Matrix Representations

Transcription

1 Matri Rprsntations C74b C74b Matri Rprsntations A matri is an arra of numbrs: { a } A Eampl: In gnral: A 8 a a A M am row a a a a M M M a a m m column a n an M a mn Indics m n tlls us th ordr of th matri C74b Matri Rprsntations

2 Not: transpos of matri A { a } { a } Vctors in a p-dimnsional spac ar spcifid b a p column vctor. ji Gomtrical intrprtation: th giv th orthogonal coordinats of on nd of th vctor if th othr nd is at th origin of th coordinat sstm Matri Algbra Matrics can b addd subtractd multiplid and dividd. a Addition and subtraction: A ± B C b Multiplication b a scalar α α c αa ± αb c a ± b C74b Matri Rprsntations c Matri multiplication: matrics must b conformabl. if C A B thn th numbr of columns of A numbr of rows of B if th ordr of A and B ar (i j and (j k th ordr of C is: (i j(j k (i k Each lmnt in C can b computd b: c il k a ik b kl row column Not: AB BA ncssaril C74b Matri Rprsntations 4

3 C74b Matri Rprsntations 5 d d Matri division B A B A whr B - is th invrs of B E B B BB O E E idntit matri which is oftn dnotd b I. C74b Matri Rprsntations 6 If two matrics ar block diagonal th corrsponding blocks of idntical ordr can b multiplid individuall

4 Dfinition: For a squar matri its charactr or trac Χ sum of its diagonal lmnts Proprtis:. if C A B and D B A χ a jj j χ C χ D. Conjugat matrics rlatd b a similarit transformation hav idntical charactrs - if A X B X χ χ A B. If C A B χ C χ χ A B C74b Matri Rprsntations 7 Considr th vctor shown blow: î kˆ r r P ĵ C74b Matri Rprsntations 8 4

5 In gnral: r bra kt... n r r... rn r row matri basis st column matri coordinats Both { i } and {r i } ma b compl dfin Hrmitian scalar product of vctors u and v as: r r + u * v u v whr suprscript + dnots adjoint or transposd compl conjugat C74b Matri Rprsntations 9 r u * r v u* * v u* M v i j u * M v i j h squar matri M * mtric of th linar vctor spac. M * *... n *... n * * * * M M M and M * ( * * M * If M M i j i * j j i j i ji + (Hrmitian or slf-adjoint matri M * δ basis st is orthonormal or unitar and thrfor M E C74b Matri Rprsntations 5

6 Configuration spac -D spac in which phsical objcts (atoms molculs crstals ist R Points in R ar dscribd with rspct to a sstm of right-handd orthogonal as { } Right-handd mans a right-handd scrw advancing from th origin; rotats Will us activ rprsntation: as rmain fid but th whol of configuration spac is rotatd. Rotation carris with it all vctors in configuration spac including a st of coordinats { } originall coincidnt with {i j k}. C74b Matri Rprsntations Matri Rprsntation of Oprators Suppos a basis < is transformd to a nw basis < as a rsult of an oprator R R R n n { j } can b prssd in trms of th old st b writing j as a sum of its projctions: n j ir j K n i whr r componnt of j along i In matri form: Γ( R n n Γ ( R ( r r r M rn r r r M n M r n r n M r nn C74b Matri Rprsntations 6

7 7 C74b Matri Rprsntations Γ(R matri rprsntativ of th oprator R In -D configuration spac thr ar 5 oprations to dscrib th transformation of a point or points in spac: E σ i C n and S n Each can b dscribd b a matri Γ(R such that ( ( k j i R ˆ ˆ ˆ ( Γ.. Idntit E Idntit E ( ( ( E Γ ( Γ E C74b Matri Rprsntations 4.. Rflction Rflction σ If th plan of rflction coincids with a principl Cartsian plan ( or rflction changs th sign of th coordinat to plan but lavs th coordinat whos as dfins th plan unchangd. ( ( ( σ Similarl: ( ( ( σ ( ( ( σ

8 θ θ σ π/ - θ In gnral using trigonomtr: θ ( θ sin( θ ( θ cos( cos σ sin θ Θ angl with rspct to th plan C74b Matri Rprsntations 5 ( ( whn θ π/ cos π sin π σ sin( π cos( π as bfor. Invrsion i: Hr a point ( (- - - nd a ngativ unit matri i C74b Matri Rprsntations 6 8

9 4. Propr rotation C n : Considr rotation about an angl Φ about th ais φ π/ + φ φ cos φ + ( φ + sin( π cos + φ + cos φ ( φ + sin ( φ + cos( C74b Matri Rprsntations 7 C n cos sin ( φ sin( φ ( φ cos( φ 5. Impropr rotation S n : this is a C n rotation followd b rflction σ h. hrfor for th rotation in 4: - S n cos sin ( φ sin( φ ( φ cos( φ C74b Matri Rprsntations 8 9

