Exact Propagator of a Two Dimensional. Anisotropic Harmonic Oscillator in

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1 Journal of Modern Physis, 7, 8, ISSN Online: 53-X ISSN Print: Eat Propagator of a wo Dimensional Anisotropi Harmoni Osillator in the Presene of a Magneti Field Jose M. Cerveró Department de Fsia Fundamental, Faultad de Cienias, Universidad de Salamana, Salamana, Spain How to ite this paper: Cerveró, J.M. (7) Eat Propagator of a wo Dimensional Anisotropi Harmoni Osillator in the Presene of a Magneti Field. Journal of Modern Physis, 8, Reeived: January 5, 7 Aepted: Marh 3, 7 Published: Marh 6, 7 Copyright 7 by author and Sientifi Researh Publishing In. his work is liensed under the Creative Commons Attribution International Liense (CC BY 4.). Open Aess Abstrat In this paper we solve eatly the problem of the spetrum and Feynman propagator of a harged partile submitted to both an anharmoni osillator in the plane and a onstant and homogeneous magneti field of arbitrary strength aligned with the perpendiular diretion to the plane. As we shall see in the beginning of the letter, the Hamiltonian, being a quadrati form, is easily diagonalizable and the Classial Ation an be used to onstrut the eat Feynman Propagator using the Stationary Phase Approimation. he result is useful for the treatment of quasi two dimensional samples in the field of magneti effets in nano-strutures and quantum optis. he presented solution, after minor etensions, an also be used for motion in three dimensions, and in fat it has been used for years in suh ases. Also it an be used as a good eerise of a Feynman Path Integral that an be alulated easily with just the help of the Classial Ation. Keywords Quantum Mehanis, Path Integrals and Propagators. Introdution he problem of quantum harged partiles in the presene of eletri and magneti fields has always remained on target of any theoretial physiist who wishes to predit the behavior of one or a swarm of suh partiles imbedded in the struture of a solid. he eat solution in three dimensions has eluded these attempts for years. However, when the partiles remain in a two dimensional solid, the problem of the eat solution beomes both feasible and possible. Eperimentally, the last situation has only been possible to be realized until very reently in the laboratory thanks to the tehnologial means that allow the DOI:.436/jmp Marh 6, 7

2 eperimentalists to produe quasi dimensional samples of metals and semimetals. One an indue large magneti behavior ombining them with ferromagneti D materials suh as La,7Sr,3MnO3 SriO 3 and KCuF 4. Indeed we should also mention the reent disovery of Magneti Graphene whih falls in the lass of these two dimensional samples when ombined with the aforementioned substrates. he propagator an also be used in three dimensions with some little arrangements. However, I restrit myself to the two dimensional ase given the importane of two dimensional solids nowadays. However, the stationary solution of a bounded Hamiltonian is only of aademi interest as we would like to know the evolution of the partile inside of the two dimensional solid. his goal an only be ahieved if we know the propagator of the partile(s) along the time and the properties that may or may not hange in the sample: phase transitions from metal to semimetal or to an insulator phase, the behavior of the Anderson loalization, the magneto-optis properties of the solid and the like. his wealth of physial effets an be studied if we know the Feynman propagator of the system and the help of omputing algebrai alulus. he aim of this paper is to show that these results an be attained with little effort and a touh of elegane. he problem of the eat solution of the Feynman propagator of the system simplifies a great deal with the help of the Stationary Phase Approimation that turns to be eat in the ase of a quadrati Hamiltonians. In this paper, we use the most general two-dimensional Hamiltonian with magneti effets provided by a magneti field pointing in the perpendiular diretion of the plane. he rest of the interation is represented by anisotropi harmoni osillators lying in the plane along perpendiular diretions. I would like to finish this Introdution with a set of Referenes. he Feynman Path Integrals [] are well known from a historial perspetive [] to an enylopedi aount with many eamples [3]. Also this Quantum Mehanial problem has been partially treated many years ago in Referenes [4] and [5].. Quantum Mehanial Hamiltonian he Hamiltonian desribed in the Introdution has the form: ˆ e ˆ ˆ e ˆ ˆ H p ˆ ˆ A py Ay ω ω y m m m m where ω and ω are the first order frequenies on eah diretion of the plane. he set of variables are well known in eletromagneti theory. One an selet the following gauge for the magneti vetor that produes a onstant magneti field of arbitrary strength and perpendiular to the plane: ω ω A By ; Ay B ; A3 () ω ω ω ω and the Hamiltonian reads: ˆ ˆ pˆ p y ω ω ω H mω ˆ ˆ ˆˆ ˆˆ mω y ω yp ωp m m ( ω ω) ( ω ω) ω ω ( y) () (3) 5

