So as an exercise in using this notation

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So as an exercise in using this notation let’s look at The indices indicate

So as an exercise in using this notation let’s look at The indices indicate very specific matrix or vector components/elements. These are not matrices themselves, but just numbers, which we can reorder as we wish. We still have to respect the summations over repeated indices! And remember we just showed (g ) = g i. e. All dot products are INVARIANT under Lorentz transformations.

even for ROTATIONS as an example, consider rotations about the z-axis

even for ROTATIONS as an example, consider rotations about the z-axis

The relativistic transformations: suggest a 4 -vector that also transforms by so should be

The relativistic transformations: suggest a 4 -vector that also transforms by so should be an invariant!

In the particle’s rest frame: px = ? 0 E = ? mc 2

In the particle’s rest frame: px = ? 0 E = ? mc 2 p p = ? m 2 c 2 In the “lab” frame: = - mv E = = mc c so

Limitations of Schrödinger’s Equation 1 -particle equation 2 -particle equation: mutual interaction But in

Limitations of Schrödinger’s Equation 1 -particle equation 2 -particle equation: mutual interaction But in many high energy reactions the number of particles is not conserved! n p+e++ e n+p+3 e-+ p + 6 + 3 g

Sturm-Liouville Equations a class of differential equations that include: Legendre's equation i. e. a

Sturm-Liouville Equations a class of differential equations that include: Legendre's equation i. e. a class of differential eq's to which Schrodinger's equations all belong! the associated Legendre equation Bessel's equation the quantum mechanical harmonic oscillator whose solutions satisfy: 0 for different eigenfunctions, If we adopt the following as a definition of the "inner product" compare this directly to the vector "dot product" then notice we have automatically n

Recall: any linear combination of simple solutions to a differential equation is also a

Recall: any linear combination of simple solutions to a differential equation is also a solution, and, from previous slide: mn eigenvalues are REAL and different eigenfunctions are "orthogonal" Thus the set of all possible eigenfunctions (basic solutions) provide an "orthonormal" basis set and any general solution to the differential equation becomes expressible as where any general solution will be a function in the "space" of all possible solutions (the solution set) sometimes called a Hilbert Space (as opposed to the 3 -dimensional space of geometric points.

What does it mean to have a matrix representation of an operator? of Schrödinger’s

What does it mean to have a matrix representation of an operator? of Schrödinger’s equation? where n represents all distinguishing quantum numbers (e. g. n, m, ℓ, s, …) Hmn since †

H= E 1 0 0 E 2 0 0 : . 0 0 E

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

Hydrogen Wave Functions 0 1 0 0 0 : : 0 0 1 0 0 : : 0 0 0 1 0 : : 0 0 1 : : 0 0

But the sub-space of angular momentum (described by just a subset of the quantum

But the sub-space of angular momentum (described by just a subset of the quantum numbers) doesn’t suffer this complication. Angular Momentum |lmsms…> l = 0, 1, 2, 3, . . . Lz|lm> = mh|lm> for m = - l, - l+1, … l-1, l L 2|lm> = l(l+1)h 2|lm> Sz|lm> = msh|sms> for ms = -s, -s+1, … s-1, s S 2|lm> = s(s+1)h 2|sms> Of course |nℓm> is dimensional again

can measure all the spatial (x, y, z) components (and thus L itself) of

can measure all the spatial (x, y, z) components (and thus L itself) of not even possible in principal ! So, for example azimuthal angle in polar coordinates

Angular Momentum nlml… Measuring Lx alters Ly (the operators change the quantum states). The

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, . . . L 2 lm( , )R(r)= l(l+1)ħ 2 lm( , )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 2 ℓ=1 mℓ = -1, 0,

ℓ=2 mℓ = -2, -1, 0, 1, 2 2 ℓ=1 mℓ = -1, 0, 1 1 1 0 0 L 2 = 1(2) = 2 |L| = 2 = 1. 4142 L 2 = 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: Pauli

Spectra of the alkali metals (here Sodium) all show lots of doublets 1924: Pauli suggested 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

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 & Goudsmit find actually J=0, ½, 1, 3/2, 2, … are allowed!

quarks leptons spin 12 p, : n, e, , , e , , ,

quarks leptons spin 12 p, : n, e, , , e , , , u, d, c, s, t, b the fundamental constituents of all matter! ms = ± 1 2 spin “up” spin “down” s = 32 ħ = 0. 866 ħ sz = |nlm>| 1 2 >= ( ) 1 0 1 2 ħ 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

SPIN ORBITAL ANGULAR MOMENTUM fundamental property of an individual component relative motion between objects

SPIN ORBITAL ANGULAR MOMENTUM 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: Isun 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 nlmlsmsj… l = 0, 1, 2, 3, . . . Lz|lm>

Total Angular Momentum nlmlsmsj… l = 0, 1, 2, 3, . . . Lz|lm> = mħ|lm> for m = -l, -l+1, … l-1, l L 2|lm> = l(l+1)ħ 2|lm> Sz|lm> = msħ|sms> for ms = -s, -s+1, … s-1, s S 2|lm> = s(s+1)ħ 2|sms> 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 Either spin-1/2 electron in an l=2 orbital. Total J ? 3/2 or 5/2 possible

BOSONS spin 1 FERMIONS spin ½ p, n, e, Nuclei (combinations of p, n)

BOSONS spin 1 FERMIONS spin ½ p, n, e, Nuclei (combinations of p, n) can have J = 1/2, 1, 3/2, 2, 5/2, …

BOSONS FERMIONS mesons spin 0 “psuedo-scalar” spin ½ + 0 + - 0 p

BOSONS FERMIONS mesons spin 0 “psuedo-scalar” spin ½ + 0 + - 0 p , p , K , K mediators spin 1 Force spin 3/2 “vector”bosons: g, W, Z “vector” mesons r, w, f, J/ , spin 2 : spin 5/2 : quarks and leptons e, , , u, d, c, s, t, b, n Baryon “octet” p, n, Baryon “decupltet” D, S, X, W

Combining any pair of individual states |j 1 m 1> and forms the final

Combining any pair of individual states |j 1 m 1> and forms the final “product state” |j 2 m 2> |j 1 m 1>|j 2 m 2> 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. S j 1+j 2 |j 1 m 1>|j 2 m 2> = z j j 1 j 2; m m 1 m 2 | jm> j=| j 1 -j 2 | Clebsch-Gordon coefficients

Matrix Representation for a selected j J 2|jm> = j(j+1)h 2| j m >

Matrix Representation for a selected j J 2|jm> = j(j+1)h 2| j m > Jz|jm> = m h| j m > for m = -j, -j+1, … j-1, j J±|jm> = j(j +1)-m(m± 1) h | j, m 1 > The raising/lowering operators through which we identify the 2 j+1 degenerate energy states sharing the same j. adding J+ = Jx + i. Jy J- = Jx - i. Jy subtracting 2 Jx = J+ + J- Jx = (J+ + J- )/2 2 i. Jy = J+ - J- Jy = i(J- - J+)/2

The most common representation of angular momentum diagonalizes the Jz operator: <jn| Jz |jm>

The most common representation of angular momentum diagonalizes the Jz operator: <jn| Jz |jm> = lm mn 1 0 0 = 0 0 0 -1 (j=1) Jz (j=2) Jz = 2 0 0 0 1 0 0 0 -1 0 0 0 -2

For J=1 states a matrix representation of the angular momentum operators

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

Which you can show conform to the COMMUTATOR relationship you demonstrated in quantum mechanics for the differential operators of angular momentum [Jx, Jy] = i. Jz Jx Jy - Jy Jx = = i. Jz

R( 1, 2, 3) = about x-axis about y′-axis z′′-axis 1 st 2 nd

R( 1, 2, 3) = about x-axis about y′-axis z′′-axis 1 st 2 nd 3 rd These operators DO NOT COMMUTE! but as n n Recall: the “generators” of rotations are angular momentum operators and they don’t commute! Infinitesimal rotations DO commute!!