- Email: [email protected]
Greater understanding of the hydrogen atom: RTT-integrability and Yangian symmetry
27 September 1999
Physics Letters A 260 Ž1999. 484–488 www.elsevier.nlrlocaterphysleta
Greater understanding of the hydrogen atom: RTT-integrability and Yangian symmetry Mo-Lin Ge b
a,b,)
, Kang Xue
a,b,c
, Yong-Min Cho
b
a Theoretical Physics DiÕision, Nankai Institute of Mathematics, Nankai UniÕersity, Tianjin 300071, China Asia-Pacific Center for Theoretical Physics, KAIST, 207-43 Cheongryangri-dong, Seoul 130-012, South Korea c Physics Department, Northeast Normal UniÕersity, Changchun, Jilin, 130024, China
Received 14 January 1999; received in revised form 21 July 1999; accepted 21 July 1999 Communicated by C.R. Holland
Abstract We show that the hydrogen atom possesses Yangian symmetry and integrability in terms of the RTT relation. A possible experimental test of the physical effect of Yangian is proposed. q 1999 Elsevier Science B.V. All rights reserved. PACS: 02.20.Fh; 03.65.Fd
A great deal of mathematical ingenuity has gone into solutions and symmetries of quantum integrable models w1,2x. In connection with the symmetries we are very impressed with Yangian algebra which belongs to quantum groups as it is the symmetry of many notable models, such as the Haldane-Shastry model w3–5x, the one-dimensional Hubbard model w6x, the long-ranged Hubbard model w7,8x and the Calogero–Sutherland model and its extensions w9,10x. Furthermore, the Hamiltonians of most of the models can be shown to be derived through RTT relations w11–16x, which we call RTT integrability for simplicity. However, so far Yangian research has probablyly been in the field of mathematical physics. How then can we describe the ‘real’ consequence of Yangian in physics through the simplest model? In this
) Corresponding author. Fax: q86-22-2350-1532; e-mail: [email protected]
paper we shall show that there is possibly physical effect of Yangian in the hydrogen atom. The integrability of the hydrogen atom ŽHA. Žwith SO Ž4. symmetry. has been known for long time. At first glance, it seems to be no relationship between HA and the RTT-relation approach where RŽ u y Õ .ŽT Ž u. m I .Ž I m T Ž Õ .. s Ž I m T Ž Õ ..ŽT Ž u. m I . RŽ u y Õ . which guarantees for the transfer matrix T Ž u.wtrT Ž u.,trT Ž Õ .x s 0, i.e. it provides a commuting family formed by a infinite number of conserved quantities: trT Ž u. s I q S`ns1 uyn T0Ž n.. However, there should be a finite number of T0Ž n. for HA. This is why so far both HA and RTT have been viewed as independent. In order to set up the connection between them, it is necessary to show that T Ž u. should be truncated automatically for HA in accordance with the fact of the finite number of conserved quantities for HA. In this Letter we would like to point out that HA does have Yangian symmetry and RTT integrability. For sell-contain we shall first list the basic relations
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 0 3 - 4
M.-L. Ge et al.r Physics Letters A 260 (1999) 484–488
for Yangian with T0-extension related to RTT relation w17x, then show the Yang symmetry and the truncation of T Ž u. for™ HA. In the presence of ap™ plied magnetic field B0 s B0 Bz , a possibly observable effect of Yangian in the HA spectrum is proposed. It is in fact a parity-broken effect, although the Yangian provides an enhancement factor automatically such that this effect may be observed, in principle, at large principal quantum number n and large angular momentum L2 . Finally, as a simple chain-extension we discuss a HA-like chain model with Yangian symmetry. (I). The well-known RTT relation is R Ž u y Õ . T Ž1. Ž u . T Ž2. Ž Õ . s T Ž2. Ž Õ . T Ž1. Ž u . R Ž u y Õ . where T Ž1. Ž u. s T Ž u. m I, T Ž3. Ž u. s I m T Ž u., and RŽ u. s u q P, with P being the 4 = 4 representation of the permutation operator. The elements of the transfer matrix T Ž u. s 5 TaŽbn. 5 Ž a,b s 1,2. are operators. Denoting Ž n. Ž n. Ž n. Ž n. Ž n. Ž n. T0Ž n. s T11 q T22 ,T3Ž n. s T22 y T11 , Tq s T12 ,
the relations for iteration and T0Ž2. are: Ž nq1. Ž n. Tq s T3Ž2. ,Tq q
1
T3Ž nq1. s
1 2
Ž T0Ž n.Iqy T0Ž1.TqŽ n. . ,
Ž nq1. Ž nq1. Tq , Iy , Ty s Iy ,T3Ž nq1.
2
Ž 3.
T0Ž2. , Ia s 0, T0Ž2. , Jq s 2 Ž I3 Jqy J3 Iq . , T0Ž1. ,T Ž u . s 0.
Ž 4.
