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Modern Physics
Unit 14: Relativistic Kinematics
Lecture 14.2: Advanced Topics
Ron Reifenberger
Professor of Physics
Purdue University
1
What Next? – Topic I: Relativistic Schrodinger Equation
In free-space:
E=p2/2m
→
E ↔ i
∂
∂t
p ↔ −i ∇
∂
2 2

Schrodinger
(1926 ) : i Ψ = − ∇ Ψ (non − relativistic)
∂t
2m
In free space: E2=p2c2+m2c4 → E2-p2c2-m2c4=0
E −c
2
3
2
2
2 4
0
p
−
m
c =
∑ i
i =1
2
∂
Klein − Gordon (1926 ) : −  2 2 Ψ +  2 c 2 ∇ 2 Ψ − m 2 c 4 Ψ =0
∂t
Klein-Gordon describes a free, relativistic spinless particle. It
requires two initial conditions [Ψ(t=0) and ∂Ψ/∂t(t=0)] because of
2nd derivative wrt time. Therefore specifying a wavefunction Ψ at
t=0 is insufficient to uniquely specify the system’s behavior.
2
Dirac’s Relativistic Wave Equation
Dirac (1928) :
E = ( mc
)
3
2
2
i
o
=i 1 =i 1
2 2
+c
3
2
∑p
α mc + c ∑ α i pi
=
 
∂
imc 2
Ψ + cα ∇Ψ +
α o Ψ =0
∂t





=
α α1 x + α 2 y + α 3 z
 not true for numbers,

 OK for matrices !
four 1st order differential equations

ψ 1 (r , t ) 
ψ (r, t ) 
Ψ
=  2   Spinors
ψ 3 (r , t ) 
 

ψ
(
r
 4 , t )
Predicts spin and anti-particles!
3
The non-relativistic limit for H atom
Dirac’s Relativistic Wave Equation
in the non-relativistic limit takes the form
Relativistic
correction to KE
Spin-orbit
coupling
Energy correction if
electron and proton
are at same location
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What Next? – Topic II: Forces in Nature
Fundamental Forces


•
•
Gravitational
Electromagnetic
ElectroWeak
Strong (nuclear)

E
Traditional Field view: the field generated by
object No. 1 acts on object No. 2

E=
Q
r
2
4πε o r


F = q2 E
1
Q
- q
2
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Virtual particles
Object No. 1 emits a virtual (short-lived) exchange particle which
transfers momentum to object No. 2. (F=dp/dt)
Image from http://hepwww.rl.ac.uk/public/bigbang/file5.html
For every Force there is a Force carrier
 Gravitational – graviton?
 Electromagnetic – photon
ElectroWeak Force
 Weak – charged and neutral vector bosons (spin=1);
 Particles that decay by the Weak Interaction have lifetimes from 10-12 s to 10-16 s.
 Strong (between nucleons, between quarks) – pion (meson), gluons
 Particles that decay by the Strong Interaction have lifetimes from 10-16 s to 10-23 s.
 Lifetime of elementary particle measures imbalance of forces
But…….. we’re getting ahead of the historical story
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We Already Know about the Electronic Structure of Atoms
Periodic Table – Mendeleev 1869
Elements organized
by atomic number:
ℓ=0
atomic
number = Z
Organized by atomic
number AND chemical
activity:
ℓ=1
ℓ=2
How do we learn about the internal
(nuclear) structure of an atom?
7
Atomic Size
Scattering Experiments: The Basic Idea
What’s inside
the box?
B
A
Two Limiting Scenarios:
• Result A: Box is filled with soft, fluffy material
• Result B: Box is filled with hard, rigid material
8
Evidence for a Massive Nucleus:
Rutherford’s Scattering Experiments
(1911)
KEY QUESTION:
We know gold foil is made
of gold atoms that are
electrically neutral. But
how are the + and charges distributed?
(+)
Number of Events
A few of the particles are
backscattered!
Typical Rutherford Scattering
(172,000 events)
x10
4,000
3,000
2,000
1,000
0
0
45
90
135
180
Scattering angle
Conservation of energy and momentum imply the presence of
small “heavy objects”. Rutherford concludes that the “heavy
objects” are roughly ~10,000 times smaller than an atom!!
9
What we knew circa 1935
composition and approximate dimensions
Atom
(diameter ≅ 0.2 nm)
Nucleus
(diameter≅3 fm)
electrons
neutron
proton
1 fm = 1×10-15 m
class
name
hadrons
heavy composite particles
symbol
neutron (1931)
n
charge
0 (zero)
mass
spin
1.675 x 10-27 kg
1/2
proton (1919)
leptons
light elementary particle
electron (1897)
e-
P
+1.602 x 10-19 C
-1.602 x 10-19 C
1.673 x 10-27 kg
1/2
9.109 x 10-31 kg
1/2
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Classification of Sub-Atomic Particles
a) Historical – time of discovery
Positron, an anti-electron
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b) by mass
increasing mass
Leptons
(λϵπτον = thin coin; fermions)
elementary particles - no quarks!;
subject to ElectroWeak force, do
not feel strong nuclear force
Hadrons
(αδροζ = massive)
subject to strong nuclear force
Mesons
(μεσοσ = middle)
bosons
Baryons
(βαρυζ= heavy)
fermions; all baryons
decay into a proton
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Topics we will cover in the next few weeks
How to describe nuclear matter: Mass Density, Charge Density
How are nuclei held together: Isotopes, Binding Energy
How stable are nuclei: Binding Energy vs. Size
Nuclear Force and Nuclear Shell model
How do unstable nuclei disintegrate: Radioactivity, half-life
Can we predict how a nucleus will disintegrate : Q-factor
Predict whether a certain nuclear reaction will occur: Q-factor
Energetics of low energy nuclear reaction: Q-factor
Nuclear Energy: Fission/Fusion reactions
13
Up Next - Nuclear Characteristics
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