Transcript Document

Modern Physics (I)
Chap 3: The Quantum Theory of Light
Blackbody radiation, photoelectric effect, Compton effect
Chap 4: The Particle Nature of Matter
Rutherford’s model of the nucleus, the Bohr atom
Chap 5: Matter Waves
de Broglie’s matter waves, Heisenberg uncertainty principle
Chap 6: Quantum Mechanics in One Dimension
The Born interpretation, the Schrodinger equation, potential wells
Chap 7: Tunneling Phenomena (potential barriers)
Chap 8: Quantum Mechanics in Three Dimensions
Hydrogen atoms, quantization of angular momentums
Chapter 3: The Quantum Theory of Light
Emission of electromagnetic radiation by solids – continuous spectra
(Cf. emission and absorption spectra of atoms – discrete spectra)
科學發展月刊 2005/11
Wavelength of radiation near peak of emission spectrum determines color of object
500oC
700oC
1000oC
Radiation at wavelength
longer than optical
2500oC
Increase fraction in
optical wavelengths
hc
maxT 
 constant
5k B
The Problem to answer:
The problem is to predict the radiation
intensity at a given wavelength  emitted
by a hot glowing “solid” at a specific
temperature T (in thermal equilibrium)
To calculate the energy per unit volume
per unit frequency of the radiation within
the blackbody cavity
u = u (f, T )
Energy density in frequency range from f to f+df
1
u(f ,T)df   N ( f )df  
V

 : average energy per mode
P  , T   A(T )e
    P   d 
  / k BT
0
e   / k BT

k BT
Boltzmann distribution  Rayleigh-Jeans formula

 
  P  
 0

 P  
0



n 0
nhf Ae  nhf / kBT

 Ae

 nhf / k BT
hf
ehf / kBT  1
0
  nhf (n = 1, 2, 3, ......)
 Planck’s blackbody
radiation theory
Energy density in frequency range from f to f+df
2
1
8

f
hf


N
(

)
d


u(f )d
,Tdf  N ( f )df   3 hf / k BT
df
V
V
c e
1

5
1
e
hc /  k BT
1
d
To fit the data by Coblentz (1916)
Planck obtained h = 6.5710-34 J · s
uT()
u ,T  d  
8 hc
(h = 6.626 10-34 J  s)
h: a very small number which plays significant
roles in microscopic worlds
Planck thought that his concept of energy
quantization was merely a desperate calculational
device and moreover a device that applied only in
the case of blackbody radiation
Einstein elevated quantization to the level of a
universal phenomenon by showing that light itself
was quantized
 Photoelectric effect
Photoelectric Effect: Einstein’s quantization theory of light
Radiant energy is quantized and is localized in a small volume of space,
and that it remains localized as it move away from source with a speed of c
wave package
Radiant EM waves seems like many packages
(grains) of energy. Each has energy hf
Total radiant energy E = nhf
cutoff
VS
KEmax  eVs  hf  
When a light quantum hit an
electron, it can either be absorbed
completely or no reaction
0
f
1/e
2/e
3/e
metals 1, 2, 3
1 < 2 < 3
Compton Effect
(1922)
Between 1919–23, Compton showed that x-rays collide elastically with
electrons, in the same way that two particles would elastically collide
What does this tell us ?
• Light “particles” (photons) carry momentum !
P
E photon
E is the photon energy
c
c is the speed of light
• Earlier result that Ephoton= hf = hc/
P
h
h is Planck’s constant

 is the wavelength of light
graphite (carbon)
Experimental details
A beam of x-ray of wavelength o is scattered through an angle 
by a metallic foil, the scattered radiation contains a well-defined
wavelength  which is longer than o
o=0.0709nm
Photon can scatter off matter

When   0, one more peak with
 > o appears.  depends on 
    o 
h
1  cos 
mec
 0.00243 1  cos  [nm]
Collision of particles
ŷ
after
collision
before collision
x-ray
E o  hf o
Po 
hf o
c
x-ray


