Wave Function for Harmonically Confined Electrons in Time-Dependent Electric and Magnetostatic Fields

Transcription

1 Cty Unversty of New York CUNY) CUNY Aadem Works Publatons and Researh Graduate Center 4 Wave Funton for Harmonally Confned Eletrons n Tme-Dependent Eletr and Magnetostat Felds Hong-Mng Zhu Nngbo Unversty Jn-Wang Chen Nngbo Unversty Xao-Yn Pan Nngbo Unversty Vraht Sahn CUNY Graduate Center and CUNY Brooklyn College How does aess to ths work beneft you? Let us know! Follow ths and addtonal works at: Part of the Atom, Moleular and Optal Physs Commons, Bologal and Chemal Physs Commons, Condensed Matter Physs Commons, and the Quantum Physs Commons Reommended Ctaton H.-M. Zhu, J.-W. Chen., X.-Y. Pan, and V. Sahn, J. Chem. Phys. 4, 438 4). Ths Artle s brought to you by CUNY Aadem Works. It has been aepted for nluson n Publatons and Researh by an authorzed admnstrator of CUNY Aadem Works. For more nformaton, please ontat AademWorks@g.uny.edu.

2 THE JOURNAL OF CHEMICAL PHYSICS 4, 438 4) Wave funton for harmonally onfned eletrons n tme-dependent eletr and magnetostat felds Hong-Mng Zhu,,a) Jn-Wang Chen, Xao-Yn Pan,,b) and Vraht Sahn Department of Physs, Nngbo Unversty, Nngbo 35, Chna Department of Physs, Brooklyn College and The Graduate Shool of the Cty Unversty of New York, New York, New York 6, USA Reeved 3 Otober 3; aepted 3 Deember 3; publshed onlne 4 January 4) We derve va the nteraton representaton the many-body wave funton for harmonally onfned eletrons n the presene of a magnetostat feld and perturbed by a spatally homogeneous tme-dependent eletr feld the Generalzed Kohn Theorem GKT) wave funton. In the absene of the harmon onfnement the unform eletron gas the GKT wave funton redues to the Kohn Theorem wave funton. Wthout the magnetostat feld, the GKT wave funton s the Harmon Potental Theorem wave funton. We further prove the valdty of the onneton between the GKT wave funton derved and the system n an aelerated frame of referene. Fnally, we provde examples of the applaton of the GKT wave funton. 4 AIP Publshng LLC. [ I. INTRODUCTION A bas problem n quantum mehans s the determnaton of the soluton to the many-eletron system n atoms, moleules, solds, quantum wells, two-dmensonal eletron systems suh as those at semondutor heterojuntons, quantum dots, et., and of the nteraton of suh systems wth external eletromagnet felds. Systems for whh exat solutons of the Shrödnger equaton exst are unommon, and thus suh systems play a sgnfant role n the understandng of the many-body problem. The exat solutons also lead to further physal nsghts va ther use n other manfestatons of Shrödnger theory suh as densty funtonal 6 and quantal densty funtonal 7, 8 theores. In ths paper we onsder a tme-dependent TD) Hamltonan that s modfable and hene applable to dfferent physal systems, 9 3 and provde a frst-prnples dervaton of the orrespondng most general) wave funton whh we refer to as the Generalzed Kohn Theorem GKT) wave funton. Consder a system of N harmonally onfned eletrons n a magnetostat feld Br) = Ar). On applaton of a spatally homogeneous tme-dependent eletr feld Et), the Hamltonan s H ˆ t) = H ˆ ) + eet) r, ) the unperturbed Hamltonan s H ˆ ) = [ ˆp + e ) m Ar ) + ] r K r +,j) ur r j ), ) wth =,...,N denotes the eletrons;, j) denotes all pars; r = x, y, z ); ur) the eletron nteraton operator; and K s a) Present address: Natonal Astronomal Observatores, Chnese Aademy of Senes, Bejng, Chna. b) Eletron mal: panxaoyn@nbu.edu.n the fore onstant matrx ωx K = m ωy, 3) ωz and ω x,w y,w z the dretonal harmon frequenes. The orrespondng tme-dependent TD) Shrödnger equaton s ) t H ˆ t) t) =. 