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Transcript of Albert Einstein - ferenc.physics.ucdavis.eduferenc.physics.ucdavis.edu/9D/lectures.pdf ·...
Einstein’s Postulates of Relativity:
(Albert Einstein, 1905)
Postulated in an attempt to explain the laws of Classical
Electromagnetism
(Maxwell’s Equations, the constancy of the speed of light)
There are many-many experiments that
prove the consequences of the two
postulates of relativity
and
no experiments that disprove, so far,
Einstein’s Postulates of Relativity:
• Michelson- Morley Experiment, light soucre, and the medium (Discussion sessions, see Appendix A)
• Definition of an “Event”
• Some Consequences of Einstein’s Postulates:
1. Relative Simultaneity
2. Time Dilation
3. Length Contraction
1st Postulate:
The form of each physical law is the same in all
inertial frames of reference
INERTIAL FRAME = ?
The reference frame in which an
object, experiencing zero net force – a “free” object
– moves at constant velocity
v = constant
2nd Postulate:
Light moves with the same speed (c)
relative to all observers
Bob measures:
Speed of light = c
and not v+c
Anna measures:
Speed of light = c
?
Einstein’s Postulates of Relativity:
• Light Souce, Medium and Michelson- Morley Experiment (Discussion sessions, see Appendix A)
• Definition of an “Event”
• Consequences of Einstein’s Postulates:
1. Relative Simultaneity
2. Time Dilation
3. Length Contraction
Einstein’s Postulates of Relativity:
• Light Souce, Medium and Michelson- Morley Experiment (Discussion sessions, see Appendix A)
• Definition of an “Event”
• Consequences of Einstein’s Postulates:
1. Relative Simultaneity
2. Time Dilation
3. Length Contraction
AN EVENT For Bob as well!
Simultaneous Arrival
THEN:
Simultaneous
Emission is
impossible
for Bob
2nd Postulate
AN EVENT For Bob as well!
Simultaneous Arrival
BUT THEN:
Simultaneous
Emission is
impossible
for Bob
2nd Postulate
The time for the light to return:
t = 2H/c
The time for the light to return:
T > 2H/c
Longer
Path + 2nd
Postulate
The time for the
light to ‘return’:
T > 2H/c
Longer Path (L) +
2nd Postulate
L2 = H2 + (v T/2)2
L
t = 2H/c
T = 2 L/c
T = t (1 – (v/c)2 )-1/2
PRECISELY =>
Consequence 3:
Lorentz Length Contraction, or The absence of absolute distance
L = L0 (1 – (v/c)2 )1/2
Muon Lifetime
Muons are created abundantly in
elementary particle showers in the
atmosphere, initiated by energetic cosmic
rays (photons, particles and nuclei).
Muons originate from decays of particles
called “pions” (p) that are the primary
products in these showers
Muon Lifetime
Muon (m) is an elementary
particle
similar to electron, but heavier
(will learn more in Chapter 11)
CONCLUSION: According to Classical Physics,
muons should not be able to reach the ground!
BUT THEY DO
Consequence 3:
Lorentz Length Contraction, or The absence of absolute distance
L = L0 (1 – (v/c)2 )1/2
The time for the
light to return:
T > 2H/c
Longer Path (L) +
2nd Postulate
L2 = H2 + (v T/2)2
L
t = 2H/c
T = 2 L/c
T = t (1 – (v/c)2 )-1/2
PRECISELY =>
Einstein’s Postulates of Relativity:
• Michelson- Morley Experiment – NO AETHER !
• Consequences of Einstein’s Postulates:
1. Relative Simultaneity
2. Time Dilation
3. Length Contraction
SUMMARY
Viewed by Carl:
- Bob moves towards right at v = +0.98 c
- Anna moves towards left, at v = -0.98 c
- Carl sends laser pulses towards Bob
- Bob reflects those pulses towards Anna
=> What is the velocity of each laser pulse
received by Anna, when measured by Anna?
Another way of expressing
the Lorentz Transformation Equations
4-vectors
x’x’ + y’y’ + z’z’ xx + yy + zz
The LENGTH
is not INVARIANT,
i.e. it is not conserved
under Lorentz
transformations
x’x’ + y’y’ + z’z’ xx + yy + zz
Is there anything
that IS INVARIANT
under Lorentz
Transformations?
x’x’ + y’y’ + z’z’ – (ct’)(ct’) = xx + yy + zz – (ct)(ct)
4-dimensional “LENGTH”
What Follows:
• Lorentz transformations of distance and time intervals
• The Twin Paradox, revisited
• Velocity Transformation (1.7)
HOMEWORK:
• Lorentz Transformations and 4-vectors (1.11)
• The Doppler Effect (1.6)
Lorentz Transformation of
Distances and Time Intervals
IMPORTANT: space and time distances mix together
Einstein’s Postulates of Relativity:
• Michelson- Morley Experiment – NO AETHER !
