PHY 102: Quantum Physics
Topic 2
EM Radiation from atoms
•Broadband Thermal Radiation
•“Blackbody” spectrum
•Resolution of “ultraviolet catastrophe”
•Atomic line spectra
•Structure of the atom: Rutherford scattering
Thermal Radiation
•Heat is associated with vibrational thermal motion of
atoms/molecules
•General principle: accelerating charged particles
generate electromagnetic radiation (examples:
generation of radio waves by moving electrons in
antenna, generation of continuous X-ray spectrum by
electrons decelerated by interaction with atoms of metal
target)
•So, e.m. radiation is generated by the thermally
induced motion of atoms/molecules: THERMAL
RADIATION….
Thermal Radiation
•Unlike convection and conduction, transfer of heat by
thermal radiation doesn’t require a “medium”
•So, for example, heat can reach Earth from the Sun
through millions of kilometres of empty space.
•Rate at which an object, surface area A, temperature T,
radiates energy is given by Stefan’s Law
P  AeT 4
= “Stefan’s constant” = 5.67 x 10-8 Wm-2K-4
e = “emissivity” ; 0< e < 1, depending on nature of surface
For a “black body” (perfect emitter/absorber), e=1
Spectrum of emitted radiation
Black body emission
spectrum for various
temperatures
•Peak wavelength decreases
with increasing temperature
•Area under curve (total
emitted power increases with
increasing temperature
•Experimentally, the
dependence of peak
wavelength on temperature is
found to be given by:
 (m)
 pT  constant  2.90 103 m.K
“Wien’s displacement law”
Modelling the black body spectrum
•Rayleigh attempted to calculate the black body spectra from
solids by assuming the radiation to originate from classical
EM standing waves (“normal modes”) within the object
•Standing wavelength in one dimension for cubic box of side
L:
2L

n
•wavenumber:

•n = 1,2,3,………………..

2 n
k


L
Modelling the black body spectrum
•Rayleigh attempted to calculate the black body spectra from
solids by assuming the radiation to originate from classical
EM standing waves (“normal modes”) within the object
•Standing wavelength in one dimension for cubic box of side
L:
2L

n
•wavenumber:

•n = 1,2,3,………………..

2 n
k


L
Modelling the black body spectrum
•In 3 dimensions, ktotal given by:
ktotal2  k x2  k y2  k z2
•for cube of side L:

n1
n 2
n 3
kx 
;k y 
;k z 
L
L
L
•In “k-space” each allowed state occupies a “volume” 3/L3

“k-space”
How much energy is emitted?
• energy emitted in a wavenumber range
between k and k + dk,
Number of modes in wavenumber range x
energy per mode
Number of modes
“Volume” of spherical shell between
k and k + dk?
“Volume” occupied by each mode in
k-space?
4 k 2 dk
Vk 2
dN(k) 
 2  2 dk
3
8 3

L
Factor of 2 for two different independent polarizations per mode,
V=L3

Energy per mode?
According to classical equipartition theory,
each mode (classical oscillator) has energy
E = kbT
So…………….
dE(k) 
2
Vk kb T

2
dk
Changing variables to wavelength:
dE(k) 
2
Vk kb T

2
dk
Vk 2 kb T dk
V 4 2 kb T 2
dE( ) 
d 
 2 d
2
2 2


d



8Vk b T

4
dE( ) 8Vkb T
d 

4
d

Spectral Intensity (Rayleigh prediction)
• Defined as radiated power per unit
wavelength volume per unit area per unit
time:
1 dE( )
I( ) 
 c  geometrical factors
V d
• Rayleigh-Jeans result:

I( ) 
2ckT

4
Spectrum of emitted radiation
Black body emission
spectrum for various
temperatures
•Peak wavelength decreases
with increasing temperature
•Area under curve (total
emitted power increases with
increasing temperature
•Experimentally, the
dependence of peak
wavelength on temperature is
found to be given by:
 (m)
 pT  constant  2.90 103 m.K
“Wien’s displacement law”
“ULTRAVIOLET CATASTROPHE”…………….
Rayleigh classical theory doesn’t work.
Classical vs Quantum
Classical (Rayleigh-Jeans) picture:
•EM modes have continuous spread of energies
•Average energy of oscillator at temperature T = kT
Quantum (Planck) picture:
•EM modes only allowed to have energy in integer
multiples of some constant times the oscillator
frequency: E = nhf
•Average energy of oscillator at temperature T:
hf
E
hf
e
kT
1
Modelling the black body spectrum
Obtain expression for spectral intensity by taking
product of average energy per oscillator and
number of oscillator modes per unit volume…….
Planck result:
2hc 2
I ( )  5 hc kT
 e
1


