f.Itisworthytomention f thatthisoperatorwasoriginallyconsideredbyDiracinhisseminalbook. Besides,theL generatorsarerelatedwith if anewtypeofsphericalharmonicsthatresulttobePT-orthonormal. Additionally,wehaveshownthatthisquantities are conservedfor mechanicalsystems described by a central potential Hamiltonian with an additional complex term. Asaparticularcase,wehaveobtainedthesolutionsofthecorrespondingPT-Hydrogenatomthatincludesacomplex term, and we have found that a non-Hermitian Runge-Lenz vector is a conserved quantity. Considering this case, one remarkable result is that, as we have obtained the non-Hermitian generatorsof the SO(3)-Lie algebra and also a non-Hermitian Runge-Lenz vector, then we have the non-Hermitian generators of the SO(4)-Lie algebra. 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