3 with eb ω (4) m Obviously B is the strength of the onstant magneti field. Let us now all: γ ω ( ω ω ) and we sale the position and momentum operators in a sale free way in the form: (5) mγω ˆ mγω X ˆ, Yˆ yˆ ħ ħ (6) ˆ ˆ ˆ γω, y γω y Pˆ ħm p P ħ m p (7) Again, with this hange of variables the Hamiltonian takes the form: ω ω ω H ( P X ) ( Py Y ) YP XPy ( ω ω) ω ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ħωγ ω ω that looks as a quadrati form that we an easily diagonalize: λ λ ( λ ) b λ b λ where b and are onstants: ( ) ωb ω ω ω ; ; γ ( ) ω ( ω ω) ω b ω ω ω (8) (9) he non vanishing real roots of the quarti equation take the form: ± ± 4ω ωω ( ω ω) ( ω ω) ω ω ( ω ω) ( ω ) ω ω ω ω ± ω ω ( ω ω) and multiplying the Hamiltonian by { } eigenvalues: ħ ωγ (), one finally obtains the following 5

4 E ω ω ω ω ω ω 4 4 ħ ħ ± ± Ω ±Ω or in a more tratable manner: { } ħ n, n Ω Ω n Ω Ω n 4 ħ { Ω ( n n ) Ω( nn) } 4 If one diagonalizes the isotropi harmoni osillator in both artesian and angular oordinates the Landau eigenlevels are labeled respetively by the, n, m. Both eigenlevels take the form: quantum numbers ( n n ) and ( r ) ( n n ) ( n m ) where m ( n n ) r () () (3) herefore the Hamiltonian ehibits a omplete breaking of the magneti degeneray. his effet has nothing to do with the spin-orbit oupling as the spin degrees of freedom are absent from the Hamiltonian. It is in turn losely related to the Landau orbital oupling interating with the magneti field. he non trivial question of writing down the wave funtions has been largely disussed in [6] and [7]. he interested reader is invited to address himself to these referenes. 3. Lagrangian Formalism and Constrution of the Propagator he Lagrangian of the system has the obvious form: e ( i, vi) m{ y } { A ya y} mω mωy (4) and now selet a different gauge, but representing also a onstant magneti field in the perpendiular diretion to the plane: B B A y; Ay ; A3 (5) he hange of gauge auses no problems in the final solution of the propagator as it was pointed out many years ago by Bialyniki-Birula [8]. he propagator and quantum amplitudes in general are invariant under a gauge transformation. In our ase: ω ω A By ; Ay B y ω ω Λ ω ω Λ (6) It is easy to find out the atual fom of Λ ( yz,, ) z. One finds: (, y) B ( ω ω) ( ω ω ) y, obviously independent of Λ (7) whih vanishes for the isotropi limit. For a throughout and reent disussion on gauge independene of the propagator see referene [9]. 53

5 he Lagrangian looks now as: { } m (, y,, y ) y ( ω ω y ) ω ( y y ) (8) and the Equations of Motion are: ω ω y ; y ω y ω (9) where where ω and ω are the first order frequenies on eah diretion of the plane due to the potential of the net. he real solutions of these Equations are always harmoni funtions that we an write of the form: where: t Aos Ω ) t Bsin Ω) t Cos Ω ) t Dsin Ω) t y( t) Λ Asin Ω ) tbos Ω) t Λ Csin Ω ) tdos Ω) t ω Ω ) ω Ω ) Ω Ω 4ω Ω Ω 4ω ; () () Λ Λ () We shall name: t ( ), y( t ) y and t ( ), y( t ) y to the points at time equal zero, those points at an arbitrary evolution time. hus, one an see trivially that the following relationships hold: A C; y Λ B Λ D (3) Aos Ω ) Bsin Ω) Cos Ω ) Dsin Ω) y Λ Asin Ω ) Bos Ω) Λ Csin Ω ) Dos Ω) (4) (5) Net we shall be onerned with the evaluation of the lassial ation of the anisotropi net in the presene of a uniform and onstant magneti field. he various steps of the alulation are listed in Setion 4 at the end of the paper. For the moment we shall be onerned formally with the ation derived from this AMN Lagrangian S l having the form: S AMN l t ( ), y Lyy,,, dt t m t dt y y y y t { ( ω ω ) ω ( ) } where we have used (6). Net we integrate by parts the terms ẋ and leading easily to: (6) ẏ 54