•This model predicts the form of the blackbody spectrum perfectly, no
“UV catastrophe”
•First experimental “anomaly” to be explained by the need for a
quantum theory (1900)
•“h” originally introduced by Planck purely as an empirical constant to
fit data…………………………
2hc
I( )  5 hc kT
 e
1
2
2.50E+10

1.50E+10
1.00E+10
5.00E+09
9.8
9.2
8.6
8
7.4
6.8
6.2
5
5.6
4.4
3.8
3.2
2.6
2
1.4
0.8
0.00E+00
0.2
I(W/m3)
2.00E+10
Wavelength/m
Quantum theory gives excellent agreement with experiment.
Line spectra
•“Hot” solids and liquids display the continuous emission spectra
described above
•“excited” gases display something completely different: LINE SPECTRA
Line spectra
•Line spectrum of a gas of atoms/molecules is reproducible, and is
a unique “fingerprint” of the gas
•Suggests that the spectrum is somehow related to the internal
structure of the atom……….
•So, what is an atom???
The atom: a brief (incomplete) history
Leucippus of Miletus, Democritus (~450BC)
Suggest universe composed of hard, uniform, indivisible particles
and the space between them (“atom” ≈ “cannot be cut”)
Pierre Gassendi (1592-1655), Robert Boyle (1627-1691)
Matter composed of rigid, indestructible atoms, varied size and
form, different elements composed of different atoms, atoms can
combine to form molecules……….
Joseph Louis Proust (1754-1826), John Dalton (1766-1844)
“Law of definite proportions”, atomic picture of chemical
processes, stoichiometry
Lothar Meyer (1830-95), Dmitry Mendeleev (1834-1907)
Significance of atomic weights, Periodic Table of the elements
The atom: a brief (incomplete) history
So, by the 19th century, it was universally accepted that matter was composed
of atoms. But we still haven’t answered the question. What is an atom?
1897: JJ Thomson discovers electron, measures ratio e/m
1907: Millikan measures charge on electron
~1910: Thomson’s “plum pudding” model of the atom
1910-1911: Rutherford, Geiger and Marsden clarify internal structure of
atom by scattering of positively charged -particles…………..
Rutherford Scattering
Most particles
pass straight
through, or
deflected
only slightly
Some particles
deflected back
through large
angles
Rutherford Scattering
To explain results of the Rutherford scattering :
1) Atom must be mostly empty space
2) Positive charge must be concentrated in a small volume occupying a
very small fraction of the total volume of the atom…………
Christmas pudding model
doesn’t work
Nuclear model does work
Atomic radius ~ 10-10m
Nuclear radius ~ 10-14m
The Rutherford/Bohr Model
•More on line spectra
•Orbital model of the hydrogen atom
•Failure of classical model
•Quantisation of orbital angular momentum: stationary states
•Successes and failures of the Bohr Model
Line Spectrum of hydrogen
•Hydrogen has line spectrum ranging in wavelength from the
UV to the infrared
•Balmer (1885) found that the wavelengths of the spectral
lines in the visible region of the spectrum could be
EMPIRICALLY fitted to the relationship:
m2
 (nm)  364.6 2
: m  3, 4, 5.........
m 4
(The group of hydrogen spectral lines in the visible region still known
as the Balmer Series)
Line Spectrum of hydrogen
•Rydberg and Ritz subsequently obtained a more general
expression which applies to ALL hydrogen spectral lines (not
just visible), and also to certain elements (eg alkaline
metals):
1 1
 R 2  2 

 n2 n1 
1
n2, n1 integers, n2 < n1
•R is called the Rydberg constant, which changes slightly
from element to element.
•For hydrogen, RH = 1.097776 x 107 m-1
•Can a model of the atom be developed that’s consistent
with this nice, elegant formula??
Rutherford Scattering
To explain results of the Rutherford scattering :
1) Atom must be mostly empty space
2) Positive charge must be concentrated in a small volume occupying a
very small fraction of the total volume of the atom…………
Christmas pudding model
doesn’t work
Nuclear model does work
Atomic radius ~ 10-10m
Nuclear radius ~ 10-14m
Rutherford “planetary” model
Basic idea: electrons in an atom orbit the positively-charged
nucleus, in a similar way to planets orbiting the Sun
(but centripetal force provided by electrostatic attraction rather
that gravitation)
Hydrogen atom: single electron orbiting positive nucleus of
charge +Ze, where Z =1:
v
r
-e
+Ze
F
Rutherford Model: electron energy
v
r
-e
+Ze
F
From electrostatics, the potential energy of the electron is given by:
q1q2
( Ze)  (e)  Ze 2
U


