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# notes24 - 5.73 Lecture#24 24 1 J Matrices Last time...

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24 - 1 5.73 Lecture #24 revised November 15, 2001 J Matrices Last time: DEFINITION ! nonzero matrix elements and “Condon Shortley” phase choice TODAY: 1. What do the matrices look like for 2. many operators are expressed as an angular momentum times a constant — Zeeman example — density matrix 3. other operators involve things like or products of two angular momenta J = 0121 , / , ? r q Stark effect Wigner-Eckart Theorem * classify operators by commutation rule * matrix elements in convenient basis sets * transform between inconvenient and convenient basis sets. starting with JJ J J J J J ij k ijk k z xy i jm j j jm jm m jm i jm j j m m jm , / [] =+ () = −± ± ± ± h h h h ε 22 12 1 11 1 ′′ = ± ± jm jj j m jm m mm jj mm zj j m m J J J 1 1 1 h h h δδ / JJJJJJ JJJJ J 2 2 ,,,,, , zxy +− ± all stay within j all matrix elements of , , are real and positive (only those of are imaginary) zx y

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24 - 2 5.73 Lecture #24 revised November 15, 2001 A student in 1999 suggested that he could find f(x,y) such that This is possible, but f(x,y) would have to have a form that excludes it as an acceptable ψ (x,y) . Typically, the f(x,y) will have to be discontinuous or have discontinuous first derivatives. For all well behaved V(x,y), ψ (x,y) will have continuous first derivatives. The f(x,y) used to prove a commutation rule must be acceptable as a quantum mechanical wavefunction, ψ (x,y) . This is a good thing because (see Angular Momentum Handout) linear translations commute (but rotations do not) This is the basis for (or a consequence of ) ∂∂ [] 22 0 f xy f yx Thus pp ,! , = 0 ex x a ia x =+ p h 11 ea x ia x p h generates a linear translation of + in direction. JJ J ij k ijk k i , , = 0 h ε
24 - 3 5.73 Lecture #24 revised November 15, 2001 e a b c ae be ce et te e tt it iE t iE t iE t a b c −+ = [] = () = = H h h h h h hh but otherwise, need TT U E E UU THT †† /† , (, ) ( ) ( 0 00 ρ ρ , ) 0 Nonlecture prepare (excite) E evolve detect D e.g. basis set 0 1 2 ,, excite: evolve: If we are in the eigenbasis of H detect: D the “detection matrix” ρ (0) in eigenbasis of H Building Blocks e H h Ε 0 011 100 1 0 0 0 1 1 0 = = = ρ EE DD t = Trace ρ (translation in time) 0 1 2 The “excitation matrix” E creates equal amplitudes in

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notes24 - 5.73 Lecture#24 24 1 J Matrices Last time...

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