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Copyright © 2009 Pearson Education, Inc.

Chapter 36The Special Theory of Relativity


• Galilean–Newtonian Relativity

• The Michelson–Morley Experiment

• Postulates of the Special Theory of Relativity

• Simultaneity

• Time Dilation and the Twin Paradox

• Length Contraction

Units of Chapter 36


• Galilean and Lorentz Transformations

• Relativistic Momentum

• The Ultimate Speed

• E = mc2; Mass and Energy

• Doppler Shift for Light

• The Impact of Special Relativity

Units of Chapter 36


Definition of an inertial reference frame:

An object experiencing zero net force moves at constant velocity.

One in which Newton’s first law is valid.

Earth is rotating and therefore not an inertial reference frame, but we can treat it as one for many purposes.

A frame moving with a constant velocity with respect to an inertial reference frame is itself inertial.

36-1 Galilean–Newtonian Relativity


Relativity principle:

The basic laws of physics are the same in all inertial reference frames.


A reference frame that moves with constant velocity with respect to an inertial frame is itself also an inertial frame. Newton’s laws hold in it as well.

When we say that we observe or make measurement from a certain reference frame, it means that we are at rest in that reference frame.

Laws are the same but paths may be different in a different reference frame. (example – dropping a coin)

In classical mechanics, space and time are absolute: their measurement does not change from one reference frame as in another. The mass of an object as well as all forces are assumed to be unchanged by a change in inertial reference frame.

Position and velocity are different in different reference frames, but length is the same.

Laws of mechanics are the same in all inertial reference frames implies that on one inertial frame is special in any sense.

Thus, all inertial frames are equivalent.

Einstein published THREE papers of extraordinary importance.

• Analysis of Brownian motion

• Photoelectric effect

• Special theory of relativity, proposing the drastic revisions in the Newtonian concepts of space and time.

In special theory of relativity, Einstein based it on TWO postulates.

1. Law of physics are the same in all inertial frames of reference.

2. The speed of light is the same in all inertial frames of reference.

Implications of these proposition:

1. Events that are simultaneous for observer may not be simultaneous for another.

2. When two observers moving relative to each other measure a time interval or length, they may not get the same results.

3. For the conservation principles of momentum and energy to be valid in all inertial frames, Newton’s second law and the equations for momentum and kinetic energy need to be revised.

1. A passenger in a train at 30 ms-1 passes a man standing on a platform at t = t’ = 0 s. 20 s after the train passes him, the man on the platform determines that a bird flying along the tracks in the same direction as the train is 800 m away. What is the distance of the bird as determined by the passenger? [200 m]

2. 5 s later, the man on the platform determines that the bird is 850 m away. Find the velocity of the bird as determined by the man i. on the platform[+10 ms-1]ii. on the train.[-20 ms-1]

3. A sample of radioactive material, at rest in the lab, ejects two electrons in opposite direction. One of the electrons has a speed of 0.6c and the other has a speed of 0.7c . What is the speed of one electron as measured from the other?[-1.3c]

4. An observer, at rest observes the following collision: m1 = 3 kg moves with velocity u1 = 4 ms-1 along the x axis approaches m2 =1 kg moving with velocity u2 = - 3 ms-1 along the x axis. After head-on collision m2 has velocity v2 = 3 ms-1 along the x axis. Find v1 of m1 after collision.[2 ms-1]

5. A second observer , who is walking with a velocity of 2 ms-1 relative to the ground along the x axis. What are the velocities before and after the collision.[2 ,-5,0,1]


This principle works well for mechanical phenomena.

However, Maxwell’s equations yield the velocity of light; it is 3.0 x 108 m/s.

So, which is the reference frame in which light travels at that speed?

Scientists searched for variations in the speed of light depending on the direction of the ray, but found none.



36-2 The Michelson–Morley Experiment

This experiment was designed to measure the speed of the Earth with respect to the ether.

The Earth’s motion around the Sun should produce small changes in the speed of light, which would be detectable through interference when the split beam is recombined.



The Michelson interferometer is sketched here, along with an analogy using a boat traveling in a river.