x̂
e-
e-
E e  m ec 2
E  hf
E'e  me2c4  Pe2 c2
Pe   0
P  hf
Pe  0
Conservation
of momentum
c
hf o hf
 cos   Pe cos 
c
c
Conservation of energy
hf
sin   Pe sin 
c
hf o  mec2  hf  me2c4  Pe2 c2
Summary:
Planck: energy quantization of oscillators
in the walls of a perfect radiator
Einstein: extension of energy
quantization to light in the photoelectric
effect
Compton: further confirmation of the
existence of the photon as a particle
carrying momentum in x-ray scattering
experiments
Rutherford’s model of the Nucleus
The Bohr Atom
Constituents of atoms (known before 1910)
There are electrons with measured charge and mass
There are positive charge to make the atom electrical neutral
The size of atom is known to be about 10-10 m in radius
Rutherford’s scattering experiment
Projectile:  particle with charge +2e
Target: Au foil
KE  5 MeV !
Rutherford’s -particle Scattering
Experiment (1911)
To probe the distribution of the
positive charge with a suitable
projectile
How is the mass of the positive charge distributed within
the atom?
Experimental results: (Geiger and Marsden)
99% of deflected  particles have deflection angle   3o
However, there are 0.01% of  particles have larger angle  > 90o
Rutherford’s model of the structure of the atom to explain the
observed large angle scattering
A single encounter of  particle with a massive charge
confined to a volume much smaller than size of the atom
Nucleus
-14
r ~ 10
4
m  10 R
All positive charges and essentially all its mass are assumed to
be concentrated in the small region
Rutherford’s scattering model
(r, )
(m,v)
fast, massive
 particles
r  r (t )
   (t )
b


b: impact parameter
Trajectory of  particle (r, )
Ze+
 2e  Ze  rˆ  m  d 2r  r  d 2  rˆ
Deflection due to
Coulomb interaction:
4 o r
P( )d
2
 2
 dt

 
 dt  
# of  particles detected by
detector at scattering angle 
 d
N D ( )d  NP(b)db 
1
sin 4  / 2 
# of  particles detected by detector
at scattering angle 
N D ( )d 
ND
1
sin 4  / 2 
sin 4  / 2
In the case when the KE of the  particle is so high that the
equation begins to fail, this distance of the closest approach
is approximately equal to the nuclear radius
mv 2 (2e)( Ze)

2
4 o D
D  5  1015 m
(Rutherford assumed that  particles do not penetrate the nucleus)
Rutherford Scattering:
Rutherford’s calculations and procedures laid the
foundation for many of today’s atomic and nuclear
scattering experiments
By means of scattering experiments similar in
concepts to those of Rutherford, scientists have
elucidated (1) the electron structure of the atom,
(2) the internal structure of the nucleus, and even
(3) the internal structure of the nuclear
constituents, protons and neutrons
Einstein:
Splitting the atom by bombardment is like
shooting at birds in the dark in a region where
there are few birds
Schematics of energy levels and radiated spectrum of H atom
1890
Rydberg & Ritz formula
1
nm
1 
 1
 R  2  2 
n m 
R  1.0968  107 m 1
n, m integers with n < m
Bohr’s quantum model of the Atom (1913)
Four postulates：
1. An electron in an atom moves in a circular orbit about the
nucleus under the influence of the Coulomb attraction
between the electron and the nucleus
2. The allowed orbit is a stationary orbit with a constant energy E
3. Electron radiates only when it makes a transition from one stationary
state to another with frequency
f 
Ei  E f
h
4. The allowed orbit for the electron: L  n  n
h
. The quantum
2
number n labels and characterizes each atomic state
n = 1, 2, 3, …… (“quantum number”)
Bohr atom
e-
Consider an atom consists of nucleus with +Ze protons
and a single electron –e at radius r
r
1 Ze2 mv 2

2
4 o r
r
Ze
Coulomb attraction
Orbital angular momentum
L
n
v

mr mr
Centripetal force
Ln r mv
n = 1, 2, 3, …
n 2  4 o 2   n 2 
  ao
Radius of allowed orbit: r 


2
Z  e m Z 
Quantized
orbits !!
ao  Bohr radius = 0.529 Å
For n = 1 and Z = 1,
r = ao = 0.529 10-10 m
Correct prediction for atomic size !!
e-
Ze
Total Energy of the electron
1 Ze2
mv 2 
1 ze2 
E  KE  U 
 
   8 r
2
o
 4 o r 
Z 2  e2 
Z2
En   2 
   2 Eo
n  8 oa0 
n
Eo  13.6 eV
 n2 
r   ao
Z
(3)
f 
Ei  E f
h
Z 2 Eo 1
1
c


