J Matrices. nonzero matrix elements and Condon Shortley phase choice. δ δ. jj mm

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1 5.73 Lectue #4 4 - Last time: Matices stating wit [ i, j]= i Σε = ± i ± ± x k ijk jm = j j + jm jm = m jm k [ ] ± / jm = j( j + ) m( m ± ) jm DEFINITION! noneo matix elements and Condon Sotle pase coice jm jm = jj+ δ δ j m jm = mδ δ j j m m jj mm [ ] jm jm = jj ( + ) mm (,, x,, +, ) all sta witin j / δ δ ± jj mm ± all matix elements of,,, ae eal and positive (onl tose of ae imagina) x ± TODAY:. Wat do te matices look like fo = 0, /,?. man opeatos ae expessed as an angula momentum times a constant Zeeman example densit matix 3. ote opeatos involve tings like q o poducts of two angula momenta Stak effect Wigne-Eckat Teoem * classif opeatos b commutation ule * matix elements in convenient basis sets * tansfom between inconvenient and convenient basis sets. evised Novembe 5, 00

2 5.73 Lectue #4 4 - [ px, p]= 0 A student in 999 suggested tat e could find f(x,) suc tat f f Tus [ px, p] 0! x x Tis is possible, but f(x,) would ave to ave a fom tat excludes it as an acceptable ψ(x,). Tpicall, te f(x,) will ave to be discontinuous o ave discontinuous fist deivatives. Fo all well beaved V(x,), ψ(x,) will ave continuous fist deivatives. Te f(x,) used to pove a commutation ule must be acceptable as a quantum mecanical wavefunction, ψ(x,). Tis is a good ting because (see Angula Momentum Handout) iap e x x = x + a iap e x geneates a linea tanslation of + ain xdiection. linea tanslations commute (but otations do not) Tis is te basis fo (o a consequence of ) [ pi, pj]= 0 [ i, j]= iσεijk k k evised Novembe 5, 00

3 5.73 Lectue #4 4-3 Nonlectue pepae (excite) evolve detect E e iht D e.g. basis set 0,, excite: Ε 0 = = Te excitation matix E ceates equal amplitudes in two excited eigenstates: ρ( 0)= E0 0E 0 evolve: If we ae in te eigenbasis of H ie a t a ae iht ie t e b b be = (tanslation in time) ie c t c ce it HTt/ but otewise, need [ Te T ] = U ( t, 0 ) it HTt + it HTt ρ() t = Te TE0 0E Te T = U(, t 0) ρ ( 0) U ( t, 0) ρ(0) in eigenbasis of H detect: D te detection matix D t = Tace( ρd) Building Blocks evised Novembe 5, 00

4 5.73 Lectue #4 4-4 Man QM opeatos ave te fom f Zeeman e.g. Zeeman effect H γb Bis magnetic field Otes ave te fom f( q) Stak e.g. Stak effect H = ee q ( E is electic field) = Otes ave te fom of e.g. spin - obit f (, ) H SO = al S We ae going to want to be able to wite matix epesentations of tese opeatos. Let us begin b witing matices fo,, x,, +,. j = 0 onl basis state is matix jm = 00 j = / 00 = 0 same fo all components and matices ( / ) ( / ) ( / ) + 3 = = 0 0 = 0 0 e.g. = + m = +/ = 0 (/ ) + = = 0 (/ ) + = m = / evised Novembe 5, 00

5 5.73 Lectue #4 ( / ) = ( / ) x = ( + + )= ( / ) i = ( + )= i = 0 i i veif tat = x = 4 0 x = i i = = 4 i 0 i ( / ) = An amaing amount of insigt gained fom tis complete set of matices I = 0 0 σ σ σ x = 0 x 0 0 i = i 0 0 = 0 ( / ) ( / ) ( / ) 3 matices wit eigenvalues ± CTDL, pages Pauli Matices. Diagonaliation of 3. Geometic intepetation of ρ in tems of fictitious spin / 4. spin / ρ 5. magnetic esonance Wat is [σ x, σ ] =? evised Novembe 5, 00