Show that



This interferometer was able to measure interference shifts as small as 0.01 fringe, while the expected shift was 0.4 fringe.

However, no shift was ever observed, no matter how the apparatus was rotated or what time of day or night the measurements were made.

The possibility that the arms of the apparatus became slightly shortened when moving against the ether was considered, but a full explanation had to wait until Einstein came into the picture.


1. The laws of physics have the same form in all inertial reference frames.

2. Light propagates through empty space with speed c independent of the speed of source or observer.

This solves the ether problem – the speed of light is in fact the same in all inertial reference frames.

36-3 Postulates of the Special Theory of Relativity


One of the implications of relativity theory is that time is not absolute. Distant observers do not necessarily agree on time intervals between events, or on whether they are simultaneous or not.

Why not?

In relativity, an “event” is defined as occurring at a specific place and time. Let’s see how different observers would describe a specific event.

36-4 Simultaneity


Thought experiment: lightning strikes at two separate places. One observer believes the events are simultaneous – the light has taken the same time to reach her – but another, moving with respect to the first, does not.

36-4 Simultaneity


Here, it is clear that if one observer sees the events as simultaneous, the other cannot, given that the speed of light is the same for each.

36-4 Simultaneity


A different thought experiment, using a clock consisting of a light beam and mirrors, shows that moving observers must disagree on the passage of time.

36-5 Time Dilation and the Twin Paradox


Calculating the difference between clock “ticks,” we find that the interval in the moving frame is related to the interval in the clock’s rest frame:



The factor multiplying t0 occurs so often in relativity that it is given its own symbol, γ:

We can then write


.



Example 36-1: Lifetime of a moving muon.

(a) What will be the mean lifetime of a muon as measured in the laboratory if it is traveling at v = 0.60c = 1.80 x 108 m/s with respect to the laboratory? Its mean lifetime at rest is 2.20 μs = 2.2 x 10-6 s.(b) How far does a muon travel in the laboratory, on average, before decaying?


To clarify:

• The time interval in the frame where two events occur in the same place is t0.

• The time interval in a frame moving with respect to the first one is Δt.




Example 36-2: Time dilation at 100 km/h.

Let us check time dilation for everyday speeds. A car traveling covers a certain distance in 10.00 s according to the driver’s watch. What does an observer at rest on Earth measure for the time interval?



Example 36-3: Reading a magazine on a spaceship.

A passenger on a high-speed spaceship traveling between Earth and Jupiter at a steady speed of 0.75c reads a magazine which takes 10.0 min according to her watch. (a) How long does this take as measured by Earth-based clocks? (b) How much farther is the spaceship from Earth at the end of reading the article than it was at the beginning?


It has been proposed that space travel could take advantage of time dilation – if an astronaut’s speed is close enough to the speed of light, a trip of 100 light-years could appear to the astronaut as having been much shorter.

The astronaut would return to Earth after being away for a few years, and would find that hundreds of years had passed on Earth.



This brings up the twin paradox – if any inertial frame is just as good as any other, why doesn’t the astronaut age faster than the Earth traveling away from him?

The solution to the paradox is that the astronaut’s reference frame has not been continuously inertial – he turns around at some point and comes back.




Conceptual Example 36-4: A relativity correction to GPS.

GPS satellites move at about 4 km/s = 4000 m/s. Show that a good GPS receiver needs to correct for time dilation if it is to produce results consistent with atomic clocks accurate to 1 part in 1013.


If time intervals are different in different reference frames, lengths must be different as well. Length contraction is given by

or

Length contraction occurs only along the direction of motion.

36-6 Length Contraction


36-6 Length ContractionExample 36-5: Painting’s contraction.

A rectangular painting measures 1.00 m tall and 1.50 m wide. It is hung on the side wall of a spaceship which is moving past the Earth at a speed of 0.90c. (a) What are the dimensions of the picture according to the captain of the spaceship? (b) What are the dimensions as seen by an observer on the Earth?


36-6 Length ContractionExample 36-6: A fantasy supertrain.

