# -18- PHY 133 Section 5: Special Relativity

```-18PHY 133
Section 5: Special Relativity
e&quot;
Einstein's postulates:
1. The laws of nature are the same in all inertial reference frames. (No preferred frame of
reference.)
Inertial reference frame: one which isn't accelerating.
2. c (speed of light in a vacuum) is the same in all inertial reference frames.
These are suggested by the Michelson - Morley experiment (see text) and also theoretical
considerations.
Notice #2 means
Speed = (distance)/(time). So, for c to be the same, distance and time must be different in different
reference frames. For example, the time interval between the bulb flashing and the light reaching
the detector:
They agree on d because there is no motion in the y direction.
(c Δt)2 = (v Δt)2 + d2
(c Δt)2 = (v Δt)2 + (c Δtp)2
(c Δt)2 - (v Δt)2 = (c Δtp)2
Δt2 - (v/c Δt)2 = Δtp2
[1 - (v/c)2](Δt)2 = Δtp2
-19Δt =
√
let
γ=
( )
Time Dilation:
√
( )
Δt = γ Δtp
(More time between the events in the frame where light has to catch up to the detector.)
The Lorentz transformation:
x'= γ(x - vt)
y'= y
z'= z
t'= γ[t - (v/c2)x]
Inverse transformation:
Replace v with -v.
Velocity transformation (Differentiate
with respect to time.):
Length in observer's rest frame: L = x2 - x1
Length in object's rest frame: Lp = x2'-x1' = γ(x2-vt) - γ(x1-vt) = γ(x2 - x1)
Lorentz - Fitzgerald Contraction:
Lp = γL
Δtp = Measured in frame where “clock” is at rest.
Lp = Measured in frame where length is at rest.
(Not necessarily the same frame, unless you’re talking about the length of the clock.)
-20Ex. 5-1:
What is the speed of ship 2 relative to Earth?
Ex. 5-2: How fast must something go for relativistic effects to make a 1% difference in time,
length, etc?
Ex. 5-3: The average lifetime of a stationary muon is 2.2 μs. If one is created 8.0 km above sea
level (by a cosmic ray hitting an air molecule), and it travels at .998c, find
a. its lifetime as seen by us, and
b. the distance to sea level in its reference frame.
Simultaneity: Events that are simultaneous to one observer are not to another. That is, events
separated only by space to one observer are separated by space and time to another. So, space and
time must all be part of the same thing. (“spacetime”)
Momentum:
In a collision between objects thrown from different frames of reference, p is conserved only if
redefined as
⃑
⃑
where m = mass when object is at rest, ⃑ = its velocity
-21-
mrel  m
γm is sometimes called &quot;relativistic mass,&quot;
(As v  c, an object acts, in some ways, like it gets heavier.)
Newton's 2nd law must then be rewritten as ⃑ =
⃑
Energy:
Work = F dx = (dp/dt)dx = ....
Put in p = γmV, make some clever substitutions, integrate. The result is that the work needed to get
from rest to a speed V is:
KE = γmc2 - mc2
KE = (mrel - m)c2: If you set an object in motion, the energy gained equals the mass gained times a
constant. This is just like changing units: Inches times a constant equals feet; kilograms times a
constant equals joules. A kilogram, then, is just another unit of energy; mass and energy are
equivalent:
Total energy:
Rest energy:
Kinetic energy:
E = γmc2
ER = mc2
KE = E - ER
= energy equivalent of the rest mass.
(KE = 1/2 mv2 is true only if v &lt;&lt; c.)
Why no object can reach c: As v  c, mrel = γm =
approaches m/0 = .
√
( )
Increasing a body’s kinetic energy is equivalent to increasing its mass, in the sense of increasing its
inertia. Near c, the inertia becomes too great for any further significant acceleration. As you add
more energy, the particle gets &quot;heavier&quot; not faster.
Ex. 5-4: an electron is accelerated through a million volts, starting from rest. Find its final speed.
Ex. 5-5: In atomic mass units (u), the mass of a proton = 1.007825, mass of a neutron = 1.008665,
mass of a deuteron (21H nucleus) = 2.014102. What is the binding energy of the deuteron?
-22Section 6: Wave - Particle Duality:
- Experimental graph of Intensity vs. λ .
- Wein’s displacement law. (λ of experimental curve’s peak.)
- Rayleigh-Jean’s law, the prediction of classical physics, doesn’t match experimental graph.
