Lecture 24: Tensor Product States

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Clinton Randall
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1 Lectue 4: Teso oduct States hy85 Fall 009

2 Basis sets fo a paticle i 3D Clealy the Hilbet space of a paticle i thee diesios is ot the sae as the Hilbet space fo a paticle i oe-diesio I oe diesio, X ad ae icopatible If you specify the wave fuctio i coodiatespace, x, its oetu-space state is copletely specified as well: p dx p x x You thus specify a state by assigig a aplitude to evey possible positio OR by assigig ad aplitude to evey possible oetu ( x ) I thee diesios, X, Y, ad Z, ae copatible. Thus, to specify a state, you ust assig a aplitude to each possible positio i thee diesios. This equies thee quatu ubes So appaetly, oe basis set is: x ( x, y, z) x, y, z { x,y,z } o x, y,z (x,y,z) R 3 ( p) p

3 Defiitio of Teso poduct Suppose you have a syste with 0 possible states Now you wat to elage you syste by addig te oe states to its Hilbet space. The diesioality of the Hilbet space iceases fo 0 to 0 The syste ca ow be foud i oe of 0 possible states This is a su of two Hilbet sub-spaces Oe quatu ube is equied to specify which state Istead, suppose you wat to cobie you syste with a secod syste, which has te states of its ow The fist syste ca be i of its 0 states The secod syste ca be i of its 0 states The state of the secod syste is idepedet of the state of the fist syste So two idepedet quatu ubes ae equied to specify the cobied state The diesioality of the cobied Hilbet space thus goes fo 0 to 0x000 This cobied Hilbet space is a (Teso) oduct of the two Hilbet sub-spaces

4 Foalis Let H ad H be two Hilbet spaces We will tepoaily tag states with a label to specify which space the state belogs to () () H φ H Let the Hilbet space H be the tesopoduct of spaces H ad H. H H H H is the teso poduct of H ad H The Teso poduct state () () belogs to H. The KEY OINT TO GET IS: Bas ad kets i the sae Hilbet space attach. () () BUT, Bas ad kets i diffeet Hilbet spaces do ot attach () () () ϕ ϕ () They just slide past each othe

5 Schidt Basis The easiest way to fid a good basis fo a teso poduct space is to use teso poducts of basis states fo each sub-space If: { () } ;,,,N is a basis i H { () } ;,,,N is a basis i H It follows that: {, () };, () () () is a basis i H. If Syste is i state: ad Syste is i state: () () N N a b () () The the cobied syste is i state:, () () () a b, () Schidt Decopositio Theoe: All states i a teso-poduct space ca be expessed as a liea cobiatio of teso poduct states () N N N N c, ()

6 Etagled States The essece of quatu `weidess lies i the fact that thee exist states i the tesopoduct space of physically distict systes that ae ot teso poduct states A teso-poduct state is of the fo () () () Teso-poduct states ae called factoizable The ost geeal state is () N N c, () This ay o ay-ot be factoizable No-factoizable states ae called etagled Fo a `etagled state, each subsyste has o idepedet objective eality

7 Cofiguatio Space The state of a quatu syste of N paticles i 3 diesios lives i cofiguatio space Thee ae thee quatu ubes associated with each paticle It takes 3N quatu ubes to specify a state of the full syste Coodiate Basis: x,y,z, x, y,z,..., x N,y N,z N Wavefuctio: (We do t kow about spi yet) { } o we could just wite,,..., { } N (,..., ),,...,, N N This fo is the coodiate syste idepedet epesetatio To specify a state of N paticles i d diesios equies d N quatu ubes So coutig quatu ubes ight be a good way to check if you ae usig a valid basis

8 Teso oducts of Opeatos THEOREM: ROOF: Let A () act i H, ad B () act i H, The the teso poduct opeato C () A () B () acts i H. () A a a () a () () B b b () b () A () B () () a b a, a () b () () () a b a b, b () ( )( () () a b ) () C a, b a (), b a, b () The actio of C () o a teso-poduct state: () : (), () () () C a, (), b a, b a b () ()

9 Geeal fo of Opeatos i Teso-poduct spaces The ost geeal fo of a opeato i H is: C () c,, ; ', ' (), ', ' () c () (), ; ', ' :, C, () Hee, ay o ay ot be a teso poduct state. The ipotat thig is that it takes two quatu ubes to specify a basis state i H A basis that is ot foed fo tesopoduct states is a etagled-state basis I the begiig, you should always stat with a teso-poduct basis as you physical basis The all opeatos ae well-defied Just expad states ad opeatos oto tesopoduct states The atch up the bas ad kets with thei pope pates whe takig ie poducts