A very fast train with a proper length of 500 m is passing through a 200-m-long tunnel. Let us imagine the train’s speed to be so great that the train fits completely within the tunnel as seen by an observer at rest on the Earth. That is, the engine is just about to emerge from one end of the tunnel at the time the last car disappears into the other end. What is the train’s speed?


36-6 Length Contraction

Conceptual Example 36-7: Resolving the train and tunnel length.

Observers at rest on the Earth see a very fast 200-m-long train pass through a 200-m-long tunnel (as in Example 36–6) so that the train momentarily disappears from view inside the tunnel. Observers on the train measure the train’s length to be 500 m and the tunnel’s length to be only 78 m. Clearly a 500-m-long train cannot fit inside a 78-m-long tunnel. How is this apparent inconsistency explained?


36-8 Galilean and Lorentz Transformations

A classical (Galilean) transformation between inertial reference frames involves a simple addition of velocities:


36-8 Galilean and Lorentz TransformationsFor an object that is moving at speed u parallel to the x axis in frame S , its velocity in S is again given by addition.



Relativistically, the conversion between one inertial reference frame and another must take into account length contraction:

For the moment, we will leave γ as a constant to be determined; requiring that the speed of light remain unchanged when transforming from one frame to the other shows γ to have its previously defined value.



We can then solve for t, and find that the full transformation involves both x and t . This is called the Lorentz transformation:

.



The Lorentz transformation gives the same results we have previously found for length contraction and time dilation.

Velocity transformations can be found by differentiating x, y, and z with respect to time:


36-8 Galilean and Lorentz TransformationsExample 36-8: Adding velocities.

Calculate the speed of rocket 2 with respect to Earth.


The expression for momentum also changes at relativistic speeds:

36-9 Relativistic Momentum


36-9 Relativistic MomentumExample 36-9: Momentum of moving electron.

Compare the momentum of an electron when it has a speed of (a) 4.00 x 107 m/s in the CRT of a television set, and (b) 0.98c in an accelerator used for cancer therapy.



The formula for relativistic momentum can be derived by requiring that the conservation of momentum during collisions remain valid in all inertial reference frames.


Gamma and the rest mass are sometimes combined to form the relativistic mass:


Care must be taken not to use the relativistic mass inappropriately; it is valid only as a component of the relativistic momentum.


A basic result of special relativity is that nothing can equal or exceed the speed of light. Such an object would have infinite momentum – not possible for anything with mass – and would take an infinite amount of energy to accelerate to light speed.

36-10 The Ultimate Speed


At relativistic speeds, not only is the formula for momentum modified; that for energy is as well.

The total energy can be written:

In the particle’s own rest frame,

36-11 E = mc2; Mass and Energy



Example 36-10: Pion’s kinetic energy.

A π0 meson (m = 2.4 x 10-28 kg) travels at a speed v= 0.80c = 2.4 x 108 m/s. What is its kinetic energy? Compare to a classical calculation.



Example 36-11: Energy from nuclear decay.

The energy required or released in nuclear reactions and decays comes from a change in mass between the initial and final particles. In one type of radioactive decay, an atom of uranium (m = 232.03714 u) decays to an atom of thorium (m = 228.02873 u) plus an atom of helium (m = 4.00260 u), where the masses given are in atomic mass units (1 u = 1.6605 x 10–27 kg). Calculate the energy released in this decay.



Example 36-12: A 1-TeV proton.

The Tevatron accelerator at Fermilab in Illinois can accelerate protons to a kinetic energy of 1.0 TeV (1012 eV). What is the speed of such a proton?


Combining the relations for energy and momentum gives the relativistic relation between them:



All the formulas presented here become the usual Newtonian kinematic formulas when the speeds are much smaller than the speed of light.

There is no rule for when the speed is high enough that relativistic formulas must be used – it depends on the desired accuracy of the calculation.



The predictions of special relativity have been tested thoroughly, and verified to great accuracy.

The correspondence principle says that a more general theory must agree with a more restricted theory where their realms of validity overlap. This is why the effects of special relativity are not obvious in everyday life.

