Energy is quantized as a consequence of the wave nature of matter

bounded by a confining potentialonly 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 systemshould 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.

Zeeman Effect(Dipole Interaction) Interaction of the nuclear magnetic dipole moment with the external applied magnetic field on the nucleus. 

As you sill see later there is a nuclear counter part:

can measure all the spatial (x,y,z) components

(and thus L itself) of vmrL

not even possible in principal !

rixyx

irL

,,

ix

yy

xiLzSo, for

example

azimuthalangle inpolar

coordinates

Angular Momentumnlml…

Lz lm(,)R(r) = mħ lm(,)R(r)

for m = l, l+1, … l1, l

L2lm(,)R(r)= l(l+1)ħ2lm(,)R(r)

l = 0, 1, 2, 3, ...

Measuring Lx alters Ly (the operators change the quantum states).The best you can hope to do is measure:

States ARE simultaneously eigenfunctions of BOTH of THESE operators!We can UNAMBIGUOULSY label states with BOTH quantum numbers

azrea

z /23

100

1

aZrea

Zr

a

Z 2/23

200 21

2

1

iaZr erea

Z

sin

8

1 2/25

121

cos2

1 2/25

210aZrre

a

Z

iaZr erea

Z sin

8

1 2/25

211

aZrea

rZ

a

Zr

a

Z 3/2

2223

300 21827381

1

Hydrogen Wave Functions

ℓ = 2mℓ = 2, 1, 0, 1, 2

L2 = 2(3) = 6|L| = 6 = 2.4495

ℓ = 1mℓ = 1, 0, 1

L2 = 1(2) = 2|L| = 2 = 1.4142

2

1

0

1

0

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

SPINORBITAL ANGULAR

MOMENTUMfundamental property

of an individual componentrelative motionbetween 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 particlesof 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

EVpm

2

2

1 total energy = sum of KE + PE

with the replacement of physical variables with “operators”

i

p

tiE

t

iVm

2

2though 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

yzxxz

xyzzy

zxyyx

LiLLLL

LiLLLL

LiLLLL

may be the more fundamental definition

Such “Commutation Rules”are recognized by mathematicians as

the “defining algebra” of a non-abelian

(non-commuting) group[ Group Theory; Matrix Theory ]

Reserving L to represent orbital angular momentum, introducing the more generic operator J to represent any or all angular momentum

yzxxz

xyzzy

zxyyx

JiJJJJ

JiJJJJ

JiJJJJ

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))2,1(),,;,,(

; 22211122222

;11111

zyxzyxj

smsmn

smsmn

under pairwise interchanges:

),,;,,(;)1,2()2,1( 11122211111

;22222

zyxzyxjs

msmns

msmn

Shouldn’t these state be indistinguishable?

Yes, but notice that only means must remain unchanged!2||

i.e. )2,1()1,2( iewith a distinguishing phase change!

222)2,1()2,1()1,2()2,1( ie

where obviously = 0,

)1,2(

)1,2()2,1(

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:

),;(;),;(; 12......21......11

;2222

;11

trrjtrrj nnnn

That’s OK for symmetric states, but for the anti-symmetric states:

)1,2()2,1(

)1,2()2,1(

with

0)2,1(

spin : 12p, n, e, , , e , , , u, d, c, s, t, b

leptons quarks

the fundamental constituents of all matter!

ms = ± 12

spin “up”spin “down”

s = ħ = 0.866 ħ 3 2

sz = ħ 12

Total Angular Momentumnlmlsmsj… l = 0, 1, 2, 3, ...

Lz|lm> = mħ|lm> for m = l, l+1, … l1, lL2|lm> = l(l+1)ħ2|lm>

Sz|lm> = msħ|sms> for ms = s, s+1, … s1, sS2|lm> = s(s+1)ħ2|sms>

In any coupling between L and S it is the TOTAL J = L + s that is conserved.

ExampleJ/ particle: 2 (spin-1/2) quarks bound in a ground (orbital angular momentum=0) stateExamplespin-1/2 electron in an l=2 orbital. Total J ?

Either3/2 or 5/2possible

ℓ = 2mℓ = 2, 1, 0, 1, 2

ℓ = 1mℓ = 1, 0, 1

2

1

0

1

0While ℓx and ℓy are

not absolutely certain, mℓ is!

When two components (ℓ1 and ℓ2) form a system,

their angular momentum must combine to preservethe 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 |

BOSONS FERMIONS

spin 1 spin ½, e,p, n,

Nuclei (combinations of p,n) can have

J = 1/2, 1, 3/2, 2, 5/2, …

BOSONS FERMIONS

spin 0 spin ½

spin 1 spin 3/2

spin 2 spin 5/2 : :

“psuedo-scalar” mesons

quarks and leptonse,, u, d, c, s, t, b,

Force mediators“vector”bosons: ,W,Z“vector” mesonsJ

Baryon “octet”p, n,

Baryon “decupltet”

Examplespin-1/2 electron in an l=2 orbital. Total J ?

Either3/2 or 5/2possible

Particle properties/characteristicsspecifically their interactions

are often interpreted in terms ofCROSS SECTIONS.

Ei , pi

Ef , pf

EN , pN

recoilNfiEEE

,

recoilNfippp

,

The simple 2-body kinematics of scattering fixes the energy of particles scattered through .

For elastically scattered projectiles:The recoilingparticles areidentical to

the incomingparticles but

are in differentquantum states

The initialconditions

may bepreciselyknowable

onlyclassically!

Nuclear Reactions

Besides his famous scattering of particles off gold and lead foil, Rutherford observed the transmutation:

OHHeN 17

8

1

1

4

2

14

7

OpN 17

8

14

7 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

• pickup reactionsincident projectile collects additional nucleons from the target O + d O + H (d, 3H)

Ca + He Ca + (3He,)

•inelastic scatteringindividual collisions between the incoming projectile and a single target nucleon; the incident particle emerges with reduced energy

2311

2412

Na + He Mg + d

16 8

15 8

31

4120

32

4020

32

9040

9140

Zr + d Zr + p (d,p)(3He,d)

•stripping reactionsincident projectile leaves one or more nucleons behind in the target

BB

BeC

LiN

O

HeO

pF

10

5

10

5

8

4

12

6

6

3

14

7

16

8

3

2

17

8

19

9

BB

BB

BeC

BeC

LiN

LiN

O

HeO

HF

dF

Ne

nNe

pF

11

5

9

5

10

5

10

5

9

4

11

6

8

4

12

6

7

3

13

7

6

3

14

7

16

8

3

2

17

8

3

1

17

9

18

9

20

10

19

10

19

9

2010[ Ne]*

Predicting a final outcome is much likerolling dice…the process is random!