6 { } AMN m t m t S l ( ), y ( ) { yy } dt { ω ωy} { y ω y ω} y t (7) t he integral of the right hand side is trivially zero as it ontains only the equations of motion (7). his property has been used in other ontets in referene [] for the general ase of kineti energy plus an arbitrary potential depending just upon the oordinates. One finally founds: AMN m S l, y { yy yy } (8) and substituting the values of the oordinates and their derivatives one finally obtains the following epression for the ation S AMN l, y : AMN m Sl, y { a ( ) a ( y y ) b D( ) (9) b yy 4 y y y y } he Amplitude of the Propagator is now alulated with the help of the VanVlek-DeWitt-Morette Determinant. As it is well known the form of the Determinant is: d d AMN AMN Sl A ( ) DetM Det (3) πiħ πiħ i yj where d is the number of spatial dimensions of the problem. In our ase d. Furthermore the form of the Determinant is: AMN AMN Sl Sl y M. AMN AMN S y S y y he result leads to: A AMN ( ) l l DetM πiħ m b b 4 πiħd m ( ωω )Ω ) πiħ D Substituting D ( ) we obtain (with t ): AMN mωω ωω A ( t) π t t (3) iħ Ω Ω ( ω ω) Ω sin ( ωω) Ω sin and the propagator takes the final form: AMN AMN i AMN G, y,, y, t A t ep S l [, y,, y, t] (33) ħ where AMN m Sl [, y,, y, t] ( t) { a ( t)( ) a ( t) ( y y ) b ( t) D (34) b t yy t y y t y y } (3) 55

7 he epressions for a( t), a( t), b( t), b( t), ( t) and ( t) an be found t as in the D ( ) in Setion 4 but now we have substituted everywhere epression. As a final word one an say that these seemingly tedious alulations simplify greatly when one uses any of the available pakages of algebrai alulations suh as MAHEMAICA or MAPLE. Also limits in the ase of vanishing magneti fields or isotropi harmoni osillator (i.e. ω ω) have been disussed in []. One of the main goals of this work has been preisely to put at work these software appliations to the non trivial field of path integrals and its use in material siene and quantum optis: for instane propagating a gaussian wave paket in two dimensional magneti solids when the gaussian represents harged partiles or intense femtoseond light pulses. A partiular ase is the one of a Shrödinger at propagating in the net. he wave funtion an be simulated as two widely separated gaussians harged with -e-harge as in a Cooper pair []. ( 8πσ a ) 8 e ( a ) ( a ) Ψ ep ep 4 σ 4σ 4σ (35) where a ħ me in atomi units ontaining the eletri harge and σ is the variane of eah paket. Although these projets for Master hesis or Graduate Courses an be onsidered as a little umbersome, they might attrat the interest of the young physiists to the ondensed matter field in whih muh of the researh is being done nowadays. 4. he Main Result and Calulations with Useful Definitions of Interest o obtain { ABC,,, D } in Equations (3)-(5) as funtions of {,,, } have used for the alulation of the inverse 4 4 MAHEMAICA pakage: y y. I matri with the help of the A u u u3 u4 B u u u3 u4 y C u3 u3 u33 u 34 D u u u u y (36) where the matri elements and the determinant of (4) are listed below: a Λ Λ os Ω ) os Ω) Λ sin Λ sin Ω Ω Ω Ω a Λ os Ω ) sin Ω) Λ sin Ω ) os Ω) 56

8 a3 ΛΛ sin ) sin Ω Ω) a4 Λ sin ) Ω Λ sin Ω) a Λ Λ sin Ω ) os Ω) Λ os sin Ω Ω Ω Ω a Λ sin Ω ) sin Ω) Λ os Ω ) os Ω ) Λ 3 a Λ Λ sin Ω Ω Λ sin Ω) a4 Λ sin ) sin Ω Ω) a3 Λ Λ os Ω ) os Ω) sin sin Λ Ω Ω Ω Ω Λ a3 Λ os Ω ) sin Ω) Λ sin Ω ) os Ω) a33 ΛΛ sin ) sin Ω Ω) a34 Λ sin ) Ω Λ sin Ω) a4 Λ Λ sin Ω ) os Ω) Λ os sin Ω Ω Ω Ω a4 Λ sin Ω ) sin Ω) Λ os Ω ) os Ω ) Λ 43 a Λ Λ sin Ω Ω Λ sin Ω) a44 Λ sin ) sin Ω Ω) and the determinant takes the form: ( ΛΛ) sin Ω ) sin Ω) ΛΛ ( os Ω) (37) 57