36-13 The Impact of Special Relativity


• Inertial reference frame: one in which Newton’s first law holds.

• Principles of relativity: the laws of physics are the same in all inertial reference frames; the speed of light in vacuum is constant regardless of speed of source or observer.

• Time dilation:

Summary of Chapter 36

.


• Length contraction:

• Gamma:



• Relativistic momentum:


• Lorentz transformations:

.


• Mass–energy relation:

• Kinetic energy:

• Total energy:


• Relationship between energy and momentum:


Chapter 37Early Quantum Theory and

Models of the Atom


• Planck’s Quantum Hypothesis; Blackbody Radiation

• Photon Theory of Light and the Photoelectric Effect

• Energy, Mass, and Momentum of a Photon

• Compton Effect

• Photon Interactions; Pair Production

• Wave–Particle Duality; the Principle of Complementarity


• Wave Nature of Matter

• Electron Microscopes

• Early Models of the Atom

• Atomic Spectra: Key to the Structure of the Atom

• The Bohr Model

• de Broglie’s Hypothesis Applied to Atoms


All objects emit radiation whose total intensity is proportional to the fourth power of their temperature. This is called thermal radiation; a blackbody is one that emits thermal radiation only.

The spectrum of blackbody radiation has been measured; it is found that the frequency of peak intensity increases linearly with temperature.

37-1 Planck’s Quantum Hypothesis; Blackbody Radiation

The heat energy Q radiated in a time interval Δt by an object with surface area A and temperature T is givenby

Where is the Stefan-Boltzmann constant.The parameter e is the emissivity of the surface, a measure of how effectively it radiates.The value of e ranges from 0 to 1. A perfectly absorbing – thus perfectly emitting - object with e = 1 is called a blackbody. Charcoal is an example of a blackbody.If we measure the spectrum of a blackbody at a certain temperature, a continuous curve will appear.

If we measure the spectrum of a blackbody at a certain temperature, a continuous curve will appear. There are FOUR important features of the spectra:i. All blackbodies at the same temperature emit exactly the same spectrum. The spectrum depends on only an object’s temperature, not the material of which it is made.Ii. Increasing the temperature increases the radiated intensity at all wavelength. Making the object hotter causes it to emit more radiation across the entire spectrumiii. Increasing the temperature causes the peak intensity to shift towards shorter wavelengths. The higher the temperature, the shorter the wavelength of the peak of the spectrum.Iv. The visible rainbow that we see is only a small portion of the continuous blackbody spectrum. Much of the emission is infrared, and very hot objects also radiate violet wavelengths.


This figure shows blackbody radiation curves for three different temperatures. Note that frequency increases to the left. The relationship between the temperature and peak wavelength is given by Wien’s law:




Example 37-1: The Sun’s surface temperature.

Estimate the temperature of the surface of our Sun, given that the Sun emits light whose peak intensity occurs in the visible spectrum at around 500 nm.



Example 37-2: Star color.

Suppose a star has a surface temperature of 32,500 K. What color would this star appear?


The observed blackbody spectrum could not be reproduced using 19th-century physics. This plot shows the disagreement.


Low Frequency Radiation The classical and quantum expressions for radiation from a hot object are Making use of the series expansion of the exponential:

http://hyperphysics.phy-astr.gsu.edu/hbase/cseri.html#c2


A solution was proposed by Max Planck in 1900. He suggested that the energy of atomic oscillations within atoms cannot have an arbitrary value; it is related to the frequency:

The constant h is now called Planck’s constant.



Einstein suggested that, given the success of Planck’s theory, light must be emitted in small energy packets:

These tiny packets, or particles, are called photons.

37-2 Photon Theory of Light and the Photoelectric Effect

.


The photoelectric effect: if light strikes a metal, electrons are emitted. The effect does not occur if the frequency of the light is too low; the kinetic energy of the electrons increases with frequency.


Several features of the photoelectric effect:

1. Dependence of photoelectron kinetic energy

on light intensity.

Classical prediction: Electrons should absorb energy

continuously from the EM waves. As the light

intensity incident on a metal is increased,

energy should be transferred into the metal at

a higher rate and the electrons should be

ejected with more kinetic energy.

