Lecture 9' Overview Ch' 13

1
Lecture 9. Overview Ch. 1-3

• Relativistic Kinematics
• Relativistic Dynamics
• Photons Photoelectric and Compton effects
• de Broglie waves
• Uncertainty Principle (both momentum-position
  and energy-lifetime)

2
Problem (Relativistic Dynamics)
An unstable particle of mass m3.34?10-27 kg is
initially at rest. The particle decays into two
fragments that fly off with velocities of 0.987c
and -0.868c. Find the rest masses of the
fragments.
fragment 2
fragment1
momentum conservation
after
before
energy conservation
3
Problem (Relativistic Dynamics)
The nuclear reaction pd?3He? (dprotonneutron)
can occur even when the initial particles have
zero kinetic energy. If the gamma photon has
energy 5.5MeV when the reaction occur with the
initial particles at rest, what is the mass of
the 3He nucleus? mp1.6724x10-27kg,
md3.3432x10-27kg
4
Problem (Relativistic Dynamics)
Find the minimum energy a proton must have to
initiate the reaction
The minimum energy when the products of
reaction are at rest in the center of mass
reference frame all the incoming energy is
transformed into the rest energy.
We can apply the same logic to a system of
particles!
For colliding beam accelerators (e.g., LHC),
the center-of-mass frame and the lab frame are
the same (each proton should have E2mc2)
5
Problem (Relativistic Dynamics, Midterm 1, 2008)
(20) Find the minimum energy a gamma photon must
have to initiate the reaction
if the target proton is at rest.
The minimum energy when the products of
reaction are at rest in the center of mass RF
This invariant for the system of particles will
be the same in the lab RF
6
Problem (Doppler)
A spaceship approaches an asteroid and sends out
a radio signal with proper frequency 6.5x109 Hz.
The signal bounces off the asteroids surface and
returns shifted by 5x104 Hz. What is the relative
speed of the spaceship and the asteroid?
In this situation, there Doppler shift occurs
twice firstly, the original frequency is
received by an asteroid as
secondly, the spaceship receives the reflected
signal with the frequency
7
Problem (Doppler, Midterm 1, 2008)
(20) A quasi-stellar object (quasar) exhibits a
Doppler shift such that
where ? is the wavelength that would be measured
by an observer at rest relative to the object,
?obs - the wavelength measured on the Earth. Is
the object moving towards us or away from us?
Assuming that the object moves either directly
away from or directly towards us, what is its
speed relative to us?
Red shift the quasar moves away from us.
8
Problem (Photoelectric Effect)
When iron is illuminated with ultraviolet
light with a wavelength of 250nm, the maximum
voltage developed between the plates in the
experiment shown in Figure is 0.46V. Find the
voltage difference between the plates if the
ultraviolet light wavelength is changed to 220nm.
Also find W for iron.
anode
- - - - - - - -
light
V

cathode
9
Problem (photons Compton, Midterm 1, 2008)
An electron and positron (particles of equal mass
and opposite charge) move toward each other
along a straight line. Each has kinetic energy
equal to mec2 511keV. The electron and positron
annihilate each other producing two photons. (5)
(a) Find the wavelengths of these photons. (15)
(b) One of the emitted photons collides with an
electron, initially at rest, and is scattered
through an angle of 600. What is the energy of
the scattered photon in keV? What is the recoil
energy of the electron?
(a)
(b)
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Problem (Compton)
A beam of monochromatic X-rays is directed at
electrons at rest. After a collision between an
X-ray photon and an electron, it was observed
that the electron had a kinetic energy of 400keV
while the scattered photon had a wavelength
exactly twice its wavelength before the
collision. (a) (10) Calculate the wavelength and
the energy of the photon before the collision
(use conservation of energy). (b) (5) Calculate
the angle by which the photon was deviated from
its original direction. (c) (5) Calculate the
total energy E and the momentum p of the electron
after the collision. (d) (5) Find the angle
between the directions of the recoil electron and
the incident photon.
(a)
(b)
(c)
(d)
11
Problem (de Broglie waves, relativity)

• Can we consider an electron as a relativistic
  particle if
• its momentum is comparable to the momentum of a
  visible-light photon with hf2eV
• its de Broglie wavelength is comparable to the
  size of a hydrogen atom (1nm)
• its kinetic energy is of an order of 1MeV.
• Calculate and explain!

(a)
non-relativistic (vltltc)
(b)
still non-relativistic
K is twice the rest energy of course,
relativistic!
(c)
12
Problem (de Broglie waves)
Find the de Broglie wavelength of an electron
with kinetic energy of 10MeV.
Because the kinetic energy is much greater than
the rest energy, we use the relativistic approach
to estimate p.
For what kinetic energy will a particles de
Broglie wavelength equal its Compton wavelength?
13
Problem (Uncertainty Principle)
An electron microscope is designed to resolve
objects as small as 0.1nm. What energy electrons
must be used in this instrument? Express your
answer in eV.
for comparison
thus, we can use non-relativistic approach
14
Problem (Uncertainty Principle, de Broglie waves)
Calculate the de Broglie wavelength of a
5.4MeV ? particle (the nucleus of 4He 2
protons2 neutrons) emitted from an 241Am
nucleus. Could this particle exist inside the
241Am nucleus (diameter 1.6?10-14 m)?
- the energy is much smaller than the rest energy
of the ? particle, thus we can apply the
non-relativistic approach.
- the nucleus is too small for such a low-energy
? particle
15
Problem (Uncertainty Principle, Length
contraction, Midterm 1, 2008)
(15) A subatomic particle produced in a nuclear
collision is found to have a mass M such as
Mc21,200MeV, with an uncertainty of
?60MeV. Estimate the lifetime of this particle
and, assuming that the particle travels with a
speed of 2.8?108m/s, calculate how far the
particle will travel before it disintegrates.
(a)
(b)
(10) To how small a region must an electron be
confined for its total energy to be