(&quot;Ultraviolet catastrophe&quot;)
Planck's solution (1900): Assume molecules of cavity walls can only vibrate with certain
energies. &quot;Jumping&quot; between energy states, they emit (or absorb) light in &quot;quanta&quot; of energy
E = hf
h = Planck's constant, f = frequency
(Called &quot;photons&quot; today.)
From this, he derived
I(λ,T) =
2πhc2 _
λ (ehc/(λkT) - 1)
which matches the observed data.
5
Ex. 6-1: A simplified derivation of Wein’s displacement law from the Planck radiation law.
The Photoelectric Effect:
Maximum KE of photoelectrons depends on light's frequency, not its intensity. Classical
theory of electromagnetic waves predicts the opposite.
KEmax is measured by finding the stopping potential, V0. (Voltage needed to turn back all
electrons before they reach the opposite plate, stopping the current.)
KEmax = eV0
-23Einstein's solution (1905): Light is a stream of photons,
each with energy hf. A photon is absorbed &quot;all or nothing&quot;
by one electron on the plate they hit.
- KE of e- depends on energy of photon that hit it, and thus
on its frequency.
- KE independent of intensity because each e- gets its
energy from just one photon.
KEmax = hf - 
 = work function = energy needed to
tear an e- off of the surface.
Ex. 6-2: What is the energy of a photon with a 500 nm wavelength?
Ex. 6-3: Sodium has a work function of 2.46 eV. Find
a. the maximum energy of photoelectrons when λ = 3000 .
b. the cutoff λ (above which there is no emission).
de Broglie wavelength.
Ex 6-4: A beam of neutrons going 1000 m/s falls on a crystal with atomic planes 4.5 apart. What
will be the angle between incident and reflected beams for first order Bragg reflection?
If all objects have this dual wave/ particle nature, why don't we see wavelike behavior for baseballs,
airplanes, etc?
λ = h/(mv)  λ is very small if m is very big. So, effects like diffraction are too small to measure
for macroscopic objects.
How can something be both a wave and a particle, when the behavior of waves and particles is as
different as black and white?
A blob of correction fluid on clear plastic looks white when front lit and black when back lit. It's
black or white, depending on how you look at it. Similarly, light is a wave or a particle, depending
on whether you look at it in a diffraction experiment, or a photoelectric experiment. (Either model
is oversimplified.)
Heisenberg's Uncertainty Principle:
Ex: What are x and px of an electron as it goes through this slit?
A convenient (but not really &quot;true&quot;) picture of wave/ particle duality is to think of a pointlike
particle &quot;surfing&quot; on some kind of wave.
-24-
px might be anywhere between these extremes.
(&quot;Particle&quot; can be anywhere &quot;wave&quot; has a significant
amplitude.)
Narrower slit (reduced uncertainty in x) diffracts waves more. This could throw particle further to
the side, increasing uncertainty in px. From a more detailed, general analysis:
Δx Δpx &gt;
Similarly,
where
= h/(2π)
ΔE Δt &gt;
The uncertainty principle is a useful &quot;rule of thumb&quot; for rough estimates:
Ex. 6-5: The electron's momentum in hydrogen's ground state is 1.99 x 10-24 kgm/s. If this is Δpx
(px could be anything between
.), what is Δx?
Ex. 6-6: (Quantum tunneling): With an order of magnitude estimate, I will explain how an alpha
particle with ~10 MeV of energy can escape from a nucleus surrounded by a ~20 MeV energy
barrier.
(I also briefly mention some other applications of tunneling: tunnel diode, Josephson junction,
scanning tunneling microscope.)
-25Sec. 7:
The Bohr model of the atom (1913), as later explained by de Broglie's ideas (1924):
Hydrogen &amp; H - like atoms (He+, Li++, etc: 1 electron):
I will explain, line by line on the board, how energy levels follow from Coulomb's law and the
electron's wave nature.
Atomic spectra:
An atom gives off light when an electron drops from a higher to a lower energy level. The energy it
loses is given off as a photon (E = hf).
13.6eV
13.6eV
In hydrogen, E0  
Ef  
2
n0
n 2f
 1
1
Ephoton = –ΔEelectron = – (Ef - E0)  13.6eV  2  2 
n

 f n0 
also, Ephoton = hf = h(c/λ)  E/(hc) = 1/λ 
Rydberg
1 13.6eV


hc
 1 1
  
 n2 n2 
0 
 f
constant: R = 1.097 x 107 m-
1
(
)
Allowed wavelengths in the spectrum of hydrogen.
This model explains line spectra:
Example,
hydrogen:
-26(Each element has its own pattern of wavelengths.)
Ex. 7-1: Find wavelength, frequency and energy per photon for the first line in the Paschen series.