Experimental result: The maximum kinetic energy of

photoelectrons is independent of light intensity

with both curves falling to zero at the same

negative voltage.

2. Time interval between incidence of light and

ejection of photoelectrons.

Classical prediction: At low light intensities, a

measurable time of interval should pass

between the instant the light is turned on

and the time an electron is ejected from the

metal. This time interval is required for the

electron to absorb the incident radiation

before it acquires enough energy to escape

the metal.

Experimental result: Electrons are emitted from the

surface of the metal almost instantaneously,

even at very low light intensities.

3. Dependence of ejection of electrons on light

frequency.

Classical prediction: Electrons should be ejected

from the metal at any incident light

frequency, as long as the light intensity is

high enough, because energy is transferred

to the metal regardless of the incident light

frequency.

Experimental result: No electrons are emitted if

the incident light frequency falls below some

cutoff frequency f0, whose value is

characteristic of the material being

illuminated. No electrons are ejected below

this cutoff frequency regardless of the light

intensity.

4. Dependence of photoelectron kinetic energy

on light frequency.

Classical prediction: There should be no

relationship between the frequency of the

light and the electron kinetic energy. The

kinetic energy should be related to the

intensity of light.

Experimental result: The maximum kinetic energy

of the photoelectron increases with the

increasing light frequency.

Mathematical description

The maximum kinetic energy Kmax of an ejected electron

is given by

where h is the Planck constant,

f is the frequency of the incident photon, and

φ = hf0 is the work function

(sometimes denoted W), which is the minimum energy

required to remove

a delocalised electron from the surface of any given metal.

The work function, in turn, can be written as

http://en.wikipedia.org/wiki/Kinetic_energy

http://en.wikipedia.org/wiki/Planck_constant

http://en.wikipedia.org/wiki/Work_function

where f0 is called the threshold frequency for the metal.

The maximum kinetic energy of an ejected electron is

Because the kinetic energy of the electron must be

positive, it follows that the frequency f of the incident

photon must be greater than f0 in order for the

photoelectric effect to occur.[6]

http://en.wikipedia.org/wiki/Kinetic_energy

http://en.wikipedia.org/wiki/Photoelectric_effect#cite_note-Fromhold1991-5


If light is a wave, theory predicts:

1. Number of electrons and their energy should increase with intensity.

2. Frequency would not matter.



If light is particles, theory predicts:

• Increasing intensity increases number of electrons but not energy.

• Above a minimum energy required to break atomic bond, kinetic energy will increase linearly with frequency.

• There is a cutoff frequency below which no electrons will be emitted, regardless of intensity.



The particle theory assumes that an electron absorbs a single photon.

Plotting the kinetic energy vs. frequency:

This shows clear agreement with the photon theory, and not with wave theory.




Example 37-3: Photon energy.

Calculate the energy of a photon of blue light, λ = 450 nm in air (or vacuum).



Example 37-4: Photons from a lightbulb.

Estimate how many visible light photons a 100-W lightbulb emits per second. Assume the bulb has a typical efficiency of about 3% (that is, 97% of the energy goes to heat).



Example 37-5: Photoelectron speed and energy.

What are the kinetic energy and the speed of an electron ejected from a sodium surface whose work function is W0 = 2.28 eV when illuminated by light of wavelength (a) 410 nm and (b) 550 nm?


The photoelectric effect is how “electric eye” detectors work. It is also used for movie film sound tracks.



Clearly, a photon must travel at the speed of light. Looking at the relativistic equation for momentum, it is clear that this can only happen if its rest mass is zero.

We already know that the energy is hf; we can put this in the relativistic energy-momentum relation and find the momentum:

37-3 Energy, Mass, and Momentum of a Photon


37-3 Energy, Mass, and Momentum of a Photon

Example 37-6: Photon momentum and force.

Suppose the 1019 photons emitted per second from the 100-W lightbulb in Example 37–4 were all focused onto a piece of black paper and absorbed. (a) Calculate the momentum of one photon and (b) estimate the force all these photons could exert on the paper.


