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E 1 0 0 0 0 . . . H = 0 E 2 0 0 0 . . . 0 0 E 3 0 0 . . . 0 0 0 E 4 0 . . . :. 1 0 0 : ·. 0 0 1 : ·. 0 1 0 : ·. with the “basis set” :. ,. ,. ,.

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## E 1 0 0 0 0 . . . H = 0 E 2 0 0 0 . . .

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**E1 0 0 0 0 . . .**H = 0 E2 0 0 0 . . . 0 0 E3 0 0 . . . 0 0 0 E4 0 . . . : . 1 0 0 : · 0 0 1 : · 0 1 0 : · with the “basis set”: ... , , , This is not general at all (different electrons, different atoms require different matrices) Awkward because it provides no finite-dimensional representation That’s why its desirable to abstract the formalism**Hydrogen Wave Functions**1 0 0 0 0 0 1 0 0 0 : : 0 0 1 0 0 : : 0 0 0 1 0 : : 0 0 0 0 1 : : : : 0 0 0 0 0 : :**But the sub-space of angular momentum**(described by just a subset of the quantum numbesrs) doesn’t suffer this complication. Angular Momentum |nlmsms…> l = 0, 1, 2, 3, ... Lz|lm> =mh|lm> for m = -l, -l+1, … l-1, l L2|lm> = l(l+1)h2|lm> Sz|lm> = msh|sms> for ms = -s, -s+1, … s-1, s S2|lm> = s(s+1)h2|sms> Of course |nℓm> is dimensional again**Classically**can measure all the spatial (x,y,z) components (and thus L itself) of Quantum Mechanically not even possible in principal ! azimuthal angle in polar coordinates So, for example**Angular Momentum**nlml… Measuring Lx alters Ly (the operators change the quantum states). The best you can hope to do is measure: l = 0, 1, 2, 3, ... L2lm(,)R(r)= l(l+1)ħ2lm(,)R(r) Lz lm(,)R(r) =mħ lm(,)R(r) for m = -l, -l+1, … l-1, l States ARE simultaneously eigenfunctions of BOTH of THESE operators! We can UNAMBIGUOULSY label states with BOTH quantum numbers**ℓ = 2**mℓ= -2, -1, 0, 1, 2 ℓ = 1 mℓ= -1, 0, 1 2 1 0 1 0 L2 = 1(2) = 2 |L| = 2 = 1.4142 L2 = 2(3) = 6 |L| = 6 = 2.4495 Note the always odd number of possible orientations: A “degeneracy” in otherwise identical states!**Spectra of the**alkali metals (here Sodium) all show lots of doublets 1924:Paulisuggested electrons posses some new, previously un-recognized & non-classical 2-valued property**Perhaps our working definition of angular momentum was too**literal …too classical perhaps the operator relations Such “Commutation Rules” are recognized by mathematicians as the “defining algebra” of a non-abelian (non-commuting) group may be the more fundamental definition [ Group Theory; Matrix Theory ] Reserving L to represent orbital angular momentum, introducing the more generic operator J to represent any or all angular momentum study this as an algebraic group Uhlenbeck & Goudsmitfind actually J=0, ½, 1, 3/2, 2, … are all allowed!**quarks**3 2 1 2 1 2 1 2 1 2 1 2 leptons spin : p, n,e, , , e , , ,u, d, c, s, t, b the fundamental constituents of all matter! spin “up” spin “down” s = ħ = 0.866 ħ ms = ± sz = ħ ( ) 1 0 | n l m > | > = nlm “spinor” ( ) ( ) ( ) the most general state is a linear expansion in this 2-dimensional basis set 1 0 0 1 = + with a 2 + b 2 = 1**ORBITAL ANGULARMOMENTUM**SPIN fundamental property of an individual component relative motion between objects Earth: orbital angular momentum: rmv plus “spin” angular momentum: I in fact ALSO “spin” angular momentum: Isunsun but particle spin especially that of truly fundamental particles of no determinable size (electrons, quarks) or even mass (neutrinos, photons) must be an “intrinsic” property of the particle itself**Total Angular Momentum**l = 0, 1, 2, 3, ... Lz|lm> =mħ|lm> for m = -l, -l+1, … l-1, l L2|lm> = l(l+1)ħ2|lm> Sz|lm> = msħ|sms> for