It is noted that besides the relations Eqs. Ž1. – Ž4. there are still other redundant ones that can be obtained by Eqs. Ž1. – Ž4. together with Jacobi identities. Also, Eqs. Ž1. and Ž2. generate the commutation relations of YŽslŽ2.. w18,19x, whereas Eqs. Ž3. and Ž4. are T0-extension to Refs. w15,17x. (II). For the hydrogen atom the Hamiltonian H0 s Ž1r2. ™ py2™ y Ž qrr . commutes with both angular mo™ mentum L and the Pauli–Lunge-lenz vector A s ™ ™ ™ ™ 1r2Ž L = p y p = L. q q Ž ™ rrr . . For the bound state, ™ ™ putting B s 1rŽ y 2 H0 . A there are soŽ4. relations:
(
Ll , Lm s i elmn Ln , Ll , Bm s i elmn Bn , Bl , Bm s i elmn Ln .
Ž n. Ž n. Tq s T21
Ž 5.
It can be directly verified that
then on substituting T Ž u. s I T0Ž n . q T3Ž n. Ž n. Tq
Ž n. Ty
1 2
T0Ž n. y T3Ž n.
™
Ž 1. h
Ž3. Tq s
2
4
Iq Ž Jq I3 y Iq J3 .
Ž 2.
and other independent relations as shown in Ref. w15x, where A "s A1 " A 2 for any operators. Introducing
ž / 2
1 2
ž
, TlŽ2. Ž l s ",3 .
™
T0Ž 2 . y L 2 y
1 q
yh
Ž 7.
i.e. the hydrogen atom possesses YŽslŽ2.. symmetry. Substituting Eq. Ž6. into Eq. Ž3. we have
Il , Im s i elmn In , Il , Jm s i elmn Jn ,
Ž l , m ,n s 1,2,3 .
™
H0 , I s H0 , J s 0
into RTT relation yields
Jq , w J3 , Jq x s
™ ™
I s L, J s y
2
qS`ns 1 uyn
ih ™ ™ ™ L = B q FI Ž 6. 4 satisfy Yangian Eqs. Ž1. and Ž2., where F satisfies ™ ™ w F, L x s w F, B x s 0, and of course it can be zero. Obviously, we have ™
1
Il s TlŽ1. , Jl s
485
h
ž
T0Ž 1 . q
8 h
1 4
2 Hy1 0 q y
/
F Jq .
8 h2
F 2 Lq
/
Ž 8.
Since T0Ž1. commutes with everything, by choosing T0Ž1. s yŽ8rh. F s H0 or ™ T0Ž1. s 0, F s 0, we find Ž2. 1 2 that the relation T 0 s L2 q Ž1r4. H y 0 q q 2 2 Ž3. Ž8rh . F satisfies Eq. Ž4. and leads to Tq s 0, Ž n. Ž3. hence Ta s 0 Ž a s ", 3., so Ta s 0 Ž n G 3.,
M.-L. Ge et al.r Physics Letters A 260 (1999) 484–488
486
namely, T Ž u. is truncated: T Ž u. s I q uy1 T Ž1. q uy2 T Ž2. for HA. (III). We now move on to the physical effect of Yangian given by Eq. Ž6.. When the transfer matrix T Ž u. ™ F Ž u. s b 0 s 3 q T Ž u. where s 3 refers to the Pauli matrix and b 0 is constant, T0 Ž u. s trT Ž u. does not change, although the term b 0 s 3 will definitely alter the commutation relations. Choosing T0Ž1. s F s 0 we obtain
˙
L1 ,F 0Ž2.
s i b 0 T2Ž2. s i L1 ,
L2 ,F 0Ž2.
s yi b 0 T1Ž2. s iL2 ,
L3 ,F 0Ž2.
s0
Ž 9.
and here the Heisenberg equation has been used. As a result of Eq. Ž9., F 0Ž2. ŽF 0 s trF Ž u.. can be written as
F 0Ž2. s T0Ž2. q HI q f Ž H0 .
Ž 10 . TaŽ3. s 0.
T0Ž2.
where satisfies both Eq. Ž4. and A Ž H0 . can be free to choose. Obviously, function f ™ w L,T0Ž2. x s 0. From Eq. Ž9. it follows that ™
™ ™
HI s g B 0 P J , B 0 s B 0 ™ e3
Ž 11 .
where b s g B0 and g is a constant. Since when ™ ™0 ™ . still satisfies J ™ g L q J Žg is an arbitrary constant ™ Yangian, it is natural to identify B0 with a magnetic field by comparison with the familiar interaction ™ ™ Hamiltonian B0 P L. However, with such an understanding the free Hamiltonian H0 should subject to a gauge fromsformation. The modified Hamiltonian H is well-known: H l s H 0 q a L 3 s H 0 q Hl
Ž 12 .
where a s qB0rŽ2 mc . in CGS. In Eq. Ž12. the contribution of spin, i.e. 2 a s3 , may be added if necessary for spin commutes with everything appeared in the above discussion. Under Eq. Ž12. the perturbative interaction Hamiltonian HI should be added to H . On substituting Eq. Ž6. with F s 0 into Eq. Ž11., we obtain:
l HI q C.C.s
™
'y 2 E ™
B0 ™
™
™
P L2 P q PL2 q 2 q Ž Pr q rP
ž
™
™
./
Ž 13 .
where P s yi= and l is a™constant. We see that ™ there is a correction form B0 P P which should be
very small as it breaks the parity. However, meanwhile there also appears the enhancement factor nl Ž l q 1. where L2 s l Ž l q 1. which is quite large for large n and l. This strengthening factor provides, in principle, the possible observable effect. To see how this idea works for HA, let us consider the perturbation correction to be El s E0 q ´ l , that is, nothing but the Zeeman effect for angular momentum. Setting H < n,l : s Ž E0 q ´ l . < n,l : .
Ž 14 .
Since HI does not change the principal quantum number n, the transition element due to HI is given by ² n,lX < HI < n,l : s Ž const . n ² n,lX < Ž L2 q 2 qr . P3 qP3 Ž L2 q 2 qr . < n,l : 4 .
Ž 15 .
Using the Heisenberg equation we obtain ² n,lX < HI < n,l : s Ž const . n Ž ´ lX y ´ l . lX Ž lX q 1 . q 2 qgn ,lX ql Ž l q 1 . q 2 qgn ,l 4 ² n,lX < x 3 < n,l :
Ž 16 .
where gn,l s ² n,l < g < n,l : is average radius of the Ž n,l .-orbit, and ² n,lX < x 3 < n,l : is the usual dipole excitation referring to the anglular part. When semi-classical states are taken into account, i.e. n and l are large, as lX small the element is proportional to v nŽ l 2 q 2 qrn,l ., where v s Ž ´ lX y ´ l .r". Suppose that n F 100 Žthe maximum of current observation. and l ; 100, there is enhancement factor ; 10 6 before the usual dropped interaction ™ ™ B0 P P. It is noted that this factor is a peculiar contribution due to Yangian. As an intuition toward understanding the significance of possible measurements of the effect, let us note the following. Ža. The Yangian interaction does not change the transition frequency which determined by Hl for the quantum number n keeping fixed in the process because w H0 , HI x s 0. What the Yangian contributes is a huge strengthen factor to the intensity in such a transition for large l-states. Žb. When n and l are large, we may encounter the ambiguity of quantum chaos, which is popularly explained by the classical chaos with quantum fluc-
M.-L. Ge et al.r Physics Letters A 260 (1999) 484–488
tuation that may be related to the anomaly spectrum of HA. However, the Yangian effect is related to the intensity depending on l in a nonlinear manner and completely does not affect the frequencies. Žc. The effect looks like the measurement of the Weinberg angle w20x which violates parity in atomic physics. The effect is very small. However, because of the strengthening factor, it was observed in bismuth vapor by laser more than 20 years ago. ™ With the conserved H and L2 Žas well as L3 ., meanwhile, the quantum determinant Žsee Ref. w21x, and references therein, for the primary discovery. detT Ž u. s T11Ž u.T22 Ž u y 1. y T12 Ž u.T21Ž u y 1. s ™ S`ns 0 uyn Cn is also truncated; C2 s L 2 q Ž1r4. = y1 2 H0 q . We find C0 s 1, C1 s H and C2 s Ž1r2. H02 2 q Ž1r8. H0 q Ž1r4. Hy1 0 q . We thus verify that the hydrogen atom does have YŽslŽ2.. symmetry and RTT integrability. More importantly, the effect of Yangian may be observed through measuring the intensities of spectrum of HA at large n and l.(IV). The above idea can be extended to a frozen hydrogen-like atom ŽHLA. chain ŽFLHAC.. Suppose that N HLA are put at N sites to form a chain, and the Hamiltonian Hchain s H0 q HI still has the global soŽ4. symmetry for N
H0 s
Ý is1
ž
1 ™2 qi p y s Ý H0 i 2 i ri
/
and N
™
™
N
ž
™
™
™
™
HI s Ý f i , j L i P Bj q Ý g i , j L i P L j q Bi P Bj i/j
i/j
/
Ž 17 .
487
considering a simple case where f i, j s g i, j , then Hchain becomes N
™
™
q Hchain s H0 q 2 Ý f i , j Iq i P Ij
Ž 18 .
i/j
which possesses YŽslŽ2.. symmetry. On account of ™ 2 . is still a good quantum number, it the fact that Ž Iq i takes values 0 Žcorresponding to n s 1 in the hydrogen atom., 1r2 Žcorresponding to n s 2., 1 Žto n s 3., etc.. The interaction makes the energy levels Ž; 1rn2 . split into bands. Suppose that f i, j s d i,iq1 , then the n s 2 level becomes the band formed by the ™ . chain, whereas the n s equivalent ‘spin y 1r2’ Ž Iq i 3 level becomes the band formed by ‘spin y 1’ chain. When N ™ `, the n s 2 band is continuous, although the n s 3 band does have the Haldane gap w10x, and references therein. ™ As for the case where quantum number I22 s 0 Ž n s 1 band. it is very complicated. Following the discussion by Faddeev and Korchemsky w22–25x it is related to the infinite dimensional representation of slŽ2.. Because of the Haldane gap, as N ™ ` the transition between n s 2 and n s 3 bands is expected to have unusual behavior. We have got used to meet the infinite number of conserved quantities for various chain models w26– 30x. This is because the interaction is nonlinear and one-dimensional. In this letter we have seen that for hydrogen atom the RTT relation is automatically truncated and the strengthening factor associated with Yangian interaction may be measurable. The discussed example may help to shed a new light on the application of the RTT integrability and Yangian to the elementary quantum mechanics.