Compton did experiments in which he scattered X-rays from different materials. He found that the scattered X-rays had a slightly longer wavelength than the incident ones, and that the wavelength depended on the scattering angle:

37-4 Compton Effect


This is another effect that is correctly predicted by the photon model and not by the wave model.

37-4 Compton Effect

In physics, Compton scattering is a type

of scattering that X-rays and gamma rays

undergo in matter. The inelastic scattering

of photons in matter results in a decrease

in energy (increase in wavelength) of an X-

ray or gamma ray photon, called the

Compton effect. Part of the energy of the

X/gamma ray is transferred to a scattering

electron, which recoils and is ejected from

its atom (which becomes ionized), and the

rest of the energy is taken by the scattered,

"degraded" photon.

http://en.wikipedia.org/wiki/Physics

http://en.wikipedia.org/wiki/Scattering

http://en.wikipedia.org/wiki/X-rays

http://en.wikipedia.org/wiki/Gamma_rays

http://en.wikipedia.org/wiki/Inelastic_scattering

http://en.wikipedia.org/wiki/Energy

http://en.wikipedia.org/wiki/Wavelength

http://en.wikipedia.org/wiki/X-ray

http://en.wikipedia.org/wiki/Gamma_ray

http://en.wikipedia.org/wiki/Photon

http://en.wikipedia.org/wiki/Ionized

Since the wavelength of the scattered light is different

from the incident radiation, Compton scattering is an

example of inelastic scattering, but the origin of the

effect can be considered as an elastic collision between

a photon and an electron. The amount the wavelength

changes by is called the Compton shift. Although

nuclear compton scattering exists[1], Compton scattering

usually refers to the interaction involving only the

electrons of an atom. The Compton effect was observed

by Arthur Holly Compton in 1923.

http://en.wikipedia.org/wiki/Inelastic_scattering

http://en.wikipedia.org/w/index.php?title=Nuclear_compton_scattering&action=edit&redlink=1

http://en.wikipedia.org/wiki/Compton_scattering#cite_note-0

http://en.wikipedia.org/wiki/Electron

http://en.wikipedia.org/wiki/Atom

http://en.wikipedia.org/wiki/Arthur_Holly_Compton

The effect is important because it demonstrates that light

cannot be explained purely as a wave phenomenon.

Light must behave as if it consists of particles to explain the

low-intensity Compton scattering. Compton's experiment

convinced physicists that light can behave as a stream of

particle-like objects (quanta) whose energy is proportional to

the frequency.

http://en.wikipedia.org/wiki/Wave

The interaction between electrons and high energy

photons (comparable to the rest energy of the electron,

511 keV) results in the electron being given part of the

energy (making it recoil), and a photon containing the

remaining energy being emitted in a different direction

from the original, so that the overall momentum of the

system is conserved. If the photon still has enough

energy left, the process may be repeated. In this

scenario, the electron is treated as free or loosely

bound.

http://en.wikipedia.org/wiki/Electron


http://en.wikipedia.org/wiki/Electronvolt

http://en.wikipedia.org/wiki/Momentum

If the photon is of lower energy, but still has sufficient

energy (in general a few eV to a few KeV,

corresponding to visible light through soft X-rays), it

can eject an electron from its host

atom entirely (a process known as the photoelectric

effect), instead of undergoing Compton scattering.

Higher energy photons (1.022 MeV and above) may be

able to bombard the nucleus and cause

an electron and a positron to be formed, a process

called pair production.

http://en.wikipedia.org/wiki/Visible_light

http://en.wikipedia.org/wiki/Photoelectric_effect

http://en.wikipedia.org/wiki/Pair_production

Description of the phenomenon

A photon of wavelength λ comes in from the left,

collides with a target at rest,

and a new photon of wavelength λ′ emerges at an angle θ.

http://en.wikipedia.org/wiki/File:Compton-scattering.svg

http://en.wikipedia.org/wiki/File:Compton-scattering.svg

By the early 20th century, research into the interaction

of X-rays with matter was well underway. It was known

that when a beam of X-rays is directed at an atom, an

electron is ejected and is scattered through an angle θ.