10 Not: all Γ(R for th smmtr oprations ar ral orthogonal matrics. whr Γ(R transpos of Γ(R R Γ( Γ( R E Γ(R - Γ(R is radil calculatd. Not: r Rr r R r r Γ( R r r Smmtr transformations ar rigid. h lngth of all vctors and angls btwn thm rmain unchangd. C74b Matri Rprsntations 9 Asid: can show that if Γ(R is compl thn Γ(R is a unitar matri dfind as Γ(R - Γ(R Γ(R * Propr and impropr rotations can b distinguishd b thir dtrminant. Γ( Γ Qdt AB dt A dt B R Γ( R E Γ( R ( R ( ( ( dt( Γ( R Γ( R dt( Γ( R dt( Γ( R ( dt( Γ( R dt( Γ( ± R Ral orthogonal matrics with dtrminant + impl propr rotations spcial orthogonal matrics hos with dtrminant - impl impropr rotations C74b Matri Rprsntations

11 C74b Matri Rprsntations h ffct of a smmtr oprator R on a point R mapping h rsult is calld a Jons smbol A map (propr or impropr rotation of a basis with rspct to fid as which carris all of R and {r} in R with it. his is important sinc it mans vr point smmtr oprator is quivalnt to a propr or impropr rotation. C74b Matri Rprsntations Effct of a smmtr oprator R on th componnts { } of an vctor P r can b dtrmind b finding Γ(R of R from: ( R Γ and thn us Γ(R to calculat r from r using: ( r r r Γ or Γ R (

12 Group Rprsntations If {A B C.} for a group G thn th st of matri rprsntativs {Γ(A Γ(B Γ(C } form an isomorphic group with G calld a group rprsntation. and if AB C thn Γ(AΓ(B Γ(C h matri rprsntativs ob th sam multiplication tabl as th oprators Eampl: in C v th oprations ar E C σ v σ v For H O plac C along th -ais and lt σ v σ C74b Matri Rprsntations σ σ v coming out of th pag C σ σ v H O H C74b Matri Rprsntations 4

13 C74b Matri Rprsntations 5 σ v σ v C : φ π cos( sin( sin( cos( π π π π C and E C74b Matri Rprsntations 6 Eampl: Eampl: Considr th product σ v C σ v O H H O H H O H H C F v σ v Now: σ v σ C σv σ

14 ransformation of Scalar Functions Rlvant for th undrstanding of how atomic orbitals transform undr smmtr oprations If f f( it mans that f has a dfinit valu at ach point P( with coordinats t ( {} and ( { } If an oprator transforms P( P( { } { } Γ( C74b Matri Rprsntations 7 *But a smmtr oprator lavs a sstm in an indistinguishabl configuration. hrfor th proprtis of th sstm ar unaffctd b transforms f into a nw function f in such a wa that: ({ } f ({ } ˆf ˆ function oprator Mans: th valu of th nw function f valuatd at th transformd point { } is th sam as th valu of th original function at th original point {} Mans: whn a smmtr oprator acts on a configuration and function f is simultanousl transformd in to a nw function f. C74b Matri Rprsntations 8 4

15 5 C74b Matri Rprsntations 9 Qustion: How to calculat Qustion: How to calculat f? Undr smmtr oprator point P P that is: ( ( P P ( ( P P { } ( ( { } ( ˆ f f f Drop th prims sinc this applis to an point P ( { } ( { } ( f f ˆ C74b Matri Rprsntations t R(π/ on d-orbital d g(r Eampl: Eampl: g(r function of r onl and contains angular dpndnc ( Γ cos sin sin cos 4 π π π π C ( [ ] ( [ ] Γ Γ

16 ˆ ( { } d ( d d g( r d C 4 d d - -d C74b Matri Rprsntations Quantum Mchanical Considrations a Q.M. wavfunction Ψ({} strictl rquirs multiplication b a phas factor which is arbitrar. his phas factor has no ffct on phsical proprtis. hrfor choos th phas factor. f ( ({ } f { } ˆ can b usd for Q.M. wavfunctions b Function oprators corrsponding to smmtr oprators ar unitar oprators ˆ + ˆ ˆ ˆ + E C74b Matri Rprsntations 6

17 c Whn acts on a phsical sstms (atom molcul tc a Q.M. oprator M corrsponding to a dnamical variabl bcoms: Mˆ M ˆ ˆ + Epctation valus ar invariant undr smmtr oprators. [ ˆ Mˆ ] ˆ ˆ ˆ + M Mˆ Q.M. oprator for th nrg of a sstm is th Hamiltonian oprator H. his mans must commut with H. h st of all function oprators {} that lavs H invariant and which form a group isomorphic with th smmtr oprators {} is known as th group of th Hamiltonian or th group of th Schrodingr quation. C74b Matri Rprsntations d If th dnamical variabl is an obsrvabl with oprator M this mans Ψ is an ignfunction of M with ignvalu m. his mans <M> m valu of phsical quantit in stat Ψ. M is invariant undr a smmtr oprator. hrfor Ψ and Ψ rprsnt th sam stat; that is th ar dgnrat. C74b Matri Rprsntations 4 7