Acknowledgements where functions f i, j and g i, j are symmetric and dependent on the distance between™the ith and ™ jth ™ ™ sites, L i s™ ri =™ pi and A s Ž1r2.Ž L i =™ pi y ™ pi = L i . q qi Ž™ rirri . and the origin™of ™ ri has been taken at ith ™ site, Bi s 1rŽ y 2 H0 i . A i . ™ ™ ™ N ™ Defining globally I ' S is1 L i s L and B s ™ N Ž6.. It can be S is1 Bi , we get the ™ Yangian equation ™ w x w x proved that Hchain , I s Hchain , J s 0. i.e. H has YŽslŽ™ 2.. ™symmetry. Note that because at the same ., Eq. Ž17. is valid site L P B s 0 Žwithout monopole ™" ™ ™ Ž only for i / j. Defining Ii s 1r2.Ž L i " Bi . and
We would like to thank Prof. C.N. Yang for the enlighten discussions. This work was in part supported by NSF of China and APCTP in Seoul.
(
References w1x M. Jimbo ŽEd.., Yang-Baxter Equation in Integrable Systems World Scientific, Singapore, 1990. w2x D.C. Mattis ŽEd.., Many-Body Problem, World Scientific, Singapore, 1992.
488
M.-L. Ge et al.r Physics Letters A 260 (1999) 484–488
w3x F.D.M. Haldane, Phys. Rev. Lett. 60 Ž1988. 635. w4x S. Sharstry, Phys. Rev. Lett. 60 Ž1988. 639. w5x F.D.M. Haldane, Z.N.C. Ha, J.C. Talstra, D. Bernard, V. Pasquier, Phys. Rev. Lett. 69 Ž1992. 2021. w6x D.B. Uglov, V.E. Korepin, Phys. Lett. A 190 Ž1994. 238. w7x F. Gohmann, V. Inozemtsev, Phys. Lett. A 214 Ž1996. ¨ 161–166. w8x D.F. Wang, Q.F. Zhang, P. Coleman, Phys. Rev. B 48 Ž1993. 8476. w9x M. Wadati, Phys. Rev. Lett. 60 Ž1988. 635. w10x F.D.M. Haldane, Physics of the Ideal Semion Gas: Spinions and Quantum Symmetries of the Integrable Haldnade-Shastry Spin-Chain, Proc. 16th Taniguch Symp. Condensed Matter, Berlin, Springer, 1994. w11x D. Bernard, M. Gaudin, F.D.M. Haldane, U. Pasquier, J. Phys. A 26 Ž1993. 5219. w12x V.I. Inozemtsev, J. Stat. Phys. 59 Ž1990. 1143. w13x J.C. Talstra, F.D.M. Haldan, J. Phys. A 28 Ž1996. 2369. w14x M.L. Ge, Y.W. Wang, Phys. Rev. E 51 Ž1995. 2919. w15x Z.F. Wang, M.L. Ge, K. Xue, J. Phys. A 30 Ž1997. 5023– 5036. w16x M.L. Ge, K. Xue, J. Phys. A 29 Ž1996. 6881. w17x M.L. Ge, K. Xue, Y.M. Cho, Phys. Lett. A 249 Ž1998. 358–362.
w18x V. Drinfled, Sov. Math. Dokl. 32 Ž1985. 254. w19x V. Drinfled, Sov. Math. Dokl. 36 Ž1988. 212. w20x M. Cvetic, P. Langacker ŽEds.., Testing the Standard Model, World Scientific, Singapore, 1991. w21x V.E. Korepln, N.M. Bogoliubov, A.G. Izergin, Quantum Inverse Scattering Method and Correlation Function, Cambridge University Press, 1993. w22x L.D. Faddeev, G.P. Korchemsky, Phys. Lett. B 342 Ž1995. 311–322. w23x L.N. Lipatov, Phys. Lett. B 251 Ž1990. 284. w24x L.N. Lipatov, Phys. Lett. B 309 Ž1993. 394. w25x L.N. Lipatov, JETP Lett. 59 Ž1994. . w26x P.P. Kulish, E.K. Sklyanin, in: L.D. Faddeev, Lecture Notes, Springer, Phys. 151 Ž1982. 61. w27x E.K. Sklyanin, Quantum Inverse Scattering Methods, selected topics, in: M.L. Ge ŽEd.., Quantum Groups and Quantum Integrable Systems, World Scientific, Singapore, 1991, pp. 63—88. w28x N.M. Bogoliubov, A.G. Izergin, Quantum Inverse Scattering Method and Correlation Function, Cambridge Univiversity Press, 1993. w29x L.D. Faddeev, Les Houches, Session 39, 1982. w30x J.B. Zuber, R. Stora ŽEds.., Recent Adv. in Field Theory and Stat. Mech., Elsevier, 1984, pp. 563—608.