2
2 
h n f ni

1
1
1
 Z R 2  2

n f ni
Allowed transition
Rydberg constant
2
Eo
R 
 1.097  107 m 1
hc
Good to describe the observed spectra of any Hydrogen-like atom
with nucleus charge +Ze and a single orbital eH, He+, Li2+, …
The Bohr atom—
“Bohr’s original quantum theory of spectra was one of the
most revolutionary, I suppose, that was ever given to
science, and I do not know of any theory that has been
more successful …… I consider the work of Bohr one of the
greatest triumphs of the human mind.” (Lord Rutherford)
“This is the highest form of musicality in the sphere of
though.” (Einstein)
「他不但具有關於細節的全部知識，而且還始終堅定的注視著
基本原理。」(Einstein)
Franck-Hertz Experiment
(1914)
To observe current I to
collector as a function of
accelerating voltage Va
(6 V)
Accelerating
voltage (0–40 V)
Retarding
voltage (1.5 V)
Vdrop  n  4.9 V   Vo
When the tube is filled with low
pressure of mercury vapor, there
are collisions between some
electrons and Hg atoms  Peaks
in current I with a period of 4.9 V
Low-energy electrons ( a few tens
of volt)
Incoming
electron
Orbital e-
nuclear
Scattered
electron
Inelastic collision, 4.9 eV of KE of
incident electron raises Hg electron
from the ground state to the first
excited state
Inelastic collision leaves electron with
less than Vs, so the electron cannot
contribute to current
Energy levels of outer electron of Hg atom
E=0
10.4 eV
6.7 eV
2nd excited state
1240 eV nm

 253 nm
4.9 eV
1st excited state
4.9 eV
Ground state
Confirmed by emission
of single photons !!
Significance of the Franck-Hertz Experiment
The Franck-Hertz provided a simpler and more direct
experimental proof of the existence of discrete energy levels
in atoms
The experiment confirmed the universality of energy
quantization in atoms, because the quite different physical
processes of photon emission (optical line spectra) and
electron bombardment yielded the same energy levels
Summary:
D
Rutherford’s scattering of  particles from gold atoms
Bohr’s model provides the explanation of the motion
of electrons within the atom and of the rich and
elaborate series of spectral lines emitted by the atom
Chapter 5: Matter Waves
de Broglie’s intriguing idea of “matter wave” (1924)
Extend notation of “wave-particle duality” from light to matter
For photons, P  E  hf  h
c
Suggests
for matter,
c

The wavelength is detectable
only for microscopic objects
h

de Broglie wavelength
P
P: relativistic
momentum
E
f 
de Broglie frequency
h
E: total relativistic
energy
The Davisson-Germer Experiment (1927)
a clear-cut proof of wave nature of electrons
1
h
h
2
mev  eV   

2
mev
2eVme
d sin   q
Applying Ni atoms as a reflecting
diffraction grating
a constructive
peak
in excellent agreement
with the de Broglie
formula !!
50o
Kept detector at a fixed angle  and varied the accelerating voltage V
d sin   1  qq
(q = 1, 2, 3, ……)
1
h
q  
q
2meVq
Constructive peaks occur at
wavelengths: q = 1 /q


h
Vq  q 
  q  const.
 1 2me 
Experiment of Davisson and Germer confirmed that low-energy
electrons with mass (v << c) do have wave-like properties
Wave groups and Dispersion
Toward a Wave description of Matter
Particle
(波群與色散)
“wave packet”
m
v
Large probability to be found in a small region
of space at a specific time t
vg = v
Wave representation
“Wave group” or summed collection of waves
with different wavelengths: amplitudes and
relative phases chosen to produce
constructive interference in small region
Δx
the group velocity of the matter wave = the velocity of the particle
a k  



f ( x )e ikx dx

波
數
值
越
密
集
，
波
週包
期在
性空
越間
大的
f ( x) 
ikx
a
k
e
dk




Matter waves are represented by wavefunctions:
(x,y,z,t )
(a solution for the Schrodinger equation)
Matter waves is not measurable; they require no medium
for propagation
(x,y,z,t ) is a complex number and is used to calculate
the probability of finding the particle at a given time in a
small volume of space
The statistical view (Max Born): the probability of finding
a particle is directly proportional to ||2 = 
The Heisenberg Uncertainty Principle
(1927)
It is impossible to determine simultaneously with unlimited
precision the position and momentum of a particle
If a measurement of position x is made with an uncertainty x
and a simultaneous measurement of momentum Px is made
within an uncertainty Px, then the precision of measurement
is inherently limited by
Px x  /2 (momentum-position uncertainty)
Similarly,
E t  /2 (energy-time uncertainty)
Double-slit electron
diffraction experiment
first minimum:

D sin min   / 2
wave
properties
While the electrons are detected as particles at a
localized spot at some instant of time, the
probability of arrival at that spot is determined by
finding the intensity of two interfering matter waves
particle
properties
Accumulated results with
each slit closed half the
time
The experimental result
contradicts this sum of
probability !!
A thought experiment: Measuring through which slit
the electron passes
py
Once one measures unambiguously which slit the
electron passes through, the act of measurement
disturbs the electron’s path enough to destroy the
interference pattern
Summary
The existence of matter waves (de Broglie)
Davisson-Germer experiment (electron diffraction from Ni crystal)
Constructing “wave packets” by superposition of matter waves
with different frequencies, amplitudes, and phases
Uncertainty principles
Wave-particle duality; double-slit electron diffraction experiment
Need a new mechanics that incorporates both
wave and particle natures of subatomic objects