[ ] matrice densità. ( t) ( t) ( t) ( )
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1 marce desà La marce desà è ua rappreseazoe alera4va dello sao d u ssema qua4s4co per cu abbamo precedeemee u4lzzao la fuzoe d'oda. Ache se descrvere u ssema qua4s4co co la marce desà equvale a u4lzzare la fuzoe d'oda, l suo u4lzzo compora oevol vaagg pra4c ella descrzoe d alcu problem dpede4 dal empo - par4colare l rlassameo e la spe@roscopa o leare fase codesaa. s defsce formalmee come l prodo@o esero ra ua fuzoe d oda ed l suo complesso cougao: ( ) ( ) ( ) ψ ψ. cò mplca che se s specfca uo sao χ, l egrale χ χ rappresea la probablà d rovare la par4cella ello sao χ. l suo ome derva dal fa@o che goca u ruolo ella desà d probablà: A ( ) AP A da [ ] A ψ Aψ Tr A. (*)
2 marce desà [ ] la relazoe A ψ Aψ Tr A mplca che è possble calcolare valor d a@esa (e osservabl) calcolado la racca d u prodo@o d marc. è uo srumeo u4le quado s devoo descrvere sa4 ms4 ( mxed saes ): - pure saes: cara@erzza4 da ua sola fuzoe d oda - mxed saes: mscela sa4s4ca su cu s hao formazo mperfe@e e per la quale è ecessaro calcolare mede sa4s4che per descrvere gl osservabl. è u esemble d sa4 seza relazoe d fase. daa ua mscela sa4s4ca descr@a dalla probablà p k d rovare occupao lo sao ψ k co p k 1, l valore d a@esa d u operaore è ceramee semplfcao a@raverso k l uso della marce desà: () ψ () ˆψ () A ˆ pk k A k k ( ) p ( ) ( ) k ψk ψk k () ˆ () A ˆ Tr A.
3 marce desà: propreà 1) herm4aa: * m 2) ormalzzazoe: Tr ( ) 1 3) 2 1 for puresae Tr ( ) < 1 for mxed sae propreà delle racce: Tr ( ABC) Tr ( CAB) Tr ( BCA) ( ) Tr ( A) Tr S AS varaza cclca varaza rspe@o alla rasformazoe uara
4 eleme4 della marce desà uo sao mso (o mscela sa4s4ca) può essere erpreao come u seme d molecole (1,2, N) descr@e co lo sesso se d base ma co le probablà d occupare ques4 sa4 che possoo varare da molecola a molecola. Lo sao d ua cera molecola s può esprmere come: ψ c allora gl eleme4 della marce desà dveao: m cc, m * m ψ c ψ m ( c ) * m eleme% dagoal (): rappreseao la probablà che lo sao sa popolao. per queso soo den popolazo cc p * eleme% fuor dagoale ( m): soo compless e hao u fa@ore d fase dpedee dal empo che descrve l evoluzoe d sovrapposzo coere4. per queso soo den coereze () () c c c c, * * ω m m e
5 marce desà all equlbro eq e Z β Ĥ ( ) Z Tr e β Ĥ ( eq ) 1 e Z β E e δ Z p δ β Hˆ m.
6 evoluzoe emporale della marce desà l evoluzoe emporale della marce desà segue dre@amee dalla sua defzoe e dall eq. d Schrodger dpedee dal empo ψ H ψ ψ ψ H h h ψ ψ ψ ψ + ψ ψ H ψ ψ + ψ ψ H h h h [ H, ] equazoe d Louvlle-Vo Neuma
7 evoluzoe emporale della marce desà la soluzoe è: ( ) U( ) U. che s onee egrado l eq. d Luovlle: () ( ) dτ H( τ), ( τ) h e sos4uedo era4vamee dero l egrale. S arrva allo sesso rsulao sos4uedo ella precedee l espressoe gà rcavaa per l operaore evoluzoe emporale: U exp dτ H( τ ) + h s o4 che daa l varaza cclca della racca, l valore d a@esa d u operaore può essere calcolao propagado l operaore (Hesemberg represea4o) oppure la marce desà (Schrodger o Ierac4o represea4o): () ˆ () A ˆ Tr A Tr AU ˆ U Tr Aˆ ()
8 marce desà: evoluzoe co HH se l ssema è descr@o da hamloao o dpedee dal empo: HH, allora: () () ( ) ( ) ψ ψ ψ ψ m m U U m () e ω ( ) ( ) le popolazo soo cosa4 e le coereze oscllao co frequeza corrspodee all eergy splng ω
9 marce desà: evoluzoe co HH +V S è gà vso che quado l hamloao s può separare due par4, cu u@a la dpedeza emporale è coceraa ella perurbazoe V, allora la rappreseazoe d erazoe è par4colarmee coveee. per la marce desà s applca sesso procedmeo vso co l eq. d Schrodger ella vsoe d erazoe: ( ) + ( ) H H V ψ I U ψ S I VI (), I () h VI U VU $ L la soluzoe è: I () I ( ) d VI ( ), I ( ) h sos4uedo era4vamee I () sé sessa, s onee u espasoe sere perurba4va:
10 marce desà: vsoe d erazoe I ()! dτ 1 V I (τ 1 ), +! ! + (1) + (2) () 2 [ ] + dτ 1 dτ 2 V I (τ 2 )[ V I (τ 1 ), ] + τ 1 dτ 1 dτ 2... dτ V I (τ ), V I (τ 1 ),... V I (τ 2 ) V I (τ 1 ), τ 1 τ 1 [ ]
[ ] matrice densità. ( t) ( t) ( t) ( )
matrce destà La matrce destà è ua rappresetazoe altera4va dello stato d u sstema qua4s4co per cu abbamo precedetemete u4lzzato la fuzoe d'oda. Ace se descrvere u sstema qua4s4co co la matrce destà equvale
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