Physics Letters A 260 Ž1999. 484–488 www.elsevier.nlrlocaterphysleta
Greater understanding of the hydrogen atom: RTT-integrability and Yangian symmetry Mo-Lin Ge b
a,b,)
, Kang Xue
a,b,c
, Yong-Min Cho
b
a Theoretical Physics DiÕision, Nankai Institute of Mathematics, Nankai UniÕersity, Tianjin 300071, China Asia-Pacific Center for Theoretical Physics, KAIST, 207-43 Cheongryangri-dong, Seoul 130-012, South Korea c Physics Department, Northeast Normal UniÕersity, Changchun, Jilin, 130024, China
Received 14 January 1999; received in revised form 21 July 1999; accepted 21 July 1999 Communicated by C.R. Holland
Abstract We show that the hydrogen atom possesses Yangian symmetry and integrability in terms of the RTT relation. A possible experimental test of the physical effect of Yangian is proposed. q 1999 Elsevier Science B.V. All rights reserved. PACS: 02.20.Fh; 03.65.Fd
A great deal of mathematical ingenuity has gone into solutions and symmetries of quantum integrable models w1,2x. In connection with the symmetries we are very impressed with Yangian algebra which belongs to quantum groups as it is the symmetry of many notable models, such as the Haldane-Shastry model w3–5x, the one-dimensional Hubbard model w6x, the long-ranged Hubbard model w7,8x and the Calogero–Sutherland model and its extensions w9,10x. Furthermore, the Hamiltonians of most of the models can be shown to be derived through RTT relations w11–16x, which we call RTT integrability for simplicity. However, so far Yangian research has probablyly been in the field of mathematical physics. How then can we describe the ‘real’ consequence of Yangian in physics through the simplest model? In this
) Corresponding author. Fax: q86-22-2350-1532; e-mail: [email protected]
paper we shall show that there is possibly physical effect of Yangian in the hydrogen atom. The integrability of the hydrogen atom ŽHA. Žwith SO Ž4. symmetry. has been known for long time. At first glance, it seems to be no relationship between HA and the RTT-relation approach where RŽ u y Õ .ŽT Ž u. m I .Ž I m T Ž Õ .. s Ž I m T Ž Õ ..ŽT Ž u. m I . RŽ u y Õ . which guarantees for the transfer matrix T Ž u.wtrT Ž u.,trT Ž Õ .x s 0, i.e. it provides a commuting family formed by a infinite number of conserved quantities: trT Ž u. s I q S`ns1 uyn T0Ž n.. However, there should be a finite number of T0Ž n. for HA. This is why so far both HA and RTT have been viewed as independent. In order to set up the connection between them, it is necessary to show that T Ž u. should be truncated automatically for HA in accordance with the fact of the finite number of conserved quantities for HA. In this Letter we would like to point out that HA does have Yangian symmetry and RTT integrability. For sell-contain we shall first list the basic relations
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 0 3 - 4
M.-L. Ge et al.r Physics Letters A 260 (1999) 484–488
for Yangian with T0-extension related to RTT relation w17x, then show the Yang symmetry and the truncation of T Ž u. for™ HA. In the presence of ap™ plied magnetic field B0 s B0 Bz , a possibly observable effect of Yangian in the HA spectrum is proposed. It is in fact a parity-broken effect, although the Yangian provides an enhancement factor automatically such that this effect may be observed, in principle, at large principal quantum number n and large angular momentum L2 . Finally, as a simple chain-extension we discuss a HA-like chain model with Yangian symmetry. (I). The well-known RTT relation is R Ž u y Õ . T Ž1. Ž u . T Ž2. Ž Õ . s T Ž2. Ž Õ . T Ž1. Ž u . R Ž u y Õ . where T Ž1. Ž u. s T Ž u. m I, T Ž3. Ž u. s I m T Ž u., and RŽ u. s u q P, with P being the 4 = 4 representation of the permutation operator. The elements of the transfer matrix T Ž u. s 5 TaŽbn. 5 Ž a,b s 1,2. are operators. Denoting Ž n. Ž n. Ž n. Ž n. Ž n. Ž n. T0Ž n. s T11 q T22 ,T3Ž n. s T22 y T11 , Tq s T12 ,
the relations for iteration and T0Ž2. are: Ž nq1. Ž n. Tq s T3Ž2. ,Tq q
1
T3Ž nq1. s
1 2
Ž T0Ž n.Iqy T0Ž1.TqŽ n. . ,
Ž nq1. Ž nq1. Tq , Iy , Ty s Iy ,T3Ž nq1.
2
Ž 3.
T0Ž2. , Ia s 0, T0Ž2. , Jq s 2 Ž I3 Jqy J3 Iq . , T0Ž1. ,T Ž u . s 0.
Ž 4.