6 5.73 Lectue #4 abita M= m m = m m 4-6 m + m m m m + m I+ + + i m m σ σ σ x M scala pat of M = a I+ a σ 0 vecto pat of M a0 = T( M) Cente of Gavit a = T( Mσ) M ρ ax = T( Mσx) Infomation in ρ is epackaged into a 3 a = T( Mσ) component vecto. Visualiation of dnamics! a = T( Mσ) Tis povides a basis fo taking apat te dnamics of an abita ρ into dnamics of x,, fictitious spin-/ components. Beat te S = / Zeeman poblem to deat and use it as basis fo undestanding dnamics of an space. evised Novembe 5, 00

7 5.73 Lectue #4 4-7 = A set of 3 3 matices () = = = () / + = () / = () / x () / = i 0 i 0 i 0 i = ( ) + = 0 Fo poblem (e.g. = /), needed 4 independent matices (because tee ae 4 elements in a matix) to epesent abita matix. fo 3 3poblem, need 9 independent 3 3 matices ( x,,, x,,, x, x, ) (because tee ae 9 elements in a 3 actuall scala (I), vecto, tenso s+ p+ d 3 matix) [fo it was s + p = 4]. Can ou wite out eac of te (3/) matices (6 4 4 matices)? evised Novembe 5, 00

8 5.73 Lectue # basis matices is not neal so nice as te 4 basis matices fo poblem. But tis tuns out to be wat is needed to undestand and pictue spin = sstems. similal fo j = 3/,, etc. Tee ae lovel consequences of being able to take an abita matix and ewite it as sum of matices.. If M is te matix of an opeato a tem in te Hamiltonian ten it is clea tat tis opeato ma be e-expessed as a sum of opeatos, eac of wic beaves exactl like a (combination of) component(s) of evaluated in te jm basis set. ( j) M = a I+ a + c 0 li i j 3ijk i j k i ijk,, and Wigne Eckat Teoem fo evaluation of matix elements.. especiall fo level sstems, if M = ρ and a is defined fom M as on page 4-6, ten we ave a vecto pictue to undestand pepaation, evolution, detection + basis fo classification of opeatos into T m k a π/ pulse evolution of vecto, fictitious B-fields π pulse evised Novembe 5, 00

9 5.73 Lectue #4 4-9 Now let s do some = examples Zeeman effect fo an l =, (p obital) state q e - L= q p ( up out of page) µ L q p down into page µ L µ L L L γ cuent on a cicula wie classical eneg field stengt E = B µ = Bkˆ ( γl)= γbl L field exclusivel along fo L= sstem: Zeeman H = γb case () Let Ψ( 0)= LM L = 0 0 ρ = Ψ Ψ = = E E Tace LM = = L ρh = = γbt γb evised Novembe 5, 00

10 5.73 Lectue #4 Wat about? case () E ρ = = Tace (ρh) = 0 E =+ γb no motion of E / Let Ψ()= [ ] / Ψ()= t e + 0 e ie t i0t 4-0 ρ()= t e + 0 e iωt e 0 ()= iω t ρ t e ie ( t + ie t ) noting at location of coeence in ρ H( B + )= γb E()= t H = Tace( Hρ)= γb () no motion of E Looked at cases:. pue state E = γb, B E = γb. mixed state -/ + 0, B alwas mixed state gives time independent E NMR: oscillating B x, B, cw B evised Novembe 5, 00

11 5.73 Lectue #4 4 - Stak Effect: Electic field classical E ε q q + e p so we will need matix elements of ε x,, in jm basis set. How? Based on [ L, j ]= iσεjkqk vecto opeato definition late k Ote angula momenta. l electon obital ang. mom.. s electon spin 3. I nuclea spin Tese sepaate angula momenta inteact wit eac ote spin - obit: ζ() λ s Zeeman: -γb L + gs + gi pefine: a I S s I coupled and uncoupled basis sets: lm sm jlsm l s j evised Novembe 5, 00

12 5.73 Lectue #4 4 - case (3) / same Ψ( 0)= + 0 but H is fo B x H = γb x / one at coeence in ρ Someting subtle is intentionall wong ee. Can ou find it? B () γ x Et + iωt iωt E()= t T( Hρ)= γbx e + e + 0 = γb cosω t x [ ] t evised Novembe 5, 00