Transcript Physics 320: Astronomy and Astrophysics – Lecture V

Physics 320: Astronomy and
Astrophysics – Lecture V
Carsten Denker
Physics Department
Center for Solar–Terrestrial Research
NJIT
The Interaction of Light and Matter
 Spectral
Lines
 Photons
 The
Bohr Model of the Atom
 Quantum Mechanics and Wave–Particle
Duality
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Electromagnetic Spectrum
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Spectral Lines
 Auguste
Comte 1835 in Positive Philosophy:
We see how we may determine their forms
their distances, their bulk, their motions,
but we can never know anything of their
chemical or minerological structure.
 William Wollaston, Joseph Fraunhofer,
Robert Bunsen, Gustav Kirchhoff, … 
spectroscopy
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Kirchhoff’s Laws
 A hot
(< 0 K), dense gas or solid object
produces produces a continuous spectrum
with no dark spectral lines.
 A hot, diffuse gas produces bright spectral
lines (emission lines).
 A cool, diffuse gas in front of a source of a
continuous spectrum produces dark
spectral lines (absorption lines) in the
continuous spectrum.
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Spectroscopy
Prisms
 Diffraction gratings



Transmission grating
Reflection grating
d sin   n and n  1, 2, 3, ...
 

nN

 nN

Resolving
power
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October 1st, 2003
Photoelectric Effect
Ephoton  h 
hc

K max  Ephoton    h   
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hc


October 1st, 2003
Compton Effect
Ephoton  h 
hc

 pc
h
   f  i 
1  cos 
me c
Compton wavelength
c 
h
 0.0243 Å
me c
In a collision between a photon and an
electron initially at rest, both the (relativistic)
momentum and energy are conserved.
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October 1st, 2003
The Bohr Model of the Atom
 Wave–particle
duality of light
 Rutherford 1911   Au: It was quite the
most incredible event that ever happened to me
in my life. It was almost as incredible as if you
fired a 15–inch shell at a piece of tissue paper
and it came back an hit you.  discovery of a
minute, massive, positively charged
atomic nucleus
 Proton: mp = 1836  me
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Group Assignment
Problem 5.7
 Verify
that the units of Planck’s constant
are the units of angular momentum!
 m
m2 
L  mvr  kg m = kg

s
s


E J
m
m2 
E  h  h 
 -1 = Js = Nm s = kg 2 m s = kg

 s
s
s 
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October 1st, 2003
Hydrogen Atom
1 1 
 RH   2  and RH  109677.585  0.008 cm 1

4 n 
1
1 
 1
 RH  2  2  and m  n

m n 
1
Planetary model of
the hydrogen atom?
m=1
UV [122, 103, 97, …] nm
Lyman
m=2
Visible [656, 486, 434, …] nm
Balmer
m=3
IR [1875, 1282, 1094, …] nm
Paschen
m=4
IR [4051, 2625, 2165, …] nm
Brackett
m=5
IR [7458, 4652, …] nm
Pfundt
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October 1st, 2003
Bohr’s Postulates
 Only
orbits are stable, where the angular
momentum of the electron is quantized
L = nh/2=nħ, and will not radiate in spite
of the electron’s acceleration.
 Every allowed orbit corresponds to a
distinct energy level and the transition
from a distant orbit to an orbit closer to
the nucleus Ephoton = Ehigh – Elow results in
the emission of an energy quantum, i. e., a
photon.
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Bohr Atom
1 q1q2
FE 
r
3
4 0 r
Coloumb’s law
me mp
(me )(1836me )


 0.9994556me
me  mp
me  1836me
M  me  mp  me 1836me  1837me
M
mp and 
Reduced mass
Total mass
me
1 q1q2
v2
1 e2
v2
F  a 
r   2 r 

3
2
4 0 r
r
4 0 r
r
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
1 2
1 e2
1 e2
 K  v 
and U  
 2K
2
8 0 r
4 0 r
1 e2
 E  K  U  K  2K   K  
8 0 r
L   vr  n and
h
=
2
Quantization of angular
momentum
1 e
1 2 1   vr 
1 n 
K
 v 

2
8 0 r 2
2 r
2 r 2
2
rn  4 0
2
 e2
2
2
n2  a0 n2 and a0  5.29 1011 m  0.529 Å
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Bohr Atom (cont.)
1 e2
 e4 1
1
En  
  2 2   13.6 eV  2
8 0 rn
2 n
n
E1  13.6 eV
r1  a0  0.529 Å
E2  E1 / 4  3.40 eV r2  4a0  2.12 Å
Ephoton  Ehigh  Elow
  e4 1    e4 1 

  2 2  2 2 
  2 nhigh   2 nlow 
hc
 e4  1
1
 
 2
3 
2

 4 c  nhigh nlow
1
EH

 1
1
  RH  2  2

 nhigh nlow

 e4
 and RH 
3
4

c

hc
1 1
 (13.6 eV)  2  2   1.89 eV   
 6565 Å
EH
3 2 
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Kirchhoff’s Laws Revisited

A hot, dense gas or hot solid object produces a continuous spectrum
with no dark spectral lines. This is the continuous spectrum of black
body radiation, described by the Planck functions B(T) and B(T),
emitted at any
temperature above
absolute zero. The
wavelength max at
which the Planck
function B(T) obtains
its maximum is given
by Wien’s
displacement law.
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Kirchhoff’s Laws Revisited (cont.)


A hot, diffuse gas produces bright emission lines. Emission lines are
produced when an electron makes a downward transition from a
higher to a lower orbit. The energy lost by the electron is carried
away by the photon.
A cool, diffuse gas in front of a source of continuous spectrum
produces dark absorption lines in the continuous spectrum.
Absorption lines are produced when an
electron makes a transition from a lower
to a higher orbit. If the incident photon in
the continuous spectrum has exactly the
right amount of energy, equal to the
difference in energy between a higher
orbit and the electron’s initial orbit, the
photon is absorbed by the atom and the
electron makes an upward transition to
the higher orbit.
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Quantum Mechanics
and Wave–Particle Duality
E

h
De Broglie frequency
h

p
De Broglie wavelength
1
x p 
2
1
E t 
2
x p 
or
E t 
NJIT Center for Solar-Terrestrial Research
Heisenberg’s uncertainty principle
October 1st, 2003
Problem 4.5
(a)
(b)
(c )
u
1
u2
 0.8   1  2  0.6 and Lmoving  60 m
c

c
t P  Lmoving / 0.8c  0.25 μs
60 m
Lrest   Lmoving 
 100 m
0.6
1
Lmoving  Lrest  0.6  60 m   36 m

(d ) tT  100 m/0.8c  0.417 μs
(e) Lrest  Lmoving  100 m  36 m  64 m
 tT  64 m / 0.8c  0.267 μs
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Problem 4.13
v A  u  0.8c and vB  0.6c (Frame of reference @ rest = Earth)
vB 
vB  u
1  uvB / c 2
(Eqn. 4.40)
0.6c  0.8c
 0.946c
2
1  (0.8c)( 0.6c) / c
 vA  0.946c
=
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Problem 4.18
E   mc 2 (Eqn. 4.46)
 E 2  ( mc 2 )2

 E  m c   mc
2
2 4
2
 
2
 mc
2
 
2

  mc 2  mc 2  mc 2  mc 2

(Eqn. 4.48)  p 2c 2  mc 2 (1   )mc 2 (1   )  Kmc 2 (1   )
p2
p2
K 
K 
if v
m(1   )
2m
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c
October 1st, 2003
Homework Class Project
 Prepare
a 200 – 250 word abstract for one
of the five topics mentioned in the
storyline
 Important scientific facts
 Form of presentation
 Learning goals
 Homework is due Wednesday October 8th,
2003 at the beginning of the lecture!
 Exhibition name competition!
NJIT Center for Solar-Terrestrial Research
October 1st, 2003
Homework
is due Wednesday October 8th,
2003 at the beginning of the lecture!
 Homework assignment: Problems 5.4, 5.5,
and 5.15
 Late homework receives only half the
credit!
 The homework is group homework!
 Homework should be handed in as a text
document!
 Homework
NJIT Center for Solar-Terrestrial Research
October 1st, 2003