bounded by a confining potential only very specific Energy is quantized as a consequence of the wave nature of matter bounded by a confining potential only very specific (sine, cosine, exponential) functions can satisfy the boundary conditions. The geometry of 3-dimensional space forces angular momentum to be conserved! The spatial descriptions of a system should be completely symmetric in terms of the azimuthal angle  …cyclic in 2! As bizarre as this might seem, its beautifully exhibitted by the Zeeman effect!

Energy-level splitting in a magnetic field for the 2P3/2, 2P1/2, and 2S1/2 energy levels for Sodium.

As you sill see later there is a nuclear counter part: Zeeman Effect(Dipole Interaction) Interaction of the nuclear magnetic dipole moment with the external applied magnetic field on the nucleus. 

Classically (and thus L itself) of Quantum Mechanically 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

Lz lm(,)R(r) = mħ lm(,)R(r) 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, ... L2lm(,)R(r)= l(l+1)ħ2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

Hydrogen Wave Functions

ℓ = 2 mℓ = -2, -1, 0, 1, 2 ℓ = 1 mℓ = -1, 0, 1 2 1 1 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: Pauli suggested electrons posses some new, previously un-recognized & non-classical 2-valued property

ORBITAL ANGULAR MOMENTUM 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: Isunsun 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

Schrödinger’s Equation is based on the constant (conserved) value of the Hamiltonian expression total energy = sum of KE + PE with the replacement of physical variables with “operators” though amazingly accurate for many (simple) atomic systems…not relativistic!

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 all allowed!

In systems of identical particles (for example pairs) under pairwise interchanges: Shouldn’t these state be indistinguishable? Yes, but notice that only means must remain unchanged! i.e. with a distinguishing phase change! where obviously  = 0, 

Two cases: symmetric under interchange anti-symmetric under interchange Hey! What if 2 (identical) particles are in identical states? …both trapped in the same potential …co-existing with the same energy level En obviously we’d have to expect: That’s OK for symmetric states, but for the anti-symmetric states: with

spin : p, n, e, , , e ,  ,  , u, d, c, s, t, b s = ħ = 0.866 ħ quarks leptons spin : 1 2 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 3 2 ms = ± 1 2 sz = ħ 1 2

Lz|lm> = mħ|lm> for m = -l, -l+1, … l-1, l 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

ℓ = 2 ℓ = 1 mℓ = -2, -1, 0, 1, 2 mℓ = -1, 0, 1 While ℓx and ℓy are 1 While ℓx and ℓy are not absolutely certain, mℓ is! When two components (ℓ1 and ℓ2) form a system, their angular momentum must combine to preserve the total m1 + m2. If the two angular momenta actually align, ℓtot = ℓ1 + ℓ2 and mtot = -ℓ1- ℓ2 … ℓ1+ ℓ2. When the two angular momenta are oppositely directed, ℓtot = |ℓ1 - ℓ2| and mtot = -|ℓ1- ℓ2| … |ℓ1- ℓ2|. Jtotal = |ℓ1 + ℓ2| … |ℓ1- ℓ2 |

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

“psuedo-scalar” mesons BOSONS FERMIONS spin 0 spin ½ spin 1 spin 3/2 spin 2 spin 5/2 : : “psuedo-scalar” mesons p+, p-, p0, K+,K-,K0 quarks and leptons e, m, t, u, d, c, s, t, b, n 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, 

Example spin-1/2 electron in an l=2 orbital. Total J ? Either 3/2 or 5/2 possible

Particle properties/characteristics specifically their interactions are often interpreted in terms of CROSS SECTIONS.

 For elastically scattered projectiles: Ef , pf Ei , pi EN , pN The recoiling particles are identical to the incoming particles but are in different quantum states Ef , pf Ei , pi  EN , pN The initial conditions may be precisely knowable only classically! The simple 2-body kinematics of scattering fixes the energy of particles scattered through .

Nuclear Reactions or, if you prefer Whenever energetic particles Besides his famous scattering of  particles off gold and lead foil, Rutherford observed the transmutation: or, if you prefer Whenever energetic particles (from a nuclear reactor or an accelerator) irradiate matter there is the possibility of a nuclear reaction

Classification of Nuclear Reactions inelastic scattering individual collisions between the incoming projectile and a single target nucleon; the incident particle emerges with reduced energy pickup reactions incident projectile collects additional nucleons from the target O + d  O + H (d, 3H) Ca + He  Ca +  (3He,) 16 8 15 8 3 1 41 20 3 2 40 20 stripping reactions incident projectile leaves one or more nucleons behind in the target 90 40 Zr + d  Zr + p 91 40 (d,p) (3He,d) 23 11 Na + He  Mg + d 3 2 24 12

[ Ne]* 20 10 Predicting a final outcome is much like rolling dice…the process is random!