10 Upgadig Subspace Opeatos Ay opeato i H ca be upgaded to a opeato i H by takig the teso poduct with the idetity opeato i H : If A is a obsevable i H, the it is also a obsevable i H (sice it eais Heitia whe upgaded). The spectu of A eais the sae afte upgadig oof: Let A () A () () () A a I a () a () The: A () I () ( () a () ) A () () a ( ) ( I() () ) a a () () a ( () () a ) Note that is copletely abitay, but a is a eigestate of A ()

11 oduct of two Upgaded Opeatos Let A () ad B () be obsevables i thei espective Hilbet spaces Let A () A () I () ad B () I () B (). The poduct A () B () is give by A () B() oof: ( A I )( I B ) A I I B A B

12 Copatible obsevables Let A () ad B () be obsevables i thei espective Hilbet spaces Let A A () I () ad B I () B (). Theoe: [A,B]0 oof: A, B AB BA [ ] ( A I )( I B) ( I B)( A I ) ( A I ) ( I B ) ( I A ) ( B I ) A B A B 0 Coclusio: ay opeato i H, is copatible with ay opeato i H,. I.e. siultaeous eigestates exist. Let A a a a ad B b b b. Let a a ad b b Let ab a b. The AB ab ab ab.

13 Ad vesus O The teso poduct coelates with a syste havig popety A ad popety B Diesio of cobied Hilbet space is poduct of diesios of subspaces associated with A ad B Exaple: stat with a syste havig 4 eegy levels. Let it iteact with a level syste. The Hilbet space of the cobied syste has 8 possible states. Hilbet spaces ae added whe a syste ca have eithe popety A o popety B Diesio of cobied Hilbet space is su of diesios of subspaces associated with A ad B Exaple: stat with a syste havig 4 eegy levels. Add oe eegy levels to you odel, ad the diesio goes fo 4 to 6

14 Exaple # : aticle i Thee Diesios Let H be the Hilbet space of fuctios i oe diesio The pojecto is: I dx x x So a basis is: { x } The H 3 (H ) 3 would the be the Hilbet space of squae itegable fuctios i thee diesios. oof: I 3 I I I dx x x dy y y dz z z dxdydz x x y y z z dxdydz( x y z ) x y z dxdydz x, y,z x, y,z dxdydz x, y, z ( x, y, z) Note: H 3 is also the Hilbet space of ( ) x, y, z x y z ( x, y, z) x, y, z thee paticles i oe-diesioal space

15 Thee-diesioal Opeatos We ca defie the vecto opeatos: Note that: X X () I () I (3) ad y I () () I (3) so that [X, y ]0. With We ca use: z y x z Z y Y x X R z y x ˆ ˆ ˆ ˆ ˆ ˆ ( ) z y x z z y y x x z y x R,, ˆ ˆ ˆ,, Z R Y R X R 3 z y x 3 R j,r k [ ] j, k [ ] 0 R j, k [ ] ihδ jk R o

16 Exaple #: Two paticles i Oe Diesio Fo two paticles i oe-diesioal space, the Hilbet space is (H ). x, x x () x () ( x, x) x, x I dx dx x,x x,x X etc... I X I [ X j,x k ] j, k [ X j, k ] ihδ jk [ ] 0

17 Hailtoias Oe paticle i thee diesios: Each copoet of oetu cotibutes additively to the Kietic Eegy Two paticles i oe diesio: ( ) ) ( ),, ( R V Z Y X V H z y x v ), ( X X V H

18 Coclusios The take hoe essages ae: The cobied Hilbet space of two systes, diesios d ad d, has diesio d d d A physical basis set fo the cobied Hilbet space, H ca be foed by takig all possible poducts of oe basis state fo space H with oe basis state fo H. () () { } H, { } H, : () () {, } H : H H I a teso poduct space, a ba fo oe subspace ca oly attach to a ket fo the sae subspace: () () () () () ϕ ϕ Fo N paticles (spi 0) i d diesios, d N quatu ubes ae equied to specify a uit-vecto i ay basis,, K x, y N, z, x, y, z, K, Nx, Ny, Nz ()

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces