Self-Adjointness and Its Relationship to Quantum Mechanics. Ronald I. Frank 2016

Transcription

1 Ronald I. Frank 06 Adjoint In gnral thr is an oprator and a procss that dfin its adjoint *. It is thn slf-adjoint if *. Innr product spac On wa of dfining slf-adjunction for matrics and linar oprators in an innr product spac (a vctor spac with an innr product <, >) is <A, > <, A*>. [A vctor spac is a st of vctors and a st of scalars (usuall ral or compl numbrs) closd undr vctor addition and scalar multiplication. An innr product of vctors is linar and rsults in a scalar.] Innr product: Smmtric,,. Linar a, a, Positiv Dfinitnss, z, z, z, + z,, z, 0, 0 0. Compl Innr product: Conjugat Smmtric,, Conjugat Linar a, a,, a a,, z, z, z, + z,, z Positiv Dfinitnss, 0, 0 0. Pag of 0 9/8/06 Slf-AdjointnssV3.doc

2 Slf-Adjointnss and Its Rlationship to Ronald I. Frank 06 Computing a ral innr product <, > (vctors ar columns). <, > [,, n ],, n n n n n kn kk cos( ), th dot product. k (A) A A <A, > jn a,a a, a A A A ij ji ij j i j An i n i n j n C A [ A, A, A ] A a n i n A A i ij j i i i j Pag of 0 9/8/06 Slf-AdjointnssV3.doc

3 Ronald I. Frank 06 <, A > A A in A a, A a a ij ji ij, A a i A ij j i jn jn in [,, ] j j a n j ij i j j i A A A D A A A n C A A, D A, A A aij a ji a in jn jn in a a i j j i A, A ij ij j i j ij i A, Slf-adjoint A A n A. Slf-adjoint mans For ral matrics, adjoint mans transpos, A A. For compl matrics, it mans conjugat transpos. * H H. For compl matrics Slf adjoint mans * H H H Pag 3 of 0 9/8/06 Slf-AdjointnssV3.doc

4 Ronald I. Frank 06 Slf adjointnss is important bcaus th oprators hav rall nic proprtis (if th ar not dgnrat):. All ignvalus ar ral. hir ignvctors span th spac 3. h ar qual to a sum of wightd projctors on th ignvctors. (Spctral Epansion). Smmtric matri is b dfinition slf- For ral matrics, a smmtric matri, adjoint. A A A Not: Givn an mn data matri, X, X X is an (nm) * (mn) (nn) matri which is smmtric (X X) X X. Data matrics ar most oftn long and narrow (m>n) of rank n. hrfor X X is of full rank, having a full st of ignvalus and ignvctors. Hrmitian matri For compl matrics, a conjugat smmtric matri, is b dfinition slf-adjoint. i i i H, H, H H i i i Ral ampl of slf-adjointnss. A spcial -D Situation & h usual drivation * * H H H, Pag 4 of 0 9/8/06 Slf-AdjointnssV3.doc

5 Ronald I. Frank 06 A ; dt( ) (4 ) 3 ; A tr( A) ( ) 4 ; 4 & 3 3 & ( ) ; A I dt , 3 & 3 v A,, v v v v 3v v v 0 v v 3v v v 0 Av 3v v v v v v v v 0 v v v v v 0 Av v v v W choos valus that satisf th lmnt rlations abov but mak th vctors as to normaliz (mak thir norm qual to ).,,,, 0 Ronald I. Frank 06 ignvctors.vsd If v is an ignvctor, so is (k v) for an non-zro scalar k. Av v A kv kv k Av k v Notic Orthogonalit: 0. A proprt of orthogonalit: it dos NO dpnd on vctor lngth. b a b a, b a a a b a b 0 b b a b a, b a a a b a b b a b a b 0 0 Pag 5 of 0 9/8/06 Slf-AdjointnssV3.doc