ms = -s, -s+1, … s-1, s S2|lm> = s(s+1)ħ2|sms> nlmlsmsj… In any coupling between L and S it is the TOTAL J = L + s that is conserved. Example J/ particle: 2 (spin-1/2) quarks bound in a ground (orbital angular momentum=0) state Example spin-1/2 electron in an l=2 orbital. Total J ? Either 3/2 or 5/2 possible**BOSONSFERMIONS**spin 1 spin ½ e, m p, n, Nuclei (combinations of p,n) can have J = 1/2, 1, 3/2, 2, 5/2, …**BOSONSFERMIONS**spin 0 spin ½ spin 1 spin 3/2 spin 2 spin 5/2 : : quarks and leptons e, m, t, u, d, c, s, t, b, n “psuedo-scalar” mesons p+, p-, p0, K+,K-,K0 Baryon “octet” p, n, L Force mediators “vector”bosons: g,W,Z Baryon “decupltet” D, S, X, W “vector” mesons r, w, f, J/y, **Combining any pair of individual states|j1m1>and|j2m2>**forms the final “product state” |j1m1>|j2m2> What final state angular momenta are possible? What is the probability of any single one of them? Involves “measuring” or calculating OVERLAPS (ADMIXTURE contributions) or forming the DECOMPOSITION into a new basis set of eigenvectors. j1+j2 S |j1m1>|j2m2>= zj j1 j2;mm1m2| jm> j=| j1-j2 | Clebsch-Gordon coefficients**Matrix Representation**for a selected j J2|jm> =j(j+1)h2|j m> Jz|jm> = mh|j m> for m = -j, -j+1, … j-1, j J±|jm> = j(j +1)-m(m±1)h |j, m1 > The raising/lowering operators through which we identify the 2j+1degenerate energy states sharing the same j. J+ = Jx + iJy J- = Jx- iJy subtracting adding 2Jx = J+ + J- Jx = (J+ + J-)/2 Jy = i(J-- J+)/2 2iJy = J+- J-**The most common representation of angular**momentum diagonalizes the Jzoperator: <jn| Jz|jm> = lmmn 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -2 1 0 0 0 0 0 0 0 -1 (j=1) (j=2) Jz = Jz =**J±|jm> = j(j +1)-m(m±1)h |j, m1 >**J-|1 1> = <1 0| |1 0> 0 0 0 0 0 0 0 J-|1 0> = J- = <1 -1| |1 -1> J-|1 -1> = 0 J+|1 -1> = |1 0> <1 0| 0 0 0 0 0 0 0 J+|1 0> = J+ = |1 1> <1 1| J+|1 1> = 0**For J=1 states**a matrix representation of the angular momentum operators**Which you can show conform to the**COMMUTATOR relationship you demonstrated in quantum mechanics for the differential operators of angular momentum [Jx, Jy] = iJz JxJy-JyJx = = iJz**R(1,2,3) =**z z′ 1 y′ 1 y x =x′**R(1,2,3) =**z z′ z′′ 1 2 2 y′ =y′′ 1 y x =x′ 2 x′′**R(1,2,3) =**z z′ z′′ 1 z′′′ = 2 3 y′′′ 3 2 y′ =y′′ 1 y 3 x =x′ 2 x′′ x′′′**R(1,2,3) =**about x-axis about y′-axis about z′′-axis 1st 2nd 3rd These operators DO NOT COMMUTE! Recall: the “generators” of rotations are angular momentum operators and they don’t commute! but as nn Infinitesimal rotations DO commute!!**1 3 0**1 0 -2 1 0 0 R(1,2,3) = -3 1 0 0 1 1 0 1 0 20 1 0 -11 0 0 1 1 3-2 1 0 0 = -3 1 0 0 1 1 20 1 0 -11 or 1 3 0 1 0 -2 = -3 1 0 0 1 1 2-11 0 0 1 1 3-2 = -31 1 2-11**1 3-2**R(1,2,3) = -31 1 2-11 1 R(1,2,3) = 2 3 + R(1,2,3) = If we imagine building up to full rotations by applying this repeatedly N times [R(1,2,3)]N ℓim ℓim N N**R(1,2,3)**ℓim N Which we can re-write in the form by slightly re-writing the “vector” components: Check THIS out: We have found a “new” representation of Lx ,Ly ,Lz!!**Our alternate approach to motivating**where the generator was an honest-to-goodness matrix! gave representing 3-dimensional rotations with the basis: A B C which we argue A B C satisifes all the same arithmetic as Lx, Ly, Lz**Notice can try diagonalizing the zth matrix:**Eigenvalues of -1, 0, 1 We should be able to diagionalize s3 by aSIMILARITY TRANSFORMATION! Us3U = †**Us3U =**†**For J=1 states**a matrix representation of the angular momentum operators

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