It is noted that besides the relations Eqs. Ž1. – Ž4. there are still other redundant ones that can be obtained by Eqs. Ž1. – Ž4. together with Jacobi identities. Also, Eqs. Ž1. and Ž2. generate the commutation relations of YŽslŽ2.. w18,19x, whereas Eqs. Ž3. and Ž4. are T0-extension to Refs. w15,17x. (II). For the hydrogen atom the Hamiltonian H0 s Ž1r2. ™ py2™ y Ž qrr . commutes with both angular mo™ mentum L and the Pauli–Lunge-lenz vector A s ™ ™ ™ ™ 1r2Ž L = p y p = L. q q Ž ™ rrr . . For the bound state, ™ ™ putting B s 1rŽ y 2 H0 . A there are soŽ4. relations:
(
Ll , Lm s i elmn Ln , Ll , Bm s i elmn Bn , Bl , Bm s i elmn Ln .
Ž n. Ž n. Tq s T21
Ž 5.
It can be directly verified that
then on substituting T Ž u. s I T0Ž n . q T3Ž n. Ž n. Tq
Ž n. Ty
1 2
T0Ž n. y T3Ž n.
™
Ž 1. h
Ž3. Tq s
2
4
Iq Ž Jq I3 y Iq J3 .
Ž 2.
and other independent relations as shown in Ref. w15x, where A "s A1 " A 2 for any operators. Introducing
ž / 2
1 2
ž
, TlŽ2. Ž l s ",3 .
™
T0Ž 2 . y L 2 y
1 q
yh
Ž 7.
i.e. the hydrogen atom possesses YŽslŽ2.. symmetry. Substituting Eq. Ž6. into Eq. Ž3. we have
Il , Im s i elmn In , Il , Jm s i elmn Jn ,
Ž l , m ,n s 1,2,3 .
™
H0 , I s H0 , J s 0
into RTT relation yields
Jq , w J3 , Jq x s
™ ™
I s L, J s y
2
qS`ns 1 uyn
ih ™ ™ ™ L = B q FI Ž 6. 4 satisfy Yangian Eqs. Ž1. and Ž2., where F satisfies ™ ™ w F, L x s w F, B x s 0, and of course it can be zero. Obviously, we have ™
1
Il s TlŽ1. , Jl s
485
h
ž
T0Ž 1 . q
8 h
1 4
2 Hy1 0 q y
/
F Jq .
8 h2
F 2 Lq
/
Ž 8.
Since T0Ž1. commutes with everything, by choosing T0Ž1. s yŽ8rh. F s H0 or ™ T0Ž1. s 0, F s 0, we find Ž2. 1 2 that the relation T 0 s L2 q Ž1r4. H y 0 q q 2 2 Ž3. Ž8rh . F satisfies Eq. Ž4. and leads to Tq s 0, Ž n. Ž3. hence Ta s 0 Ž a s ", 3., so Ta s 0 Ž n G 3.,
M.-L. Ge et al.r Physics Letters A 260 (1999) 484–488
486
namely, T Ž u. is truncated: T Ž u. s I q uy1 T Ž1. q uy2 T Ž2. for HA. (III). We now move on to the physical effect of Yangian given by Eq. Ž6.. When the transfer matrix T Ž u. ™ F Ž u. s b 0 s 3 q T Ž u. where s 3 refers to the Pauli matrix and b 0 is constant, T0 Ž u. s trT Ž u. does not change, although the term b 0 s 3 will definitely alter the commutation relations. Choosing T0Ž1. s F s 0 we obtain
˙
L1 ,F 0Ž2.
s i b 0 T2Ž2. s i L1 ,
L2 ,F 0Ž2.
s yi b 0 T1Ž2. s iL2 ,
L3 ,F 0Ž2.
s0
Ž 9.
and here the Heisenberg equation has been used. As a result of Eq. Ž9., F 0Ž2. ŽF 0 s trF Ž u.. can be written as
F 0Ž2. s T0Ž2. q HI q f Ž H0 .
Ž 10 . TaŽ3. s 0.
T0Ž2.
where satisfies both Eq. Ž4. and A Ž H0 . can be free to choose. Obviously, function f ™ w L,T0Ž2. x s 0. From Eq. Ž9. it follows that ™
™ ™
HI s g B 0 P J , B 0 s B 0 ™ e3
Ž 11 .
where b s g B0 and g is a constant. Since when ™ ™0 ™ . still satisfies J ™ g L q J Žg is an arbitrary constant ™ Yangian, it is natural to identify B0 with a magnetic field by comparison with the familiar interaction ™ ™ Hamiltonian B0 P L. However, with such an understanding the free Hamiltonian H0 should subject to a gauge fromsformation. The modified Hamiltonian H is well-known: H l s H 0 q a L 3 s H 0 q Hl
Ž 12 .
where a s qB0rŽ2 mc . in CGS. In Eq. Ž12. the contribution of spin, i.e. 2 a s3 , may be added if necessary for spin commutes with everything appeared in the above discussion. Under Eq. Ž12. the perturbative interaction Hamiltonian HI should be added to H . On substituting Eq. Ž6. with F s 0 into Eq. Ž11., we obtain:
l HI q C.C.s
™
'y 2 E ™
B0 ™
™
™
P L2 P q PL2 q 2 q Ž Pr q rP
ž
™
™
./
Ž 13 .
where P s yi= and l is a™constant. We see that ™ there is a correction form B0 P P which should be
very small as it breaks the parity. However, meanwhile there also appears the enhancement factor nl Ž l q 1. where L2 s l Ž l q 1. which is quite large for large n and l. This strengthening factor provides, in principle, the possible observable effect. To see how this idea works for HA, let us consider the perturbation correction to be El s E0 q ´ l , that is, nothing but the Zeeman effect for angular momentum. Setting H < n,l : s Ž E0 q ´ l . < n,l : .