• Consequences of Einstein’s Postulates:
1. Relative Simultaneity
2. Time Dilation
3. Length Contraction
SUMMARY
EXPERIMENT
Next:
• Velocity Transformation (1.7)
• Is causality absolute?
• Energy and mass
HOMEWORK:
• Lorentz Transformations and 4-vectors (1.11)
• The Doppler Effect (1.6)
Lorentz Transformations
NOTE that coordinates
orthogonal to the direction
of motion stay the same
NO (Lorentz) Length Contraction in directions
other than along the direction of relative
motion
Lorentz Transformations
Relativistic velocity Transformations
BUT ALL Components of the
Velocity Vector Transform!
Lorentz Transformations
Relativistic velocity Transformations
BUT ALL Components of the
Velocity Vector Transform! WHY ?
Lorentz Transformations
Relativistic velocity Transformations
BUT ALL Components of the
Velocity Vector Transform! WHY ?
Because:
The time transforms independently
of the direction of motion,
coordinates do not,
and velocity combines both
Is there Absolute Causality?
Might cause precede effect in one reference
frame but effect precede cause in different
reference frame(s)?
e.g. can someone see you
first die, and then
see you get born?
Benjamin Button
Let’s assume that the order of events is
changed in some reference frame S’
0'
0
>
t
t
Is that Possible?
t and t’ are the time intervals between
the same two events observed in
S and S’, respectively
=
=
1'
'
2
2
t
x
c
vtt
txc
vt
v
v
g
g
Using Lorentz transformations….
0'
0
>
t
tif then
tv
cx
t
x
c
v
>
2
201
=
=
1'
'
2
2
t
x
c
vtt
txc
vt
v
v
g
g
Using Lorentz transformations….
0'
0
>
t
tif then
tv
cx
t
x
c
v
>
2
201
Impossible
Is there Absolute Causality?
Might cause precede effect in one reference
frame but effect precede cause in different
reference frame(s)?
e.g. can someone see you
first die, and then
see you get born?
Viewed by Carl:
- Bob moves towards right at v = +0.98 c in his 10 m long spaceship. He
wears a clock that blinks every 10 seconds.
- Anna stands next to Carl in her 20 m long spaceship, and doesn’t
move. She wears a clock that blinks every 10 seconds, too.
a. What is the length of Bob’s spaceship, when measured by Anna?
b. What is the length of Anna’s spaceship when measured by Bob?
c. What is the time interval between two blinks of Bob’s clock, when
measured by Carl?
d. What is the time interval between two blinks of Anna’s clock, when
measured by Bob?
e. What is the time interval between two blinks of Anna’s clock, when
measured by Carl?
New topics:
1. Relativistic Dynamics:
2. General Theory of Relativity – skipped
END of Relativistic Physics
(Next time – Quantum Physics)
2mcE =
Another way of expressing
the Lorentz Transformation Equations
4-vectors
x’x’ + y’y’ + z’z’ xx + yy + zz
The LENGTH
is not INVARIANT,
i.e. it is not conserved
under Lorentz
transformations
x’x’ + y’y’ + z’z’ xx + yy + zz
Is there anything
that IS INVARIANT
under Lorentz
Transformations?
x’x’ + y’y’ + z’z’ – (ct’)(ct’) = xx + yy + zz – (ct)(ct)
4-dimensional “LENGTH”
22222222222222
422222222
'''' cpcpcpEcpcpcpE
cmINVARIANTcpcpcpE
zyxzyx
zyx
=
==
2222
2222
2
2222
2
)(
)(
)(
mccpE
mccpE
mccpE
=
=
=
E = TOTAL
ENERGY
Revolutionary Concept
Energy Matter
Atomic Bomb (Chapter 10):
Energy is CREATED
From the Mass of
Nuclei
(Internal energy is transformed
into kinetic energy)
Cannonball Trajectory:
•Relative to the Earth
= Parabola
•Relative to the Falling
Physicist
= Straight Line
In any small, freely falling reference
frame
anywhere in our real, gravity-endowed
Universe,
the laws of physics must be the same
as they are
in an inertial reference frame in an
idealized, gravity-free universe.
