1. 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!

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

3. 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. 

4. 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

5. 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

6. Hydrogen Wave Functions

7. ℓ = 2
mℓ = -2, -1, 0, 1, 2
ℓ = 1
mℓ = -1, 0, 1 2 1 1
L2 = 1(2) = 2
|L| = 2 =
L2 = 2(3) = 6
|L| = 6 =
Note the always odd number
of possible orientations:
A “degeneracy” in otherwise identical states!

9. 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

10. 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

11. 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!

12. 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!

13. 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, 

14. 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

15. 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 = ħ 3 2
ms = ± 1 2
sz = ħ 1 2

16. 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

17. ℓ = 2 ℓ = 1 mℓ = -2, -1, 0, 1, 2 mℓ = -1, 0, 1 While ℓx and ℓy are1
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 |

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

19. “psuedo-scalar” mesons
BOSONS FERMIONS
spin spin ½
spin spin 3/2
spin 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, 

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

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

22.  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 .

23. 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

24. 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

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