4) Note that by modfyng the fore onstant matrx K, the Hamltonan of Eq. ) an represent a quantum well ω x = ω y =, ω z ); quantum dot ω x ω y, ω z = ; ω x = ω y ω z ); or a two-dmensonal eletron gas ω x = ω y = ω z = ). The absene of the magnet feld B n turn also onsttutes speal ases. In ths paper we present a frst prnples dervaton va the nteraton representaton of the soluton to the TD Shrödnger equaton. We refer to ths soluton as the GKT wave funton. In the symmetr gauge Ar) = Br) r,the soluton s the followng r,...,r N ; t) = e ī E n t+s [R m t)] P Rmt) R) n r R m t), r R m t),...,r N R m t)), 5) E n, n are the egenenerges and egenfuntons of the unperturbed Hamltonan: H ˆ ) n r,...,r N ) = E n n r,...,r N ); 6) S [R m t)] the total lassal aton: S [R m t)] = S XY t) + S Z t), 7) S XY [R t)] = M R R R + ω R J R )dt, 8) -966/4/4)/438/9/$3. 4, AIP Publshng LLC

3 438- Zhu et al. J. Chem. Phys. 4, 438 4) S Z [Zt)] = M Ż ωz Z) dt. 9) In Eq. 5), R m t) s the three-dmensonal vetor X mt) Y m t) ), and Z m t) P Rm t) the orrespondng anonal momentum see Eq. 68) below). The vetor R m t) satsfes the lassal equaton of moton m d R m t) = K R dt m t) eet) e dr m t) B. ) dt In Eq. 8), R t) s the two-dmensonal vetor Xt) Y t) ); R the transpose of the vetor R ; = ω x ω ); and J = y ). Fnally, the vetor n Eq. 5) R = N r = X Y ), Z ω = NeB = eb the ylotron frequeny, and M = Nm. M m Observe that the GKT wave funton s omprsed of a phase fator tmes the unperturbed wave funton n whh the oordnates of eah eletron are translated by a value that satsfes the lassal equaton of moton. Hene, f the unperturbed wave funton s known, then the tme evoluton of all propertes s known. In partular, observables represented by non-dfferental Hermtan operators suh as the densty ρr,t) = ˆρr), wth ˆρr) = δr r) the densty operator, possess the translatonal property ρr,t) = ρ r R m t)), ) ρ r) s the unperturbed system densty. Thus, f the unperturbed wave funton and hene densty ρ r)sknown, then so s the tme evoluton of the perturbed system densty ρr, t). Beause of the phase fator ths translatonal property s not obeyed for observables nvolvng dfferental operators suh as the physal urrent densty. We note two speal ases of the wave funton: In the absene of the external harmon potental,.e., for the unform eletron gas, the GKT wave funton derved redues to the Kohn theorem KT) wave funton; 9 n the absene of the external magnet feld, the GKT wave funton redues to that of the harmon potental theorem HPT) wave funton. 