Einstein’s Principle of Equivalence
Special Theory of Relativity:
General Theory of Relativity:
locally
Deals exclusively with
globally
INERTIAL FRAMES -
v = constant
Deals also with
Accelerating -
LOCALLY INERTIAL FRAMES
Lecture Today
1. Black Body Radiation (Max Planck 1900)
2. The Photoelectric Effect (Albert Einstein 1905)
3. The Production of X-Rays (Wilhelm Roentgen 1901)
4. The Compton Effect (Arthur Compton 1927)
5. Particle-Antiparticle Pair Production (Carl Anderson 1932)
6. Discussion: a Wave or a Particle (?) –> “Duality”
Electromagnetic Waves
behaving like
Particles
(Chapter 3 (or 2-old book))
The Energy (E) in
the electromagnetic radiation
at a given frequency (f)
may take on values restricted to
E = n h f where:
n = an integer (1, 2, 3,…)
h = a constant
(“Planck Constant”)
The Planck’s Black-Body Radiation Law:
Experimental Fact:
E = nhf
BUT, why should the energy of an
Electromagnetic wave be
“Quantized”?
(n= integer)
No explanation
until 1905
AWave is a
Continuous
Phenomenon
Experimental Fact:
E = nhf
BUT Why should the energy of an
Electromagnetic wave be
“Quantized”?
(n= integer)
No Explanation
until 1905
Albert Einstein
The Photoelectric Effect
A wave is a
Continuous
Phenomenon
metal
The Photoelectric Effect
(Albert Einstein 1905)
Phenomenon observed long time before Einstein,
and something very strange was observed:
metal
The Photoelectric Effect
(Albert Einstein 1905)
Even With Very strong light of low frequency
NO
electrons
ejected
metal
The Photoelectric Effect
(Albert Einstein 1905)
Even With Very strong light of low frequency
NO
electrons
ejected
metal
The Photoelectric Effect
(Albert Einstein 1905)
Even With Very strong light of low frequency
NO
electrons
ejected
The Photoelectric Effect
(Albert Einstein 1905)
Even With Very-Very weak light intensity,
but of high enough frequency
Electrons
ejected
metal
The Photoelectric Effect
(Albert Einstein 1905)
Even With Very strong light of low frequency
NO
electrons
ejected
Planck’s Law
( E = n h f )
Albert Einstein proposed:
Photoelectric Effect
(Threshold frequency)
The light is behaving as a collection of particles
called “photons” each of them having energy
E = h f
The Photoelectric Effect
(Albert Einstein 1905)
Even With Very-Very weak light intensity,
but of high enough frequency
Electrons
ejected nhfE
hfE
beam
photon
=
=
What happens
is that
1 PHOTON ejects 1 ELECTRON
To free an electron from the metal, one has to
“pay” a certain amount of energy
the Work Function
Also known at that time:
QUIZ 3:
a. At what speed will the kinetic
energy of a particle of mass m =
0.001 g (gram) be equal to its
internal energy ?
b. What is the ratio of that kinetic
energy and the kinetic energy of the
same particle moving at v = 100
km/h ?
1. Black Body Radiation (Max Planck 1900)
2. The Photoelectric Effect (Albert Einstein 1905)
3. The Production of X-Rays (Wilhelm Roentgen 1901)
4. The Compton Effect (Arthur Compton 1927)
5. Particle-Antiparticle Pair Production (Carl Anderson 1932)
6. Discussion: a Wave or a Particle (?) –> “Duality”
Electromagnetic Waves
behaving like
Particles
(Chapter 2)
The Production of X-Rays
(Wilhelm Roentgen 1901)
(The “reverse” of the Photoelectric Effect)
Bremsstrahlung CLASSICAL physics:
Radiation covers entire spectrum
SURPRISE:
Experiments indicate a cutoff wavelength:
Frequency f , Energy E=hf
1 photon 1 electron
(?) 1 electron 1 photon (?)
The Production of X-Rays
(Wilhelm Roentgen 1901)
(The “reverse” of the Photoelectric Effect)
Bremsstrahlung CLASSICAL physics:
Radiation covers entire spectrum
The Compton Effect
Photons carry momentum like particles
and scatter individually with other particles
1. Black Body Radiation (Max Planck 1900)
2. The Photoelectric Effect (Albert Einstein 1905)
3. The Production of X-Rays (Wilhelm Roentgen 1901)
4. The Compton Effect (Arthur Compton 1927)
5. Particle-Antiparticle Pair Production (Carl Anderson 1932)
6. Discussion: a Wave or a Particle (?) –> “Duality”
Electromagnetic Waves
behaving like
Particles
(Chapter 3, or 2-old book)
Previous Lectures
1. Black Body Radiation (Max Planck 1900)
2. The Photoelectric Effect (Albert Einstein 1905)
3. The Production of X-Rays (Wilhelm Roentgen 1901)
4. The Compton Effect (Arthur Compton 1927)
5. Particle-Antiparticle Pair Production (Carl Anderson 1932)
6. Discussion: a Wave or a Particle (?) –> “Duality”
Electromagnetic Waves
behaving like
Particles
(Chapter 3, or 2 in old book)
“PHOTONS”
Black Body Radiation
The Photoelectric Effect
The Production of X-Rays
Electromagnetic Waves
behaving like
Particles
(Chapter 2)
“PHOTONS”
Particle-Antiparticle Pair Production
PHOTONS
E = hf
The Compton Effect PHOTONS
p = hf/c = h/l
Although diffraction of light is
a wave phenomenon,
there is no smooth distribution of light
in the diffraction pattern,
but
the pattern is rather formed of many
individual hits of particles – the photons
A single photon
DOES NOT
get “disintegrated” in the
Diffraction process
to make a smooth
diffraction pattern
Amplitude =
function of (x,t)
),( tx
Probability
Amplitude
The probability density
to find
a particle
at coordinate x, at time t
2|),(| tx
2|)(| xdxP =
The probability
to find
a particle
in an interval x
Integrate over x
How does the Probability Wave Move?
Equation of Motion for y ?
The Free-Particle
Schrodinger Wave
Equation
),( txy
2* |),(|),(),( txtxtx yyy =
Probability Wave Function
Probability
Density
a complex function ),( txy
),(),(),( 21 txitxtx yyy
Probability Density = ?
),(),(),( 21
* txitxtx yyy
Complex Conjugate
1i
12 =i
),(),(),( 21 txitxtx yyy
Probability Density
),(),(),( 21
* txitxtx yyy 1=i
== ),(),(|),(| *2 txtxtx yyy
== 12212211 yyyyyyyy iiii
2211 yyyy =
),(),(|),(|2
2
2
1
2 txtxtx yyy =
The Uncertainty Relations and the
Fourier Transform
Any wave may be expressed mathematically as a
superposition of plane waves of different
wavelengths and amplitudes
Chapter 3 –
The End
Bound States -
Some Simple Cases
1. A Review of Classical Bound States (4.4)
2. The Schrodinger Equation for Interacting Particles (4.1) 3. Stationary States (4.2) 4. Well-Behaved Functions and Normalization (4.3) 5. Case I: Particle in a Box – Infinite Potential Well (4.5) 6. Expectation Values, Uncertainties and Operators (4.6) 7. Case II: The Finite Potential Well (4.7) 8. Case III: The Simple Harmonic Oscillator (4.8)
The Schrodinger Equation
for Interacting Particles
A Particle Interacting
With What?
Simplification:
The Concept of Potential (replaces all individual
particle-particle interactions
with a single smooth potential)
The Schrodinger Equation
for Interacting Particles
Try to add potential energy U(x)
For free
particles
Key Assumption:
Factorization of the wave function
Spatial Part Temporal Part
What happens with the Schrodinger equation?
The spatial part of y(x,t)
The time-independent
Schrodinger equation:
y(x) is Real,
but y(x,t) is Complex, because
(t)=e-it
NOTE:
Spatial part
Normalization of y(x,t)
The particle must be
somewhere in the universe
at any time
(the total probability should be = 1)
Smoothness of y(x,t)
Discontinuity
in y(x) dxexk ikx
= )(2
1)(~ y
py
x
)(xy
Extremely large k (or short l) >
> Infinite Momentum impossible
Bound States Some Simple Cases (Chapter 5 – The End (old book 4))
• A Review of Classical Bound States (4.4)
• The Schrodinger Equation for Interacting Particles (4.1)
• Stationary States (4.2)
• Well-Behaved Functions and Normalization (4.3)
• Case I: Particle in a Box – Infinite Potential Well (4.5)
1. Case II: The Finite Potential Well (4.7)
2. Case III: The Simple Harmonic Oscillator (4.8)
3. Read at Home: Expectation Values, Uncertainties and
Operators (4.6)
Transcendental Equation –
impossible to solve analytically, but
“NUMERICAL” Solution is possible
(e.g. “graphing solution”)
Unbound States
(Chapter 6 (old 5))
5.1 Obstacles and Tunneling
5.2 Decay and other applications
5.3 Particle-wave propagation
5.4 The Classical Limit @ home
@ home
Hydrogen Atom
(Chapter 7 (old book - 6))
6.1 The Schrodinger Equation in 3 dimensions
6.2 The 3D Infinite Potential well
6.3 Toward the Hydrogen Atom
6.4 Central Forces
Toward the Hydrogen Atom
proton
electron
Energy loss due to Bremsstrahlung
(because of centripetal acceleration)
The
Problem:
The meaning of l
Angular part of Schrodinger Eq.