40 r
40 r
40 r
Rutherford Model: electron energy
v
r
-e
+Ze
F
Centripetal force equation:
me v 2
Ze 2

r
40 r 2
Kinetic energy of electron:
me v 2
Ze 2

2
80 r
Total energy of electron = P.E. + K.E:
q1q2
( Ze)  (e)  Ze 2
U


40 r
40 r
40 r
me v 2
Ze 2

2
80 r
 Ze 2
Ze 2
Ze 2
Total energy 


40 r 80 r
80 r
But this classical treatment leaves us with a big problem………
Failure of the Classical model
The orbiting electron is an accelerating
charge.
Accelerating charges emit
electromagnetic waves and therefore
lose energy
Classical physics predicts electron
should “spiral in” to the nucleus
emitting continuous spectrum of
radiation as the atom “collapses”
CLASSICAL PHYSICS CAN’T GIVE
US STABLE ATOMS………………..
Bohr’s postulates
• Only certain electron orbits are allowed, in which the
electron does not emit em radiation (STATIONARY
STATES)
•An atom emits radiation only when an electron
makes a transition from one stationary state to
another.
•The frequency of the radiation emitted when an
electron makes a transition from a stationary state
with energy E2 to one with energy E1 is given by:
E2  E1
f 
h
Transition energies
Suppose an electron is initially in stationary state with energy E1, orbital
radius r1. It then undergoes a transition to a lower energy state E2, with
(smaller) radius r2:
Ze 2
Ze 2
Ze 2  1 1 
  
E1  E2  


80 r1
80 r2 80  r2 r1 
If Bohr’s postulates are correct, then the frequency of the radiation
emitted in the transition is given by:
E1  E2
Ze 2  1 1 
  
f 

h
8h 0  r2 r1 
Rydberg-Ritz Revisited
1 1
 R 2  2 

 n2 n1 
1
Bohr result:
c = fλ
 1 1
f  cR 2  2 
 n2 n1 
Ze 2  1 1 
  
f 
8h 0  r2 r1 
Looks promising, if we can make the connection that r is somehow proportional to
“integer squared”……………….
Quantisation of angular momentum
Bohr now makes the bold assumption that the orbital angular momentum
of the electron is quantised………
Since v is perpendicular to r, the orbital angular momentum is just given
by L = mvr.
Bohr suggested that this is quantised, so that:
nh
mvr 
 n
2
IMPLICATIONS???..........................................................................
Kinetic energy (earlier slide)
me v 2
Ze 2

2
80 r
2
Ze
v2 
40 me r
quantisation of A.M. (last slide)
nh
mvr 
 n
2
2
2
n
v  2 2
r m
2
n  40
r
2
me Ze
2
2
2
2
Ze
n
 2 2
40 me r r m
2
Bohr radius
So, introduction of the idea that angular momentum is quantised
has the desired effect: rn2. Simplifying the expression for r a bit
(Z=1 for hydrogen):
n h 0
2
r

r

n
a0
n
2
me e
2
2
a0, the radius of the n=1 orbit, is called the BOHR RADIUS
h 0
a0 
2
me e
2
We conclude that in the Bohr model only certain orbital radii (and
electron velocities) are allowed.
e2  1 1 
  
f 
8h 0  r2 r1 
e 4 me  1 1 
f  3 2  2  2 
8h  0  n2 n1 
R=1.07 x 107 m-1
How nice.
e2  1 1 
 2  2 
f 
8h 0 a0  n2 n1 
Rydberg-Ritz
 1 1
f  cR 2  2 
 n2 n1 
Origins of hydrogen spectral lines:
Bohr Model: Shortcomings
•The Bohr model does an excellent job of explaining the “gross”
features of hydrogen line spectrum
BUT
•Doesn’t work well for many-electron atoms (even helium)
•Can’t explain fine structure of spectral lines observed at high
resolution, or relative intensities of spectral lines
•Can’t explain effect of magnetic field on spectral lines (Zeeman
effects), although Sommerfeld’s modifications (elliptical orbits,
varying orientations) help to some extent
•Is fundamentally inconsistent with Heisenberg’s uncertainty
principle
THE BOHR MODEL IS WRONG