9 Likewise one an easily alulate {, y,, y } Ω ) B Ω ) D (38) Λ A Λ C (39) y Ω Ω Ω Ω Ω ) Asin Ω ) Bos Ω) Ω ) Csin Ω ) Dos Ω) (4) y Ω ) Λ Aos Ω ) Bsin Ω) Ω ) Λ Cos Ω ) Dsin Ω) (4) he derivatives then take the form: a b y f y D { } ( t) b a y f y D { } ( t) y a y b y f D { } ( t) y b y a y f D { } ( t) (4) (43) (44) (45) where: ω a { Ω ( ω ω) sin Ω Ω ( ωω) sin Ω} (46) ω a { Ω ( ω ω) sin Ω Ω ( ωω) sin Ω} (47) Ω Ω Ω Ω b ( ) ω Ω ( ω ω) sin os Ω ( ωω) os sin (48) b Ω Ω Ω Ω sin os os sin (49) ( ) ω Ω ( ω ω ) Ω ( ω ω ) Ω Ω ( ) ωωω sin sin (5) Ω Ω Ω Ω ( ) ω( ω ω)( ωω ) sin sin (5) Ω Ω Ω Ω Ω Ω f ( ) ω ω( ω ω) sin ω( ω ω) sin (5) Ω Ω Ω Ω Ω Ω f ( ) ω ω( ωω) sin ω( ω ω) sin (53) Ω Ω 58

10 D Ω Ω Ω Ω ω ω sin ωω sin Ω Ω (54) Notie the interesting relationship: f f (55) he following onstants have been defined in the body of the Paper as: ; ω ω ω ω ω ω Ω Ω (56) Ω Ω Ω ; Ω Ω Ω (57) ω Ω ) ω Ω ) Ω Ω 4ω Ω Ω 4ω ; Λ Λ (58) With these definitions the following epressions hold: 5. Conlusion ( ω ω ) ( ω ω ) Ω Ω Λ Λ ; Λ Λ (59) ωω ωω Λ Ω Λ Ω ω ω ω ω ω (6) { } ω ΩΩ Λ Ω Λ Ω (6) ω Λ Ω Λ Ω ω ω ω ω ω (6) { } ω ω ΩΩ ΛΩ Λ Ω (63) ω ω We have presented and fully solved the propagator of the anisotropi two dimensional harmoni osillator in the presene of a onstant magneti field in the perpendiular diretion of the plane. he solution is based on the Stationary Phase Approimation that is eat in this ase as the Hamiltonian ontains terms no more than quadrati terms. We believe this solution is most interesting today when a large amount of plane samples as grapheme with large mobility (large number of eletrons per unit of area) are submitted to various both strong and onstant magneti fields. Aknowledgements his researh has been arried out with the partial support of MINECO under the projet MA C--R and JCyL Projet Number SA6U3. Referenes [] Feynman, R.P. and Hibbs, A.R. () Quantum Mehanis and Path Integrals. Emended Edition, Dover Books in Physis, MGrawHill, USA. [] Brown, L.M. (5) Feynman s hesis. World Sientifi Pub., Singapore. [3] Kleinert, H. (6) Path Integrals in Quantum Mehanis, Statistis, Polymer Phys- 59

11 is, and Finanial Markets. World Sientifi Pub., Singapore. [4] Kokiantonis, N. and Castrigiano, D.L.P. (985) Journal of Physis A: Mathematial and General, 8, [5] Davies, I.M. (985) Journal of Physis A: Mathematial and General, 8, [6] Sebawe Abdalla, M. (99) Nuovo Cimento, 5B, 9-9. [7] Dippel, O., Shmelher, P. and Cederbaum, L.S. (994) Physial Review A, 49, [8] Bialyniki-Birula, I. (967) Physial Review, 55, [9] Bandrauk, A.D., Fillion-Gordeau, F. and Lorin, E. (3) Journal of Physis B: Atomi, Moleular and Optial Physis, 46, Artile ID: [] Dittrih, W. and Reuter, M. (996) Classial and Quantum Dynamis. Springer, New York. [] Urrutia, L.F. and Manterola, C. (986) International Journal of heoretial Physis, 5, [] O Connell, R.F. and Zuo, J. (3) arxiv:quant-ph/3 v. Submit or reommend net manusript to SCIRP and we will provide best servie for you: Aepting pre-submission inquiries through , Faebook, LinkedIn, witter, et. A wide seletion of journals (inlusive of 9 subjets, more than journals) Providing 4-hour high-quality servie User-friendly online submission system Fair and swift peer-review system Effiient typesetting and proofreading proedure Display of the result of downloads and visits, as well as the number of ited artiles Maimum dissemination of your researh work Submit your manusript at: Or ontat jmp@sirp.org 5