Units of angular
momentum
L ~ Magnitude of Angular
Momentum
Quantized so far:
The projection of
angular momentum to
z- axis
The magnitude of
(orbital) angular momentum
ml =
MAGNETIC QUANTUM
NUMBER
l =
ORBITAL
QUANTUM NUMBER
Ground State:
The Electron is NOT
Orbiting around the
proton
0)1( == llL eGroundStat
Classical Physics:
The Electron is Orbiting
around the Proton
New Topic
Spin and Atomic Physics
• The concept of “SPIN”
• The Stern-Gerlach Experiment
• Toward the Exclusion Principle for
“Fermions”
Let’s “create” multi-electron atoms
Our Toolkit:
The Schrodinger Equation
Quantization of Energy Levels
The Exclusion Principle
“Progressive Occupation” of Energy Levels
Fluorine
F
Like Ne with
one electron
(and proton)
less
A fluorine atom would
gladly accept one more
electron, to
“look more like Ne”
That is why it is
chemically very reactive
What maters
for Chemical
Properties is the
state of the most
loose electrons
The other, more
strongly bound
electrons are
merely passive
“placeholders”
The traditional naming
scheme
Principal
quantum
number “n” The
number of
electrons in
that
“subshell”
9.1 When atoms come together
9.5 Energy Bands
9.6 Conductors, Insulators, Semiconductors
Bonding of Atoms
Chapter 10 (old book - 9)
Insulators, Conductors, Semiconductors
Semiconductors –
insulators with a
small energy gap
(Eg < 2 eV)
LED
MASSIVE
Recombination:
Very high doping level
needed,
and
REFLECTION
Mirrors needed
The Laser Diode
Laser
The range of the force and the
mass of the mediator
Photon - the mediator of the
Electromagnetic force
Mass = 0
Range = infinite (~1/r potential)
The range of a force and the mass
of the mediator
Finite Mass Finite Range
Zero Mass Infinite Range
mcxrange
mcmcctcx
mcEt
mcE
tE
1
22
2
2
2
2
m
Relativistic Treatment is Mandatory
Because particles
may be created
and annihilated
(internal energy
plays a role; mc2)
Schrodinger Equation NON-relativistic
Relativistic Quantum Mechanics
Relativistic Quantum Mechanics:
The Klein-Gordon Wave Equation
The Dirac Equation (for Fermions)
The concept of SPIN follows naturally
Let’s “create” multi-nucleon nuclei
Our Toolkit:
The Schrodinger Equation
Quantization of Energy Levels
The Exclusion Principle
“Progressive Occupation” of Energy Levels
Rutherford Back-scattering and
The measurement of the Size of a Nucleus
Kinetic Energy = Potential Energy at Closest Approach
r” = 0
No return -
the nuclei
collide
r
r’
The Radius of a Nucleus
From Rutherford back-scattering,
and other types of measurements,
the radius of a nucleus:
Atomic number 1.2 fm (femtometer)
1 fm = 10-15 m
The Volume of a Nucleus
The volume is
proportional to the
number of nucleons
(A)
Spherical Shape
Assumed
(an excellent
approximation)
The Density of Nuclear Matter
independent of A
~ everywhere inside a nucleus
The density is constant, and:
Constant Density of Nuclear
Matter
Nucleons are closely packed in
an ~incompressible liquid-like
“droplet”
The potential well due to the (residual)
strong force
Nucleon-nucleon distance
Finite-range interaction Nucleons cannot
penetrate each other
Binding
energy per
(the least
bound)
electron in
an atom
(~10 eV)
Binding
energy per
nucleon in a
nucleus
(~ MeV)
Radioactivity
• Alpha – He nucleus
• Beta - Electron
• Gamma – Photon
• Fission – breakup
into two nuclei
Fission Explosives
U-235 and Pu-239
Exponential increase in energy release of a
supercritical assembly
Without control Nuclear (“atomic”) Bomb
Many Interesting Experiments
• Search for Proton Decay
• Cosmic Rays of unexpectedly high energies
(1020 eV)
• SUSY (Super Symmetric Particles),
annihilation in space etc.
• Distribution of the Cosmic Background Radiation
• The mass of the neutrinos
• What is Dark Matter? Dark Energy?