Motion of Wavepackets in Non-Hermitian. Quantum Mechanics

Transcription

1 Moon of Wavepaces n Non-Herman Quanum Mechancs Nmrod Moseyev Deparmen of Chemsry and Mnerva Cener for Non-lnear Physcs of Complex Sysems, Technon-Israel Insue of Technology echnon.ac..ac.l\~nmrod

2 Non-herman QM N. Moseyev, "Quanum heory of onances: calculang energes, wdhs and cross-secons by complex scalng", Physcs epors, Vol. 302, Issue 5-6, pp (1998).

3 Why do we need Non-Herman Quanum Mechancs? Smpler Descrpon of onance phenomena. Explans phenomena and expermenal uls whch canno be explaned by convenonal QM due o he complexy of he problem. Sudy of he dynamcs of laser drven sysems.

4 esonances Are localzed measable saes wh fne lfeme. In Herman Quanum Mechancs onances canno be repened by a sngle sae of he Hamlonan. The onance s depced by a large densy of saes around he onance energy. esonances are assocaed wh he complex poles of he scaerng marx.

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6 Ougong boundary condons A onance sae mples ha he parcle s localzed n he neracon regon. a onance wavefuncon has ougong boundary condons such ha he ncomng wave vanshes. The energy of he onance sae s complex E n 2 Where s he wdh of he onance sae The onance wavefuncon dverges exponenally n

7 Herman QM V( x) Non Herman QM V( x) Ae x Be x Ae x Be x E ( ) 2m 2 E 0 connuous specrum E n dscree specrum n 2 onances E 1 2 e E 1 1 E-beween onances

8 Herman QM Non Herman QM V( x) V( x) Complex scalng Ae x Be x Ae xe Be xe E ( ) 2m 2 E 0 connuous specrum E n dscree specrum n 2 onances E 1 E 2 e 1 1 e e E-beween onances connuum: e Ae o avod dvergence- xe Be xe 2 E E e

9 Complex Scalng In order o avod he dvergence of he onance wavefuncon s convenen o scale he coordnae such ha : x xe Ths can be done by he followng scalng operaor: x Sˆ e x Such ha S ˆ 0 as x The Schrodnger equaon aes he form: ( Sˆ HS ˆ ˆ 1 )( Sˆ ) E( Sˆ ) ~ H ~ E ~

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11 Egenvalues of he Non-Herman Hamlonan: The bound saes reman on he real axs. The onance energes ge complex values and are ndependen of he scalng angle hea. E n The connuum saes are roaed no he lower half of he complex plane by an angle 2 n 2 E con Ee n 2

12 Bound sae esonances E 2 Connua

13 Non Hermcy due o boundary condons ˆ ( ) 2 H lm ( x) n n n n x n n Non Hermcy wh convenonal boundary condons Complex Scalng: x ˆ ( H xe ) ( ) 2 xe n n n n ( onance ) are square negrable, le bound saes n convenonal QM The onance specrum s dscree! Im E bound onances e E oang connuum

14 New nner produc Snce he Hamlonan s no Herman a more generalzed nner produc s requred. C-produc: rgh egenfuncon: f g f g f gdr ˆ H E lef egenfuncon: In Herman QM ˆ H E ˆ ˆ H H dr all space

15 The me asymmerc problem n NH-QM A ˆ Observable are defned by: H H E e C / ) ( when ) 0 ( C E e C / ) ( when C ( 0) as ) ( Therefore he wavepace canno be propagaed from = 0 o =

16 C-PODUCT CAN NOT CAY OUT MOTION OF WAVE- PACKET CACUATIONS. ONY ESONANCE WIDTHS, PATIA WIDTHS AND COSS SECTIONS CAN BE CACUATED. ONY DYNAMICS WHICH IS CONTOED BY A SINGE ESONANCE STATE CAN BE STUDIED. THE POBABIITY DENSITY OF A ESONANCE STATE IS CONSVEED IN TIME. * E * E * () () e e 1

17 F-produc (Fne space approach ) The onance sae decays n me herefore he probably densy should decay. e e E / ( ) C e E / Tme dependen observables are calculaed by: A ( ) A ˆ

18 Wh Ido Glary & Avner Flescher : es of he new formalsm (F-produc) for calculaons of survval probables for me-ndependen problem ) (0 ) ( ) ( (0) ) ( S Herman QM 2 () () () (0) () S NH-QM E E e C e C / / ) ( ) ( () / S CCe

19 A model of cold aoms n opcal rap/ elec n QW/QD

20 Inal sae s a lnear combnaon of 2 shape ype esonances: n convenonal QM are WPs n NH-QM are egenfuncons Same rae of decays!

21 Survval probables for nal wave-pace

22 Decay of he norm of a gaussan wavepace In The F-produc formalsm: norm / ( ) ( ) C C e In Herman QM (he par of nsde he barrers): * decay ( ) ( ) dx where defnes he neracon regon. In he formalsm of nuclear physcs (e.g., oer e. al.): * 1 ' norm () () C C e (0) (0) * ( EE ) / ' ', '

23 Decay of he norm of a gaussan wavepace

24 Applcaon o cw laser drven sysems: Whn he Floque formalsm : Hˆ ( x, ) E ( x, ) QE e ( x, ) E QE / ( x, ) ( x, T ) ( x, ) Applyng CS and he F-produc Formalsm: E n' e e n * E n' e e n n' ' * n' () () E E e e n', e e n, e n' n n, n,

25 The Harmonc generaon specrum (HGS) s obaned from he Fourer ransform of he dpole acceleraon. HGS amplude (NM+M. en J. Phys. Chem. A, (2003)) (( E E ) ( n' n)) C C z * 2 ' * ' * n', ' n,, ' nn, ' [( E E' ) ( n' n)] E 2 HG 2 ( ), Hyper-aman lnes n he HGS would be obaned a: ( )/ ' ' N N n n N even or odd, depends on he pary of he and saes.

26 Pera Zdansa (2004): usng complex scaled even-empered Gaussans calculaed he specrum of non-herman Helum Hamlonan.

27 Wh Ido Glary & Pera Zdansa: The HGS of Helum were calculaed usng he new heory pened here.

28 Summary: esonances are measable saes obaned by applyng ougong boundary condons. Complex scalng of he coordnae leads o square negrable onance wavefuncons. The generalzed nner produc forbds propagaon of wavepaces n me. The new F-produc enables calculaon of me dependen observables.