# lecture27

```The photon, the quantum of light
E  pc  hf - energy
hf h
p

- moment um
c 
Photon Interactions
Photons passing through matter can undergo the following interactions:
• Photoelectric effect: photon is completely absorbed, electron is ejected
• Absorption: photon may be totally absorbed by electron, but not have
enough energy to eject it; the electron moves into an excited state
• Scattering: the photon can scatter from an atom and lose some energy
• Pair Production: the photon can produce an electron-positron pair.
In pair production, energy, electric charge, and momentum must all be conserved.
Energy will be conserved through the mass and
kinetic energy of the electron and positron;
their opposite charges conserve charge; and
the interaction must take place in the
electromagnetic field of a nucleus, which can
contribute momentum.
Example: Rank the following radiations according to their associated photon
energies, greatest first:
(a) yellow light from a sodium vapor lamp
(b) a gamma ray emitted by a radioactive
(c) a radio wave emitted the antenna of a commercial radio station
(d) a microwave beam emitted by airport traffic control radar
Example: At what rate are photons emitted by a 100 W sodium vapor lamp if
we can assume the emission is entirely at a wave-length of 590 nm?
P  100W
  590nm
rate  P / E  ?
f 
c

E  hf 
hc




P P
100W   590109 m
rate  

E hc
6.631034 J  s 3.0 108 m / s

rate  3 1020 photons/ s


Wave-Particle Duality; the Principle of Complementarity
We have phenomena such as diffraction and interference that show
that light is a wave, and phenomena such as the photoelectric effect
that show that it is a particle.
Which is it?
This question has no answer; we must accept the dual wave-particle
nature of light.
The principle of complementarity states that both the wave and
particle aspects of light are fundamental to its nature.
Indeed, waves and particles are just our interpretations of how
light behaves.
Wave Nature of Matter
Just as light sometimes behaves as a particle, matter sometimes behaves like
a wave. (Idea of the symmetry in nature - de Broglie 1927)
The wavelength of a particle of matter (De Broglie wavelength) is:
•
This wavelength is extraordinarily small.
•
The wave nature of matter becomes more important for very light
particles such as the electron.
•
Electron wavelengths can easily be on the order of 10-10 m; electrons can
be diffracted by crystals just as X-rays can.
Example1: What is the de Broglie wavelength of a 0.20kg ball moving with speed 15m/s?
h
h
6.6 1034 J  s
 

 2.2 1034 m
p m v 0.20kg  15m / s) 
Example2: Determine the wavelength of an electron that has been accelerated through
a potential difference of 100 V.
V  100V
m  9.111031 kg
e  1.60 1019 C
 ?
1
2
m v2  eV  v 

h

mv
h
2eVm
2eV
m
  1.2  1010 m
De Broglie’s Hypothesis Applied to Atoms
The correspondence principle applies here as well – when the differences
between quantum levels are small compared to the energies, they should be
imperceptible.
De Broglie’s hypothesis is the one associating a wavelength with the
momentum of a particle. He proposed that only those orbits where the wave
would be a circular standing wave will occur. This yields the same relation that
In addition, it makes more reasonable the fact
that the electrons do not radiate, as one would
otherwise expect from an accelerating charge.
quantization:
de Broglie wavelength:
2rn  n
h

mv
nh
mvr n 
2
These are circular standing waves
for n = 2, 3, and 5.
Quantum mechanics incorporates wave-particle duality, and
successfully explains energy states in complex atoms and
molecules, the relative brightness of spectral lines, and many
other phenomena.
It is widely accepted as being the fundamental theory underlying
all physical processes.
Quantum mechanics is essential to understanding atoms and
molecules, but can also have effects on larger scales.
Classical physics
v<<c
xpx  
Classical relativistic physics
xpx  
v~c
Quantum nonrelativistic physics
xpx  
v<<c
Quantum relativistic physics
xpx  
v~c
The Wave Function and Its Interpretation
Question: An electromagnetic wave has oscillating electric and
magnetic fields. What is oscillating in a matter wave?
Answer: This role is played by the wave function, Ψ.
The square of the absolute value of the wave function at any point is
proportional to the number of electrons expected to be found there.
For a single electron, the wave function is the probability of finding
the electron at that point.
In the classical mechanics we use Newton’s equations of motion to describe
particles positions and velocities, in the classical electrodynamics we use
Maxwell’s equations to describe the electric and magnetic fields.
In quantum mechanics we use Schrödinger equation to describe the
function: Ψ= Ψ(x,y,z;t).
The Wave Function and the Double-Slit Experiment
Example: the interference pattern is observed after many electrons
have gone through the slits.
If we send the electrons through one at a time, we cannot
predict the path any single electron will take, but we can
predict the overall distribution.
The Heisenberg Uncertainty Principle
Quantum mechanics tells us there are limits to measurement –
not because of the limits of our instruments, but inherently.
This is due to the wave-particle duality, and to interaction
between the observing equipment and the object being observed.
xpx  
h

2
x - theuncertainty in theposition
px - theuncertainty in themomentum
Similar:
yp y  
zpz  
The Heisenberg uncertainty principle tells us that the position and
momentum cannot simultaneously be measured with precision.
Example:
Imagine trying to see an electron with a powerful
microscope.
At least one photon must scatter off the electron
and enter the microscope, but in doing so it will
transfer some of its momentum to the electron.
The uncertainty in the momentum of the electron
is taken to be the momentum of the photon:
p 
h

In addition, the position can only be measured to
about one wavelength of the photon:
x  
Combining, we find the combination of uncertainties:
xpx  h
Two examples
Example1: An electron moves in a straight line with a constant speed
v=1.10x106 m/s which has been measured to a precision of 0.10%.
What is the maximum precision with which its position could be
simultaneously measured?
p  mv
p  103 p  103 m v


1.061034 J  s
7
x 
 3
 3

1
.
1

10
m
31
6
p 10 m v 10  9.1110 kg 1.10 10 m / s



Example2: What is the uncertainty in position, imposed by the uncertainty
principle, on 150–g baseball thrown at (93 ± 2) mph = (42 ± 1) m/s?
p  mv


1.061034 J  s
x 


 7 1034 m
p mv 0.150kg 1m / s 
The uncertainty principle for energy and time
Et  
This says that if an energy state only lasts for a limited time, its
energy will be uncertain.
It also says that conservation of energy can be violated if the time
is short enough.
Example: An object is detected by the photon
x    t 
E  hf 
x 

c
c
hc

Philosophic Implications; Probability versus Determinism
The world of Newtonian mechanics is a deterministic one. If you know the
forces on an object and its initial velocity, you can predict where it will go.
Quantum mechanics is very different – you can predict what masses of
electrons will do, but have no idea what any individual one will.
26) An electron has a wavelength of 0.123 nm. What is its energy in eV? (This
energy is not in the relativistic region.)
A) 20 eV
B) 60 eV
C) 80 eV
D) 100 eV