Classical electromagnetism predicts that the

wavelength of scattered rays should be equal to the

initial wavelength however, multiple experiments found

that the wavelength of the scattered rays was greater

than the initial wavelength.

http://en.wikipedia.org/wiki/X-ray

http://en.wikipedia.org/wiki/Classical_electromagnetism

In 1923, Compton published a paper in the Physical Review

explaining the phenomenon. Using the notion of

quantized radiation and the dynamics of special relativity,

Compton derived the relationship between the shift in

wavelength and the scattering angle:

where

λ is the initial wavelength,

λ′ is the wavelength after scattering,

h is the Planck constant,

me is the mass of the electron,

c is the speed of light, and

θ is the scattering angle.

http://en.wikipedia.org/wiki/Physical_Review


http://en.wikipedia.org/wiki/Special_relativity

http://en.wikipedia.org/wiki/Planck_constant

http://en.wikipedia.org/wiki/Speed_of_light

The quantity h⁄mec is known as the Compton wavelength of

the electron; it is equal to 2.43×10−12

m. The wavelength shift λ′ − λ is at least zero (for θ = 0°)

and at most twice the Compton wavelength of the electron

(for θ = 180°). Compton found that some X-rays

experienced no wavelength shift despite being scattered

through large angles; in each of these cases the photon

failed to eject an electron. Thus the magnitude of the shift

is related not to the Compton wavelength of the electron,

but to the Compton wavelength of the entire atom, which

can be upwards of 10 000 times smaller.

http://en.wikipedia.org/wiki/Compton_wavelength

Derivation of the scattering formula

A photon γ with wavelength λ is directed at an electron e

in an atom, which is at rest.The collision causes the

electron to recoil, and a new photon γ′ with

wavelength λ′ emerges at angle θ.

Let e′ denote the electron after the collision.

From the conservation of energy,

http://en.wikipedia.org/wiki/Conservation_of_energy

Compton postulated that photons carry momentum;[3] thus

from the conservation of momentum, the momenta of the

particles should be related by

assuming the initial momentum of the electron is zero..

The photon energies are related to the frequencies by

http://en.wikipedia.org/wiki/Compton_scattering#cite_note-taylor_136-9-2

http://en.wikipedia.org/wiki/Conservation_of_momentum


37-4 Compton Effect

Example 37-8: X-ray scattering.

X-rays of wavelength 0.140 nm are scattered from a very thin slice of carbon. What will be the wavelengths of X-rays scattered at (a) 0°, (b) 90°, and (c) 180°?

The unshifted peak λo is caused by x-rays scattered

From electrons tightly bound to the target. This

unshifted peak is also predicted by the Compton

equation if the electron mass is replaced with the mass

of a carbon atom, which is approximate 23000X the

mass of the electron. Therefore, there is a wavelength

shift for scattering from an electron bound to an

atom, but it is too small to be detected in the Compton

experiment.


Photons passing through matter can undergo the following interactions:

1. Photoelectric effect: photon is completely absorbed, electron is ejected.

2. Photon may be totally absorbed by electron, but not have enough energy to eject it; the electron moves into an excited state.

3. The photon can scatter from an atom and lose some energy.

4. The photon can produce an electron–positron pair.

37-5 Photon Interactions; Pair Production


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.


Pair production refers to the creation of an elementary

particle and its antiparticle, usually from a photon (or

another neutral boson). For example an electron and its

antiparticle, the positron, may be created. This is

allowed, provided there is enough energy available to

create the pair – at least the total rest mass energy of

the two particles – and that the situation allows both

energy and momentum to be conserved (though not

necessarily on shell).

http://en.wikipedia.org/wiki/Elementary_particle

http://en.wikipedia.org/wiki/Antiparticle


http://en.wikipedia.org/wiki/Boson


http://en.wikipedia.org/wiki/Rest_mass_energy

http://en.wikipedia.org/wiki/On_shell

Other pairs produced could be a muon and anti-muon or a tau

and anti-tau.