4 For other ndependent dervatons of the HPT wave funton va the operator, Feynman Path ntegral, and nteraton representaton methods see, respetvely, Refs. 7, 5, and 6.) We next put our work n ontext. In the orgnal work by Kohn and o-workers, 9 3 t was shown that for systems represented by the Hamltonan of Eq. ), the optal absorpton frequenes observed, whether for the unform eletron gas or when the eletrons are onfned harmonally, were dental to those of a sngle partle and ndependent of the number of eletrons and the eletron-eletron nteraton. In the lterature, the ase of the unform eletron gas s onsdered the KT, and that of the harmonally onfned eletrons ase s referred to as the GKT.) However, n spte of the fat that Yp states n hs footnote 8 that It s straghtforward to dedue the ground-state wave funton and thus also the exted states) the explt expresson for the GKT wave funton of Eq. 5) s not gven n these papers. In a later paper, Dobson 4 derved the HPT wave funton statng that ths was a slght extenson of the GKT sne the GKT only refers to the frequeny dependene of lnear response and does not address the spatal profle of the movng densty. 7 By extenson Dobson was referrng to the dervaton of the HPT wave funton. From the wave funton t beomes lear that the densty ρr, t) must satsfythetranslatonal property of Eq. ). As the densty ρr, t) s the bas ngredent of TD densty funtonal theory, 3 6 ths property of the densty ould then be employed as a rgorous onstrant to test varous approxmate aton funtonals of the densty wthn the ontext of the theory. Agan, wth the purpose of testng dfferent aton funtonals, Vgnale 8 observed that the densty ρr, t) orrespondng to the GKT Hamltonan of Eq. ) also obeyed the translatonal property of Eq. ). He arrved at ths onluson by onsderng the Shrödnger equaton n an aelerated frame of referene. As the fous of TD densty funtonal theory s the densty ρr, t), nether the GKT wave funton nor ts dervaton va ths approah were provded. The expresson for the KT wave funton now appears n the text of Ref. 9.) Thus, the expresson for the GKT wave funton does not exst n the lterature at present. Knowledge of the GKT wave funton leads dretly to the translatonal property of the densty, and to the KT and HPT wave funtons sne these onsttute speal ases. It s evdent that there are several approahes by whh the wave funton ould be derved. For example, one ould employ the operator method of Kohn and o-workers 9 3 or of the aelerated frame of referene approah due to Vgnale. 8 Here we provde a frst prnples dervaton va the nteraton representaton that dffers from these methods. Further, n the Appendx we onsder the Vgnale approah, and prove the valdty of the onneton between the wave funton derved and the Hamltonan n the aelerated frame of referene. Fnally, we provde examples of the applaton of the GKT wave funton. II. SEPARATION OF HAMILTONIAN The key to the dervaton of the GKT wave funton, and of ourse to the understandng of the optal absorpton frequenes, s that the Hamltonan of Eq. ) s separable nto ts enter of mass and relatve oordnate omponents. The enter of mass omponent s that of a sngle partle of mass Nm and harge Ne onfned harmonally by N tmes the potental of a sngle partle. The effet of the TD eletr feld also appears only n ths omponent. The eletron nteraton potental term appears only n the relatve oordnate omponent of the Hamltonan. Furthermore, the n-plane enter of mass moton omponent of the Hamltonan s separable from the moton n the plane perpendular to t whh s the assumed dreton of the magnet feld. To see ths, defne the enter of mass and relatve oordnates and momentum,, as R R ) = N r ; ) = ˆπ ; ˆπ = + e Ar ), )

