Nonlinearity versus Perturbation Theory in Quantum Mechanics

Transcription

1 Nonlneary versus Perurbaon Theory n Quanum Mechancs he parcle-parcle Coulomb neracon Glber Rensch Scence Insue, Unversy o Iceland, Dunhaga 3, IS-107 Reykjavk, Iceland

2 There are bascally wo "smple" (.e. wh classcal elds: =/= QED) ways o quanum-mechancally descrbng Coulomb neracon beween, say, wo elecrons conned n he d parabolc poenal V( x, y) = V( r) = Mω r / wh r²=x²+y² nsde he vercal slab o hegh (quanum-do helum). H z Ψ( x ') ) eher he Coulomb soluon Φ ( x) = e d x' o he classcal derenal Posson x x' equaon Φ= 4 π( e / H z ) Ψ ncludng he meaneld source erm Ψ s explcly presen n he sngle-parcle Schrödnger equaon or Ψ n addon o V(r) :

3 () r V() r e d x' () r E () r M x x' h Ψ( x ') Ψ + + Ψ = Ψ We consder he saonary case or he sake o smplcy; e.g. wo oppose-spn elecrons: c. Paul. Common d orbal waveuncon Ψ («orbal bosons»). Nonlnear egenvalue E : chemcal poenal. Then he resulng negroderenal equaon s mplc. Hence erave perurbaon seres abou orhogonal base elemens o ge an approxmae convergng denon o he Schrödnger waveuncon Ψ &/or search o he varaonal mnmum o he correspondng energy unconal.

4 Φ= 4 π( e / h z ) ) or he Posson equaon s (a leas numercally) ormally coupled o he Schrödnger equaon h M Ψ + +Φ Ψ = Ψ [ ] Ψ (,) r V() r (,) r (,) r E () r. Ths nonlnear Schrödnger-Posson (SP) derenal sysem yelds (hough only numercally...) he explc mean-eld soluon Ψ whou any approxmaon. Moreover also yelds exced saes (=/= case () ). Non-orhogonaly o he nonlnear egensaes, say a) & b), yelds quanum coherence => nererence beween he egensaes whou superposon!

5 (a b)² versus N ω e.g. reduce : «open he bowl» e / Hz N = a) = b) hω 1 * = Ψ a( ) Ψb( ) N 0 ( ) Nonlneary N : a b r r π rdr noe: π α=0.684, α=1/ In S P we had vs d d π α or S P here. d 3d π α Dashed: N = sn ; 6 ( a b)

6 Consder he saonary SP sysem wh wo opposespn elecrons => «orbal bosons» (c. Paul). In approprae dmensonless uns (d radal symmery wh angular momenum m), becomes ( E% = E/ hω): 1 m u" + u' + E% Φ % u 0, 4 = u Φ% "+ Φ % h ' + u = 0 ; = r/ ; Mω e / H Ψ = 1 = z =. h π rdr u d N ω 0 0 echncally π e = Ψ Mω

7 We wan egensae nonorhogonaly. Thereore: ma = mb (selecon rule). Indeed nner produc due o polar angle ϕ yelds (radal symmery): π ( mb ma) m m e ϕ dϕ = a b 0 ( a b) = 0. 0 n = 0; m = 0 a) : ground egensae a a (connuous) b) : 1rs exced egensae nb = 1; mb = 0 (dashed) man quanum numbers: number o nodes. n ab, u N Ψ : N 0 u<< 1 : quaslnear! % % % % c. Posson: Φ "+ Φ + u Φ "+ Φ = 0 u<< 1 ; E E E E E

8 ( ) Theorem (J. Bec, OCA 010) Wab a b = where W( ) =Φa( ) Φb( ) and E E W ab a = a W b ; poenal s resp. dened by ( ) b Φ ab, hrough Posson equaon. Consder he Schrödnger equaon or a) : 1 ua" + ua' + E% a Φ% a ua = 0 ; u" + u' + Ea a W u u" u' Ea b u 0 ; % Φ % + = + + Φ = 4 % % 4 add «perurbaon» W : u ab, «Perurbaon» W exchanges he respecve Coulomb poenals. Thereore u s no more egensae!

9 >9 => W() ~ 1/ (Coulomb).

10 ( a b ) s an nvaran u = Kˆ (, ) u ; u = a) ; u b) a pror! W ( b u) pb : when W =Φ Φ? a b h u = H% + W% u Kˆ = H + W Kˆ, h % % W 0 W W 0 W Kˆ ( ; ) = Kˆ ( ; ) h d Kˆ ( ; ) W% Kˆ ( ; ). Lowes-order n W: W W 0 W 0 W Kˆ W Kˆ W 0 n he negral: W( ; ) = W 0( ; ) 0( ; ) % 0( ; ) W W + ( ) + ( 4 ) +... Kˆ Kˆ d Kˆ W Kˆ o W o W exac bu mplc!

11 Kˆ ( ; ) e ; Kˆ ( ; ) e ; E% a( ) E% a( ) W 0 = W 0 = Kˆ ( ; ) = e W 0 E% ( ) ( b u) ( ˆ ) ( ˆ ) = b KW u = b KW a b W W 0 W 0 W 0 4 ( ) ( ) Remnd: Kˆ ( ; ) = Kˆ ( ; ) d Kˆ ( ; ) W% Kˆ ( ; ) + o W% + o W%... and (, ) (, ). ( ) E% a ( ) ( ) E ( % a E % b ) E ( % b E % a) % % ab ( ) b u = e b a W e d e + o W +... ( )( ) % = % % = τ = ( % b % a) W a b E E Theorem:. ab a b 1 Denng T & E E T : ( ) E % ( ) ( ) ( ) ( ) a T τ b u e e τ a b o a b o a b 4 = 1 sn ( e τ τ ) 1 sn 1! ( b u ) ( a b) o( a b) o( a b) ( ) oab ( ) 4 Thereore, snce W W( ): ( b u ) ( a b) o( a b) o( a b) 4 (,) = = cse + oab Golden Rule 4 =

12 We plo W % = λ Φ % Φ % bu (,) / ab ( ) ( ) ( ) : a b versus me or (C. Besse & G. Dujardn: 013 U. Llle1 & INRIA-Llle) λ = 1 Probably o ndng he -level nonlnear SP sysem n s exced sae b) s consan n me and equal o (a b) sarng n g.s. a)!

13 La mécanque quanque,c es le plus bel oul que l homme a créé pour marquer son errore enre le zéro de la maère e l nn de sa conscence Collaboraon e remercemens: Jeremy Bec, Observaore de la Côe d Azur (Nce), Vðar Guðmundsson, Scence Insue, Unversy o Iceland (Reykjavk), Gullaume Dujardn & Maxme Gazeau, Inra Llle Nord-Europe, Jean-Claude Garreau, lab «Phys. Lasers, aomes & molécules» (PhLAM), U. Llle 1, Chrsophe Besse, Insu de Mahemaques de Toulouse, U. Toulouse 3, Deer Schuch, Insu ür Theoresche Physk, U. Frankur (Germany), Jean-Mchel Ramond, Laboraore Kasler-Brossel, ENS (Pars). & everyone welcome!... Res: G. Rensch & V. Gudmundsson, Eur. Phys. J. B 84, 699 (011) G. Rensch & V. Gudmundsson, Physca D 41, 90 (01) G. Rensch, V. Gudmundsson & A. Manolescu, Physcs Leers A 378, 1566 (014) V. Gudmundsson, S. Hauksson, A. Johnsen, G. Rensch, A. Manolescu, C. Besse, & G. Dujardn, Annalen der Physk, May 014 (n press)