MODERN PHYSICS
Lesson 1: Special Theory of Relativity
A. Postulates
1. The Relativity Postulate. The laws of Physics is the same in
all inertial reference frames. No one frame is preferred over any
other.
2. The Speed of Light Postulate. The speed of light in vacuum
has the same value, c in all directions and in all inertial reference
frames.
B. Consequences of the two postulates
1. The relativity of simultaneity. If two events are in relative
motion, they will not in general agree as to whether two events are
simultaneous.
Simultaneity is not an absolute concept, but rather a relative one
depending on the motion of the observer.
2. The relativity of time.
The time interval measured in the inertial reference frame is
the proper time, to.
Time interval measured from any other inertial reference frame
are always greater (relativistic time.
5. The relationship between momentum and energy.
(classmech)
E = 1/2 mv2 and p = mv
Therefore,
Substituting K,
Considering relativistic definitions,
E = ymc
2
2
2
2. When light is emitted or absorbed by an object (matter), this
emission or absorption occurs in the atom.
and p = ymv
p2 = y2m2v2
2
2 2
2
2
2 2
p c = ymv c = ymv
2
2
2
2
𝑣2
c4 𝑐 2
4
2
2
2 2
4
1
= y m v c (1 − 𝑦2 )
4
p c = ym c - m c
Since
Thus,
E = ymc2
2
2
2
2
Mathematically:
hf = hf’ + K
K = mc2 (ӯ - 1) Where y = Lorentz factor
E = hf
(photon energy)
h = 6.63 x 10 -34 J.s = 4.34 x 10 -15 eV·s
𝑝2
E = 2𝑚
Then,
Summary:
1) the quantum of a light wave with frequency, f has the energy,
4
p c = E - m c
Particles without mass are a special case
E = pc
3) When light of frequency f is absorbed by an atom, the photon
energy hf is absorbed by that atom. The photon vanishes and the
atom absorbs it.
hf = hf’ + mc2 (y - 1)
Substituting c/λ for f,
𝒉
4) When light frequency f is emitted by an atom, the amount of
energy hf is transferred by the atom to the light. The photon
appears and the atom is said to emit it.
𝒉
=
𝝀
𝝀’
+ mc ( y - 1 )
Applying law of conservation of momentum
𝒉
B. Photoelectric Effect
If you direct a beam of light of short enough wavelength onto a
clean metal surface, the light will cause electrons to leave the
surface.
=
𝝀
0 =
𝒉
𝝀’
cos Ɵ + y mv cosФ ( x-axis)
𝒉
𝝀’
sin Ɵ + y mv sinФ ( y-axis)
To find Δλ= ( 𝝀 − 𝝀𝒐 ) by eliminating v and Ф .
′
3. The relativity of length. The length of an object measured in
the rest frame is its proper length. Measurement of the length
from any reference frame that is in relative motion parallel to that
length are always less than the proper length.
Lorentz Transformation equations relate the space and time
coordinates of two systems moving at a constant velocity relative
to each other. They formally express the relativity concepts that
space and time are not absolute; that length, time, and mass
depend on the relative motion of the observer, and that the speed
of light in a vacuum is constant and independent of the motion of
the observer or the source.
DERIVATION OF LORENTZ TRANSFORMATION
x = x’-vt
length x is different for S’
x’ = y (x-vt) (eq 1)
In reference frame S, the length is
x = y (x’-vt’) (eq 2)
E and pc can also be written:
E = ymc2 and p = ymv
Δλ =
The only way we can reconcile these last two definitions with E =
pc is to set the velocity to c. Massless particles must travel at the
speed of light. As we will learn later on, light itself is composed of
particles (photons). To travel at the speed of light, these particles
must be massless.
The quantity
wavelength.
General Theory of Relativity
A. Principle of Equivalence states that gravitation and
acceleration are equivalent.
- one cannot tell whether he is at rest on Earth or accelerating
through interstellar space at 9.8 m/s2.
- there would be no experiment that will discern the difference
between the effect of gravity and effect of acceleration.
Notable results:
1. K max = eV stop
this is Lorentz transformation factor
2. Curvature of Space
If light beam can follow a curve path, then space is curved due
to the gravitational mass. And we learned that space is entangled
with time, so it follows that space-time is curved due to gravitational
mass.