4 438-3 Zhu et al. J. Chem. Phys. 4, 438 4) and X ) = x x, X 3) = x + x x 3,..., 3) X N) = x + x x N N )x N, and smlarly for Y ),...Y N), Z ),...Z N), and ),... N). The Hamltonan an then be rewrtten as H ˆ = m + rel, 4) the enter of mass Hamltonan m t)s m t) = ˆ M + M ω x X + ωy Y + ωz Z) NeEt) R, 5) wth ˆ = R + Ne AR), 6) and M = Nm. rel s the Hamltonan of the relatve oordnates and ontans the effets of the nteraton. It an be readly shown that [ m, rel ] = so that the enter-of-mass moton and the relatve moton are separable. Therefore, the egenstates of the Hamltonan are the produt of the egenstates of the enter-of-mass moton R, t) and the relatve moton ϕ rel R ),...,R N) ): r, r,...r N,t) = R,t)ϕ rel R ),...,R N) ). 7) The relatve moton wave funton ϕ rel R ),..., R N) ) satsfes rel ϕ rel R ),...,R N) ) = E rel ϕ rel R ),...,R N) ), 8) E rel s the orrespondng egenvalue. We next fous on the m t) sne the effet of the external eletral feld appears only n the enter-of-mass moton. For smplty, hoosng the dreton of the magnet feld as the z dreton, then n the symmetr gauge AR) = BY, BX, ), and Eq. 5) an be deomposed nto two parts m t) = XY t) + Z t), 9) XY t) desrbes the n-plane moton, and Z t) = P M ˆ Z + Mω z Z + NeE z t)z, ) desrbes a trval moton n the Z-dreton wthout the nfluene of the magnet feld. Ths moton s ndependent of the n-plane moton, and as suh the enter-of-mass wave funton R, t) s just the produt of the n-plane moton wave funton and the Z-dreton wave funton. The latter s smply the one-dmensonal HPT wave funton. For the expresson of the n-plane relatve oordnate Hamltonan, see the supplementary materal. III. IN-PLANE CENTER-OF-MASS MOTION We next fous on the n-plane moton desrbed by XY t), whh may be further separated as XY t) = + t), ) the tme-ndependent part s = ˆ X M + ˆ ) Y + M ω x X + ω y Y ) = P ˆ X M + Pˆ Y ) + M ω X + ω Y ) + ω ˆL Z, ) wth ω = ω x + ω 4, ω = ω y + ω 4,ω = NeB = eb M m s the ylotron frequeny, and ˆL Z = XPˆ Y Y Pˆ X the angular momentum omponent along the Z axs. The tme dependent part s t) = Ne[E x t)x + E y t)y ]. 3) In the nteraton representaton the evoluton of state at tme t denoted by t s obtaned as t =exp ī t T exp ī t du nt u), 4) T s the tme-orderng operator, and nt t)=e t/ t)e t/ =Ne[E x t)xt) + E y t)y t)], 5) wth Xt) = e t/ Xe t/ wth = a t)x + a t)y + a 3 t) ˆ P X + a 4 t) ˆ P Y, 6) a t) = γ os λ t + γ os λ t, 7) ) sn λ t sn λ t a t) = γ 3 + γ 4 ), 8) λ λ ) sn λ t sn λ t a 3 t) = γ 5 + γ 6 ), 9) λ λ a 4 t) = γ 7 os λ t + γ 8 os λ t, 3) the oeffents ω γ = ω + ) ω, γ = ω + ), 3) γ 3 = ω 3ω + ω ), γ 4 = ω 3ω ω ), 3) γ 5 = M ω + ω ) ω, γ 6 = M + ω ω + ) ω, 33) Smlarly, γ 7 = ω Y t) = e ˆ H t/ Ye ˆ H t/ M, γ 8 = ω M. 34) = b t)x + b t)y + b 3 t) ˆ P x + b 4 t) ˆ P y, 35)

