energy is quantized as a consequence of the wave nature of matter bounded by a confining potential...
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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!
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Energy-level splitting in a magnetic field for the 2P3/2, 2P1/2, and 2S1/2 energy levels for Sodium.
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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:
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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
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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
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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
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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!
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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
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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
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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!
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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!
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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,
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)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(
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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
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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
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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 |
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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, …
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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”
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Examplespin-1/2 electron in an l=2 orbital. Total J ?
Either3/2 or 5/2possible
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Particle properties/characteristicsspecifically their interactions
are often interpreted in terms ofCROSS SECTIONS.
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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!
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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
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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
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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!