3. To let the electron escape from the target surface, the electron
must pick up a minimum energy, Ф, which is, the property of the
target material called work function. If the energy hf transferred to
the electron can exceed the work function, the electron can escape
the target.
B. Consequences of Lorentz Transformation Equations
1. TIME - If two events occur at the same place in S’ (thus Δx = 0)
but at different times ( Δt’ = 0 ), the equation reduces to
Δt = yΔt’ ( events in the same place S’)
Δt = yΔt0
(time dilation)
3. Black Holes
Since space-time is curved near massive bodies.
in Einstein’s theory, force of gravity is not acting on body, but
instead bodies and light rays move as they do because
space-time is curved.
Einstein summarized the result s of the experiment with the
equation:
hf = Kmax + Ф (photoelectric effect equation)
2. LENGTH
The Schwarzschild radius is sometimes called as the event
horizon; the surface beyond which no signals can ever reach us,
and thus inform us of events that happen.
2𝐺𝑀
R = 𝐶2
𝟏− 𝟐
𝒄
𝑳=
√1− 2
𝑐
𝑳𝒐
𝒚
( length contraction)
3. MOMENTUM
𝛥𝑥
p = mv = m 𝛥𝑡 ( classical mechanics)
To find relativistic momentum, redefine that
𝛥𝑥
p = m 𝛥𝑡0
Using time dilation equation, tr =
p=m
𝜟𝒙
𝒗𝟐
𝒕𝒓√𝟏− 𝟐
𝒄
𝑡𝑜
𝑣2
√1− 2
𝑐
= ymv
(relativistic momentum)
4. A New Look at Energy.
The special theory of relativity defines the relation of mass and
energy. This I believed has been explained in nuclear reaction. The
relation is
Eo = mc2 (mass energy or rest energy)
When the object is moving it has additional kinetic energy thus,
E = Eo + KE = mc2 + K
Relativistic energy can be also written as
E = ymc2
Particle Properties of Waves
Key Ideas:
1) An electromagnetic wave is quantized, and the quanta are
called photons.
2) When light of high enough frequency illuminates a metal
surface, electrons can gain enough energy to escape the metal by
absorbing photons in the illumination.
3) Although it is massless, a photon has momentum, which is
related to its energy, frequency and wavelength.
𝒉
𝒑=
𝝀
A. The Photon, the Quantum of Light
-quantities are in minimum (elementary) amounts or multiple of it.
-These quantities said to be quantized and the elementary mount
associated with this quantity is called the quantum of that quantity.
- Einstein ,in 1905, proposed that light, an electromagnetic
radiation, exists in elementary amounts called photon.
( 1- cos Ɵ )
( Compton Shift)
B. Consequences
1. Bending of Light.
The principle of equivalence suggests that light ought to be
bent/deflected due to the gravitational force of a massive body.
In principle of equivalence, an upwardly accelerating reference
frame is equivalent to a downward gravitational field. Thus, gravity
is expected to bend a beam of light out of its straight path.
x = ct and x’ = ct’
(eq 3 and 4)
Multiplying eq. 1 and 2,
xx’ = y2 (xx’ + xvt’ - x’vt - v2 tt’)
Substituting equation 3 and 4,
c2tt’ = y2 (c2tt’ + vctt’ - vct’t - v2 tt’)
Cancelling tt’
𝟏
1
ᵧ2 = 𝒗𝟐
Or,
ᵧ =
𝑣2
𝒉
𝒎𝒄
; where
is e is the elementary charge
2. Observations show that photoelectric effect does not occur if the
frequency is below a certain cutoff frequency f o or the wavelength
is below the cut-off wavelength, λo no matter how intense the
incident light is.
Vstop =
ℎ
𝑒
f -
Ф
𝑒
C. Photons and Momentum
P =
𝒉𝒇
𝒄
=
𝒉
𝝀
(photon momentum)
Compton Effect - The result of the experiment showed that
although there was only single wavelength in the incident x-ray
beam, scattered x-rays contain a range of wavelength with two
prominent intensity peaks as shown.
1. The direction of travel of x-ray changed, the electron must
have recoiled which means that the electron gained kinetic
energy; and
2. The energy of the
scattered photon must
be less than that of the
incident
beam
because of energy
conservation.
𝒉
𝒎𝒄
= 𝟐. 𝟒𝟑𝒙𝟏𝟎−𝟏𝟐 𝒎 is constant called Compton