6 Ronald I. Frank 06 Pr and post multipl b th matri of ignvctors. E AE E E I EE E E E AE E AE EE A EE IAI E E A A EE RANK Matri Multiplication & Projctions. A EE E E h scond trm has rows which ar th ignvctors ach multiplid b its ignvalu. h dfinition of matri multiplication CAB is: kn c a b ij ik kj. his is th innr product of rows from A, (i) and k columns from B (j). h rank on dcomposition of a matri product is: ak th columns of A and outr product thm with th corrsponding rows of B. Each trm is of rank sinc its product with a vctor ilds a multipl of th sam ignvctor. Summing ths rank on matrics givs C. i j a b c & ; i j ij all of th cij sums. is an nn matri of th first trms in Pag 6 of 0 9/8/06 Slf-AdjointnssV3.doc

7 Ronald I. Frank 06 a b c sums. ik kj ij k is an nn matri of th kth trms in th cij his shows that th matri product quals th sum of n rank products. h ar projctions on th rspctiv row or column AB C C h ar projctions sinc [taking vctors as columns] z z z k z z z K i.., multipling on ithr sid ilds onl a multipl of a vctor (rank ) b a factor. Innr Product Outr Product Projction on Right Sid Multiplication z z z Outr Product Projctions Projction Scalar on Multipl Lft sid Of Multiplication k k Scalar Multipl of Ronald I. Frank 06 Projctions&InnrProduct.vsd Pag 7 of 0 9/8/06 Slf-AdjointnssV3.doc

8 Ronald I. Frank 06 Appling this to A E E E E, w gt an pansion for A which is a sum of trms, ach of which is a projction on an ign dirction. kn kn kn kk k k k k k k k k k A P ; th k k Pk is a projctor on th k ignvctor k ; Pk P k k k k k k I k k k Pk + A his is calld th spctral mapping thorm or th spctral dcomposition of a slf-adjoint oprator. Compl ampl of slf-adjointnss. 3 5 i * i v H, H,, v i i 3 5 v tr( H ) 3, dt( H ) First Eignvalu v iv v v iv H v v iv v v iv v 0 5 v 5 iv v v v 5 K 5 v 5 v 5 v iv v Pag 8 of 0 9/8/06 Slf-AdjointnssV3.doc

9 Slf-Adjointnss and Its Rlationship to Ronald I. Frank v 5 5 K K 5 4 v K i K i v i K v Normalization v i v, i i , ~ i i 3 5 v, i i 3 5 v Normalization is a bar. Howvr, to prov orthogonalit w can us th un-normalizd vctors. W will s that both ignvalus gnrat onl on ignvctor. Scond Eignvalu v iv v v iv H v v iv v v iv v 0 5 v iv 5 v v v 5 5 v 5 v 5 v iv v Pag 9 of 0 9/8/06 Slf-AdjointnssV3.doc

10 Ronald I. Frank 06 v ˆ K ; v ˆ 3 5 v i K v v i v ˆ, ˆ 3 5 ˆ ˆ again. 3 5 i i h two ignvctors gnratd b th ignvalus ARE orthogonal., ˆ, ˆ i i i i ,ˆ QED 4 Quantum Mchanics in a Nutshll. An masurabl quantit is rprsntd b a slf adjoint Hrmitian oprator H. Its ignvctors ar takn as normalizd to. h stats of th sstm quantit ar vctors in th Hilbrt spac spannd b th ignvctors of H. A gnral sstm stat is a mitur of pur ignstats. A masurmnt projcts th mid stat onto a pur ignstat (ignvctor). h ignvalu of that stat is th valu of th masurmnt of th quantit. It is ral. h probabilit of gtting th masurd valu is th squard lngth of th projction on th ignstat (ignvctor). [Sinc th stat has lngth, th sum of squars of its orthogonal projctions is, thrfor th squard lngths ar all lss or qual, and sum up to. hrfor, th CAN b intrprtd as probabilitis.] A cntral problm in Quantum Mchanics: How dos on find th Hrmitian oprator to rprsnt th quantit of intrst? his in on of th things Phsicists do. Pag 0 of 0 9/8/06 Slf-AdjointnssV3.doc