5 438-4 Zhu et al. J. Chem. Phys. 4, 438 4) wth ) sn λ t sn λ t b t) = η + η ), 36) λ λ b t) = η 3 os λ t + η 4 os λ t, 37) b 3 t) = η 5 os λ t + η 6 os λ t, 38) ) sn λ t sn λ t b 4 t) = η 7 + η 8 ), 39) λ λ the oeffents η = ω 3ω ω + η = ω + 3ω + ω ω η 3 = ω + ), η 4 = η 5 = ), ), 4) ω ω + ), 4) ω M, η 6 = ω M, 4) η 7 = M + ω ω ) ω, η 8 = M ω + ω + ) ω. 43) Note that n above equatons = ω ω ) + ω ω + ω ), 44) λ, = ω ω + ω + ω ) ω + ω ω + ω) ). 45) The detals of how to alulate the values of the oeffents a t) and b t), =,, 3, 4 of Eqs. 6) and 35) are gven n the supplementary materal. Substtuton of Eqs. 6) and 35) nto Eq. 5) yelds nt t) = t)x + t)y + 3 t) Pˆ X + 4 t) Pˆ Y, 46) wth t) = Ne[E x t)a t) + E y t)b t)], =,, 3, 4. 47) Note that the ommutator of nt u) at dfferent tmes s a number, [ H ˆ nt u), nt v) ] = gu, v), 48) gu, v) = u) 3 v) + u) 4 v) v) 3 u) v) 4 u). 49) One then obtans T exp ī du nt u) = exp ī du nt u) exp du = exp ī du nt u) u u dv [ nt exp u), nt v ] αt), 5) αt) = du dvgu, v). 5) Combnng Eqs. 4) and 5), one obtans t =exp ī t exp ī t du nt u) exp ī αt). 5) IV. EVOLUTION OF THE EIGENSTATES OF THE IN-PLANE MOTION Sne the egenstates n of of Eq. ) form a omplete set, any state an be expanded n terms of them. We next alulate the evoluton of the egenstates of the n-plane moton,.e., wth = n, and n =En XY n we then have from Eq. 5) t =exp ī ˆ H t exp exp ī αt) n = e ī αt) e ˆ H t/ e ī Usng the denttes and ī du nt u) du ntu) e t/ e EXY n t/ n. 53) e A e B e A = e ea Be A, 54) e A Be A = B + [A, B] + [A, [A, B]]! Eq. 53) an be rewrtten as t =e ī [αt)+en XY t] Note that exp e ˆ H t/ = e ī [αt)+en XY t] exp ī + [A, [A, [A, B]]] +, 55) 3! ī du nt u) ) e t/ n du[e u t)/ u)e u t)/ ] n. 56) ˆ H u) = Ne[E x u)x + E y u)y ], 57)

6 438-5 Zhu et al. J. Chem. Phys. 4, 438 4) so that e u t)/ u)e u t)/ wth = Ne[E x u)xu t) + E y u)y u t)] = u, t)x + u, t)y + 3 u, t) Pˆ X + 4 u, t) Pˆ Y, 58) u, t) = Ne[E x u)a u t) + E y u)b u t)], =,, 3, 4. 59) Insertng Eq. 58) nto 56), we obtan t = e ī α t)x+α t) Pˆ X +β t)y +β t) Pˆ Y ) n e [EXY n t+αt)], 6) α t) = β t) = u, t)du, α t) = u, t)du, β t) = 3 u, t)du, 6) 4 u, t)du. 6) Then, the evoluton t n the oordnate representaton s X, Y t = X, Y exp ī [α t)x + α t) Pˆ X + β t)y + β t) Pˆ Y ] n exp ī [ E XY n t + αt)]. 63) Note that for operators A, B f ther ommutator [A, B] sa number, then Hene, Eq. 63) an be rewrtten as e A+B = e A e B e [A,B]. 64) X, Y t = X, Y e ī α t)x+α t) ˆ P X ) e ī β t)y +β t) Pˆ Y ) n e EXY n = exp ī [α t)x + β t)y ] t+αt)) exp α t)α t) + β t)β t)) X α t),y β t) n exp ī [ E XY n t + αt)]. 65) From the above equaton, we an mmedately see that the wave funton s shfted from the orgnal wave funton, and the phase angle s hanged. To see the physal meanng of the shft, we next nvestgate the propertes of the translaton funtons α t), β t). V. TOTAL WAVE FUNCTION AND CLASSICAL EQUATION OF MOTION The orrespondng Lagrangan for the n-plane Hamltonan of Eq. ) s L XY = MṘ R R + ω R J Ṙ ) NeE x t)x NeE y t)y, 66) ) Xt) R t) =, 67) Y t) denotes the mass enter poston vetor perpendular to the magnet feld B, R the transpose of vetor R, J = ), and = ω x ω ). From the Lagrangan one an y obtan the anonal momentum as P X = MẊ + Mω Y, P Y = MẎ Mω X, 68) and the equatons of moton Ẍ + ω Ẏ + ωx X + Ne M E xt) =, 69) Ÿ ω Ẋ + ωy Y + Ne M E yt) =. After some algebra manpulatons, one an verfy that the poston vetor, ) ) α t) Xm t) = R,m t), 7) β t) Y m t) satsfes the equaton of moton of Eq. 69) wth the ntal ondton α ) =, β ) =. Moreover, α t) = P Xm,β t) = P Ym, 7) the orrespondng anonal momentum for R, m. The lassal aton between the ponts, R, ) and t, R, t ) wthout the external eletr feld term s S XY [R t)] = M duṙ R R + ω L R J Ṙ ) Thus, = M R,t R,t R, Ṙ, + Ne S XY [R,m ] = M R,m Ṙ,m + Ne We next show αt) = Ne du R u) Eu). 7) = [α t)α t) + β t)β t)] + Ne du R,m u) Eu) du[e x u)x m u) + E y u)y m u)]. 73) du[e x u)x m u) + E y u)y m u)]. 74)

7 438-6 Zhu et al. J. Chem. Phys. 4, 438 4) Usng Eqs. 59), 6), and 6), Eq.74) an be rewrtten as u αt) = Ne) du E x u) dv[e x v)a 3 v u) + E y v)b 3 v u)] u +E y u) dv[e x v)a 4 v u)+e y v)b 4 v u)]. 75) On the other hand, from the defnton of αt) Eq.5), ombnng Eqs. 47) and 49), we only need to show αt) = Ne) t u du E x u) dv[a u) 3 v) +a u) 4 v) a 3 u) v) a 4 u) v)] + E y u) u dv[b u) 3 v) + b u) 4 v) b 3 u) v) b 4 u) v)]. 76) Usng the followng denttes, [a u)a 3 v) + a u)a 4 v) a 3 u)a v) a 4 u)a v)] = a 3 v u), 77) [a u)b 3 v) + a u)b 4 v) a 3 u)b v) a 4 u)b v)] = b 3 v u), 78) [b u)a 3 v) + b u)a 4 v) b 3 u)a v) b 4 u)a v)] = a 4 v u), 79) [b u)b 3 v) + b u)b 4 v) b 3 u)b v) b 4 u)b v)] = b 4 v u), 8) t s not dffult to verfy that Eq. 76) s true. Fnally, we have the wave funton of the n-plane moton to be X, Y t = exp [P X m X + P Ym Y ] X X m t),y Y m t) n exp ī En XY [R,m ]). 8) Hene, the total wave funton nludng the n-plane, Z-omponent and relatve motons s then gven by Eq. 5). VI. CONCLUDING REMARKS In onluson, by employng the nteraton representaton we have derved the TD wave funton for a system of harmonally onfned eletrons n the presene of a magnetostat feld and a spatally unform TD eletr feld the GKT wave funton. In the dervaton, no ansatz s assumed for ts struture, the expresson arses as a onsequene of the dervaton. In the absene of the harmon onfnng potental, the wave funton derved redues to that for the ase of the unform eletron gas the KT wave funton. Wthout the presene of the magnet feld, the wave funton s the HPT wave funton. We have also rgorously establshed the valdty of the onneton between the GKT wave funton derved and the eletrons n an aelerated frame of referene. As the KT and HPT wave funtons are speal ases, ths onneton s equally vald for the physal stuatons represented by these wave funtons. As noted n the ntrodutory remarks, wth a knowledge of the soluton to the unperturbed system, the tme evoluton of all the propertes of the system n the presene of the TD eletr feld s then exatly known va the GKT wave funton. We provde here two examples of ts applaton. In reent work, 3 t has been shown that losed form analytal solutons for the tme-ndependent ground and exted states of the two-eletron quantum dot an be derved. A typal ground state soluton 4 n a.u. e = = m = ) for the ase ω + ω L =, ω s the harmon frequeny and ω L = B/ the Larmor frequeny, s r r ) = C + r )exp r + r ), 8) r = r r and C = /π 3 + π). The ground state energy s E = 3 a.u. The orrespondng GKT wave funton s thus known, and all the propertes of the quantum dot n the presene of the TD eletr feld an be studed wthn the ontext of say the quantal Newtonan seond law 7 perspetve of Shrödnger theory and quantal densty funtonal theory. 7 Employng the unperturbed wave funton varous propertes of the quantum dot have been determned. 4 For example, the ground state densty s gven by the expresson ρr) = π3 + π) e r πe r [ + r )I r ) ) ] + r I r + + r ), 83) I x) and I x) are the zeroth- and frst-order modfed Bessel funtons. 5 The tme evoluton of the densty ρr, t)n the presene of the TD perturbaton s thus gven by the above expresson translated by the soluton of the orrespondng lassal equaton of moton. For ths ground state the physal urrent densty jr) = ρr)ar)/. The expresson for the tme evoluton of ths property jrt) = ρr, t)ar)/ s thus also known n losed analytal form. Yet another example s the three-eletron quantum dot 3 whose tme-ndependent soluton n the Wgner hgh eletron orrelaton regme s known. The orrespondng evoluton of the propertes of ths system s thus known va ts GKT wave funton. ACKNOWLEDGMENTS We thank Dr. J. J. Zhu for helpful dsussons. The work of X.P. was supported by the Natonal Natural Sene Foun-

8 438-7 Zhu et al. J. Chem. Phys. 4, 438 4) daton of Chna Grant No. 75) and the K. C. Wong Magna Foundaton of Nngbo Unversty. The work of V.S. was supported n part by the Researh Foundaton of CUNY. The ontrbutons of H.Z. were made whle vstng Nngbo Unversty. APPENDIX: RELATIONSHIP OF WAVE FUNCTION TO ACCELERATED FRAME OF REFERENCE By onsderng the system of harmonally onfned eletrons n the presene of a magnetostat feld and a TD eletr feld to be n an aelerated referene frame, Vgnale 8 arrved at the result that the densty ρr, t) was translated by a TD value that satsfed the orrespondng lassal equaton of moton. In ths appendx we prove that the GKT wave funton derved n our work satsfes the TD Shrödnger equaton n the aelerated frame of referene. Ths proves the valdty of the onneton between the GKT wave funton and the aelerated referene frame. For smplty we assume the external magnetostat feld to be along the z axs,.e., B =,, B), and onsder only the n-plane moton. The orgnal frame of referene see Se. III), the Hamltonan s ˆ H t) = m ˆp + e Ar ) ) + ur r j ) + ˆV r,t),,j) A) ur r j ) s the nteraton potental between the partles, and V h r ) + eet) A) ˆV r,t) = wth the harmon potental V h r) = r K r, K = m beng the sprng onstant matrx, r = x y ) the n-plane poston vetor, and r = x,y) ts transpose. The TD eletr feld s Et) and the magnet feld B desrbed by the vetor potental A = B r = B y,x,). The GKT wave funton ψt) satsfes the Shrödnger equaton ) t ψt) =. A3) In the absene of the external eletr feld, the Hamltonan redues to E= = [ ˆp + e ) m Ar ) + m ω x x + ω y y ) ] +,j) ur r j ). A4) Its egenstates n r, r r N ) an be obtaned by solvng E= E n ) n r, r r N ) = A5) wth E n the orrespondng egenvalues. The Hamltonan an be deomposed nto ts enter of mass and relatve moton omponents H ˆ = m + rel, wth m beng the same as of Eq. ). The expresson of rel s gven n the supplementary materal. j r j The lassal equaton of moton of the aelerated frame whose poston relatve to the orgnal referene frame s xt), s mat) = eet) K xt) e vt) B, A6) the veloty vt) = ẋt) the frst dervatve of xt) wth respet to tme, and the aeleraton at) = ẍt) s ts seond dervatve. We parameterse the tme and poston vetor n the movng frame by t and r so that r = r + xt); t = t. A7) Hene n the movng frame, the momentum operator s ˆp = r and the tme dervatve = r = ˆp, = t t + vt). r The Hamltonan n the movng frame 8 s ˆ H p, r,t ) = +,j ˆp m + e Ar ) ) ur r j ) + V r,t ), V r,t) = V r + xt),t) + mat) + e vt) B) r + gt), r A8) A9) A) A) gt) = N mvt) + e [B xt)] vt)). The wave funton ψ t ) for the Hamltonan H ˆ p, r ) satsfes the Shrödnger equaton n the movng frame: ) H ˆ t p, r,t ) ψ t ) =. A) The wave funton an be expressed n terms of a untary transformaton of the GKT wave funton: ψ t ) = ψ t) =Ût) ψt), A3) the untary operator s Ût) = [ ] [ ] û t); û t) = exp ī r dt) exp ˆp xt), A4) wth dt) = mvt) e B xt) = P xt) N, A5) and P xt) s the anonal momentum orrespondng to xt). Employng the dentty eâe ˆB = eâ+ ˆB+ [Â, ˆB], A6)

9 438-8 Zhu et al. J. Chem. Phys. 4, 438 4) whh s vald when [Â, [Â, ˆB]] = [ ˆB,[Â, ˆB]] =, we an rewrte Eq. A4) as [ ] [ ] û t) = exp ˆp xt) exp ī r dt) [ ] exp xt) dt), A7) so that [ ] [ ] û t) = exp ī dt) xt) exp dt) r [ ] exp ī ˆp xt). A8) Therefore, the nverse of the untary operator Ût)s [ ] [ ] Û t) = exp ī Ndt) xt) exp dt) NR [ ] exp ī xt) ˆP, A9) R = r j, ˆP = ˆp j. A) N j j It s easy to verfy the ommutaton relaton [ ˆP, R] =. A) Makng use of Eqs. A3), A5), and A9), we obtan [ ] ψt) =exp R xt)) Pxt) ψ r xt), r xt),, r N xt)) [ ] = exp R Pxt) ψ r, r,, r N ), A) wth R = X Y ) = N j r j = R xt). By omparson of A) wth the fnal wave funton derved n the text Eq. 5), wehave [ ] ψ r, r,, r N ) =exp xt) Pxt) [ ] [ ] exp ī E n t exp ī S [xt)] n r xt), r xt),, r N xt)). A3) Note that sne we only onsder the n-plane moton, as defned by Eq. 8) n the text, S [xt)] = S XY [xt)] = M dt[ẋ t) xt) xt) + ω xt)j ẋt)]. A4) We next show that ψ t) of Eq. A3) satsfes the Shrödnger equaton A) n the aelerated frame. Insertng Eq. A3) nto A), wehave ψ t = t + vt) r ) e ī P xt) xt) e ī E n t e ī S [xt)] e ī xt) ˆP n r,, r N ) = [P xt) vt) + Ṗ xt) xt)] + S [xt)] + e ī xt) ˆP E= e ī xt) ˆP ψ = H ˆ p, r,t) ψ. A5) Therefore, one now needs to show that H ˆ p, r ) = P xt) vt) Ṗ xt) xt) + S [xt)] + e ī xt) ˆP E= e ī xt) ˆP. A6) It s readly seen that P xt) vt) + Ṗ xt) xt) = M[ẋ t) + at) xt)], A7) whle the last term on the rhs of Eq. A6) an be alulated by usng the ommutator Eq. 55) to be e ī xt) ˆP E= e ī xt) ˆP = E= xt) M R ω ) J ˆP + M xt) xt), A8) = ω ω ). Insertng Eqs. A4), A7), and A8) nto A6), one arrves at ˆ H ˆ H E= = M vt) Mat) xt) + M [ ω ] 4 x t) + ω xt)j ẋt) xt) M R ω ) J ˆP. A9) On the other hand, from Eqs. A4) and A), one an show that H ˆ E= = ω ˆL Z ˆL Z ) + Mω 8 R R ) + mat) + e ) v B NR + gt) ˆL Z + eet) NR, A3) = X Pˆ Y Y Pˆ X. Note that ˆL Z ˆL Z = xt) J ˆP, A3) and Ne B xt) vt) = Mω xt) J vt). A3) These equatons together wth Eq. A6) then prove that Eqs. A9) and A3) are equvalent. Thus we prove the valdty of the onneton between the GKT wave funton and the dea of obtanng t va an aelerated frame of referene.

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