Ž 14 .
Since HI does not change the principal quantum number n, the transition element due to HI is given by ² n,lX < HI < n,l : s Ž const . n ² n,lX < Ž L2 q 2 qr . P3 qP3 Ž L2 q 2 qr . < n,l : 4 .
Ž 15 .
Using the Heisenberg equation we obtain ² n,lX < HI < n,l : s Ž const . n Ž ´ lX y ´ l . lX Ž lX q 1 . q 2 qgn ,lX ql Ž l q 1 . q 2 qgn ,l 4 ² n,lX < x 3 < n,l :
Ž 16 .
where gn,l s ² n,l < g < n,l : is average radius of the Ž n,l .-orbit, and ² n,lX < x 3 < n,l : is the usual dipole excitation referring to the anglular part. When semi-classical states are taken into account, i.e. n and l are large, as lX small the element is proportional to v nŽ l 2 q 2 qrn,l ., where v s Ž ´ lX y ´ l .r". Suppose that n F 100 Žthe maximum of current observation. and l ; 100, there is enhancement factor ; 10 6 before the usual dropped interaction ™ ™ B0 P P. It is noted that this factor is a peculiar contribution due to Yangian. As an intuition toward understanding the significance of possible measurements of the effect, let us note the following. Ža. The Yangian interaction does not change the transition frequency which determined by Hl for the quantum number n keeping fixed in the process because w H0 , HI x s 0. What the Yangian contributes is a huge strengthen factor to the intensity in such a transition for large l-states. Žb. When n and l are large, we may encounter the ambiguity of quantum chaos, which is popularly explained by the classical chaos with quantum fluc-
M.-L. Ge et al.r Physics Letters A 260 (1999) 484–488
tuation that may be related to the anomaly spectrum of HA. However, the Yangian effect is related to the intensity depending on l in a nonlinear manner and completely does not affect the frequencies. Žc. The effect looks like the measurement of the Weinberg angle w20x which violates parity in atomic physics. The effect is very small. However, because of the strengthening factor, it was observed in bismuth vapor by laser more than 20 years ago. ™ With the conserved H and L2 Žas well as L3 ., meanwhile, the quantum determinant Žsee Ref. w21x, and references therein, for the primary discovery. detT Ž u. s T11Ž u.T22 Ž u y 1. y T12 Ž u.T21Ž u y 1. s ™ S`ns 0 uyn Cn is also truncated; C2 s L 2 q Ž1r4. = y1 2 H0 q . We find C0 s 1, C1 s H and C2 s Ž1r2. H02 2 q Ž1r8. H0 q Ž1r4. Hy1 0 q . We thus verify that the hydrogen atom does have YŽslŽ2.. symmetry and RTT integrability. More importantly, the effect of Yangian may be observed through measuring the intensities of spectrum of HA at large n and l.(IV). The above idea can be extended to a frozen hydrogen-like atom ŽHLA. chain ŽFLHAC.. Suppose that N HLA are put at N sites to form a chain, and the Hamiltonian Hchain s H0 q HI still has the global soŽ4. symmetry for N
H0 s
Ý is1
ž
1 ™2 qi p y s Ý H0 i 2 i ri
/
and N
™
™
N
ž
™
™
™
™
HI s Ý f i , j L i P Bj q Ý g i , j L i P L j q Bi P Bj i/j
i/j
/
Ž 17 .
487
considering a simple case where f i, j s g i, j , then Hchain becomes N
™
™
q Hchain s H0 q 2 Ý f i , j Iq i P Ij
Ž 18 .
i/j
which possesses YŽslŽ2.. symmetry. On account of ™ 2 . is still a good quantum number, it the fact that Ž Iq i takes values 0 Žcorresponding to n s 1 in the hydrogen atom., 1r2 Žcorresponding to n s 2., 1 Žto n s 3., etc.. The interaction makes the energy levels Ž; 1rn2 . split into bands. Suppose that f i, j s d i,iq1 , then the n s 2 level becomes the band formed by the ™ . chain, whereas the n s equivalent ‘spin y 1r2’ Ž Iq i 3 level becomes the band formed by ‘spin y 1’ chain. When N ™ `, the n s 2 band is continuous, although the n s 3 band does have the Haldane gap w10x, and references therein. ™ As for the case where quantum number I22 s 0 Ž n s 1 band. it is very complicated. Following the discussion by Faddeev and Korchemsky w22–25x it is related to the infinite dimensional representation of slŽ2.. Because of the Haldane gap, as N ™ ` the transition between n s 2 and n s 3 bands is expected to have unusual behavior. We have got used to meet the infinite number of conserved quantities for various chain models w26– 30x. This is because the interaction is nonlinear and one-dimensional. In this letter we have seen that for hydrogen atom the RTT relation is automatically truncated and the strengthening factor associated with Yangian interaction may be measurable. The discussed example may help to shed a new light on the application of the RTT integrability and Yangian to the elementary quantum mechanics.
Acknowledgements where functions f i, j and g i, j are symmetric and dependent on the distance between™the ith and ™ jth ™ ™ sites, L i s™ ri =™ pi and A s Ž1r2.Ž L i =™ pi y ™ pi = L i . q qi Ž™ rirri . and the origin™of ™ ri has been taken at ith ™ site, Bi s 1rŽ y 2 H0 i . A i . ™ ™ ™ N ™ Defining globally I ' S is1 L i s L and B s ™ N Ž6.. It can be S is1 Bi , we get the ™ Yangian equation ™ w x w x proved that Hchain , I s Hchain , J s 0. i.e. H has YŽslŽ™ 2.. ™symmetry. Note that because at the same ., Eq. Ž17. is valid site L P B s 0 Žwithout monopole ™" ™ ™ Ž only for i / j. Defining Ii s 1r2.Ž L i " Bi . and
We would like to thank Prof. C.N. Yang for the enlighten discussions. This work was in part supported by NSF of China and APCTP in Seoul.
(
References w1x M. Jimbo ŽEd.., Yang-Baxter Equation in Integrable Systems World Scientific, Singapore, 1990. w2x D.C. Mattis ŽEd.., Many-Body Problem, World Scientific, Singapore, 1992.
488
M.-L. Ge et al.r Physics Letters A 260 (1999) 484–488
w3x F.D.M. Haldane, Phys. Rev. Lett. 60 Ž1988. 635. w4x S. Sharstry, Phys. Rev. Lett. 60 Ž1988. 639. w5x F.D.M. Haldane, Z.N.C. Ha, J.C. Talstra, D. Bernard, V. Pasquier, Phys. Rev. Lett. 69 Ž1992. 2021. w6x D.B. Uglov, V.E. Korepin, Phys. Lett. A 190 Ž1994. 238. w7x F. Gohmann, V. Inozemtsev, Phys. Lett. A 214 Ž1996. ¨ 161–166. w8x D.F. Wang, Q.F. Zhang, P. Coleman, Phys. Rev. B 48 Ž1993. 8476. w9x M. Wadati, Phys. Rev. Lett. 60 Ž1988. 635. w10x F.D.M. Haldane, Physics of the Ideal Semion Gas: Spinions and Quantum Symmetries of the Integrable Haldnade-Shastry Spin-Chain, Proc. 16th Taniguch Symp. Condensed Matter, Berlin, Springer, 1994. w11x D. Bernard, M. Gaudin, F.D.M. Haldane, U. Pasquier, J. Phys. A 26 Ž1993. 5219. w12x V.I. Inozemtsev, J. Stat. Phys. 59 Ž1990. 1143. w13x J.C. Talstra, F.D.M. Haldan, J. Phys. A 28 Ž1996. 2369. w14x M.L. Ge, Y.W. Wang, Phys. Rev. E 51 Ž1995. 2919. w15x Z.F. Wang, M.L. Ge, K. Xue, J. Phys. A 30 Ž1997. 5023– 5036. w16x M.L. Ge, K. Xue, J. Phys. A 29 Ž1996. 6881. w17x M.L. Ge, K. Xue, Y.M. Cho, Phys. Lett. A 249 Ž1998. 358–362.
w18x V. Drinfled, Sov. Math. Dokl. 32 Ž1985. 254. w19x V. Drinfled, Sov. Math. Dokl. 36 Ž1988. 212. w20x M. Cvetic, P. Langacker ŽEds.., Testing the Standard Model, World Scientific, Singapore, 1991. w21x V.E. Korepln, N.M. Bogoliubov, A.G. Izergin, Quantum Inverse Scattering Method and Correlation Function, Cambridge University Press, 1993. w22x L.D. Faddeev, G.P. Korchemsky, Phys. Lett. B 342 Ž1995. 311–322. w23x L.N. Lipatov, Phys. Lett. B 251 Ž1990. 284. w24x L.N. Lipatov, Phys. Lett. B 309 Ž1993. 394. w25x L.N. Lipatov, JETP Lett. 59 Ž1994. . w26x P.P. Kulish, E.K. Sklyanin, in: L.D. Faddeev, Lecture Notes, Springer, Phys. 151 Ž1982. 61. w27x E.K. Sklyanin, Quantum Inverse Scattering Methods, selected topics, in: M.L. Ge ŽEd.., Quantum Groups and Quantum Integrable Systems, World Scientific, Singapore, 1991, pp. 63—88. w28x N.M. Bogoliubov, A.G. Izergin, Quantum Inverse Scattering Method and Correlation Function, Cambridge Univiversity Press, 1993. w29x L.D. Faddeev, Les Houches, Session 39, 1982. w30x J.B. Zuber, R. Stora ŽEds.., Recent Adv. in Field Theory and Stat. Mech., Elsevier, 1984, pp. 563—608.