Spontaneous Emission: Electron falls on its own.
Stimulated Emission: Electron is knocked down by an incoming photon identical to the one it
emits. Both photons move off in phase. Used in lasers.
Multi-electron atoms:
Electrons are arranged in &quot;shells&quot;:
X-ray tube:
beam e-s suddenly decelerate.
Beam knocks inner e- from target atom. Then, an outer e- drops to take its place, emitting an x-ray
photon.
K series: electron falls into K shell.
L series: electron falls into L shell.
α: from the level just above.
β: from the second level above.
And so on.
-27To estimate the wavelengths:
If only one electron in atom: En = – (13.6 eV)Z2/n2
With more: Use the net charge within the electron's orbit as an effective value for Z. (Other
electrons &quot;shield&quot; the nucleus.)
For K and L electrons, Zeff = Z – 1 works pretty well.
M electrons: Zeff = Z – 9
Ex. 7-2: Electrons are accelerated through 35 kV, and strike a Mo target (atomic number = 42).
Find:
a. The wavelength of the Kα line,
b. The shortest wavelength emitted.
-28Section 8: Introduction to Quantum Mechanics
The Bohr model was successful as far as it went, but left some things unexplained. Quantum
mechanics is what replaced it.
Wave function.
Just as surfers prefer the ocean over Lake Ontario, electrons tend to go where the waves are bigger.
In a particle beam, the number per unit volume is proportional to ψ2.
However, if you turn down the intensity to one electron at a time (in an electron diffraction
experiment, for example), where that electron will go is fundamentally unpredictable. The wave
function only gives a probability of what it will do. (Like flipping coins: You can predict the
overall behavior of many - half heads and half tails. But what one individual coin will do is
unknowable.)
This is a fundamental difference from classical physics.
Ex. 8-1:
If there is a 50% chance of finding the particle between x = 0 and x = a, what is a?
Normalization.
Ex. 8-2: If ψ =
⁄
, what is A?
The Schr&ouml;dinger equation. (In the special case of a one dimensional problem, with ψ not a function
of time.)
One dimensional square well. (&quot;Particle in a box&quot;)
(I will show how the wave functions and energy levels follow from Schr&ouml;dinger's equation.)
The particle can exist only in certain quantum states. (Each different ψ the particle could have
is called a state.)
Notice there is no E0: The particle can not come to rest.
Zero-point energy = the least energy the particle can have.
Ex.8-3: In a well 2.5 &Aring; across, an electron drops from n = 2 to n = 1. What is the wavelength of the
photon given off?
-29Review for Exam 2:
1. A He-Ne laser emits light with a wavelength of 632.8 nm. If the power of the beam is .500 mW,
how many photons does it give off each second?
2. An observer on the spaceship sees Earth coming
at him at .7c, and the electrons in the beam going
toward Earth at .9c. What is the speed of the
electrons relative to Earth?
3. For an electron circling a proton:
a. Think of the electron as a wave. From the fact that a standing wave along a circular path
must have a whole number of wavelengths around its circumference (circumference = nλ)
derive Bohr's quantization condition, mvr = n .
b. Now, thinking of the electron as a classical particle, it can be shown that mv2 =
(1/4πε0)(e2/r). From this and Bohr's quantization condition, derive the expression for the radius
of the nth orbit. (Leave it in terms of n and fundamental constants: no need to fill in numbers.)
4. Show that if U = 0 everywhere (a free particle) then ψ =
, where i =  1 and A is
a constant, is a solution of the Schr&ouml;dinger equation. (Take each side of the equation separately, and
show that they are in fact equal if this is ψ.) Remember that k = 2π/λ, λ = h/p and E = p2/(2m).
(&frac12;mv2 = (mv)2/(2m).) Treat t like a constant when you differentiate.
5. Short answer, 5 points each:
a. A hydrogen atom in its ground state (n = 1) absorbs a photon. What
is the smallest possible energy this photon could have? (Refer to the
energy level diagram for hydrogen at right.)
b. The Heisenberg uncertainty principle limits the precision with which
you can measure a particle's position when its momentum is measured
at the same time. Is there any limit on the precision of a position
measurement when momentum is not measured?
c. Make a rough sketch of the intensity versus wavelength curve given
by the Planck radiation law. (I goes on the vertical axis.)
d. What is meant by an &quot;inertial reference frame?&quot;
e. Just to the left of the origin, a particle's wave
function is given by ψ = .3 - x2. Just to the right
of the origin, ψ = k, where k is a constant. What
value must k have?
```