However all other conserved quantum numbers (angular

momentum, electric charge, lepton number

(whether the particle is an electron, muon or tau)) of the

produced particles must sum to zero—thus the

created particles shall have opposite values of each (for

instance, if one particle has electric charge of +1

the other must have electric charge −1, or if one particle has

strangeness +1 then another one must have strangeness −1).

http://en.wikipedia.org/wiki/Angular_momentum

http://en.wikipedia.org/wiki/Electric_charge

http://en.wikipedia.org/wiki/Strangeness_%28particle_physics%29

In nuclear physics, this occurs when a high-energy

photon interacts in the vicinity of a nucleus. The energy

this (mass-less) photon has can be converted into mass

through Einstein's equation E=mc² where E is energy, m

is mass and c is the speed of light. Thus if the energy of

the photon is high enough so that it can make the mass of

an electron plus the mass of a positron (basically twice

the mass of an electron which is 9.11 × 10−31 kg) then an

electron-positron pair may be created.

http://en.wikipedia.org/wiki/Nuclear_physics


http://en.wikipedia.org/wiki/Atomic_nucleus

If there is more energy in the photon than just enough to

create the mass of the electron-positron pair then the

electron and positron will have some kinetic energy -

meaning they will be moving. The electron and positron

can move in opposite directions (at an angle of 180

degrees) meaning they have a total momentum of zero or

they can move at an angle of less than 180 degrees

resulting in a combined momentum which is very small

(since momentum is a vector quantity).

However, if the photon only just had enough energy to create the

mass of the electron-positron pair then the electron and positron

will be at rest. This violate the conservation of momentum since

the photon has momentum and the two resulting particles have

none if they are stationary. This means that the pair production

must take place near another photon or the nucleus of an atom

since they will be able to absorb the momentum of the original

photon, i.e since the momentum of the initial photon must be

absorbed by something, pair production cannot occur in empty

space out of a single photon; the nucleus (or another photon) is

needed to conserve both momentum and energy.

Photon-nucleus pair production can only occur if the

photons have an energy exceeding twice the rest energy

(mec2) of an electron (1.022 MeV), photon-photon pair

production may occur at 511 keV; the same

conservation laws apply for the generation of other

higher energy leptons such as the muon and tauon (for

two photons each should have the one-particle energy in

the center of momentum frame, for one photon and a

heavy nucleus, the photon needs the entire pair rest

energy).

http://en.wikipedia.org/wiki/Electronvolt

http://en.wikipedia.org/wiki/Muon

http://en.wikipedia.org/wiki/Tau_lepton



Example 37-9: Pair production.

(a) What is the minimum energy of a photon that can produce an electron–positron pair? (b) What is this photon’s wavelength?


We have phenomena such as diffraction and interference that show that light is a wave, and phenomena such as the photoelectric effect and the Compton 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.

37-6 Wave-Particle Duality; the Principle of Complementarity


The principle of complimentarity states that both the wave and particle aspects of light are fundamental to its nature. Neither model can be used exclusively to describe all properties of light. A complete understanding of the observed behavior can be attained only if the two models are combined in a complimentary manner.

In 1923 de Broglie in his doctoral thesis

postulated because photon have both wave and

particle properties, perhaps all form of matter

have both properties. This highly revolutionary

idea had no experimental confirmation at that

time. According to de Broglie, electrons, just like

light, have dual properties


Just as light sometimes behaves like a particle, matter sometimes behaves like a wave. The momentum of photon is

De Broglie suggested the wavelength of a particle of matter is given by the same expression.

In analogy with photons, de Broglie postulated that particle obey the Einstein relation E = hf.

37-7 Wave Nature of Matter

The Davisson-Germer experiment demonstrated the wave nature of the electron, confirming the earlier hypothesis of deBroglie. Putting wave-particle duality on a firm experimental footing, it represented a major step forward in the development of quantum mechanics. The Bragg law for diffraction had been applied to x-ray diffraction, but this was the first application to particle waves.

http://hyperphysics.phy-astr.gsu.edu/hbase/davger.html#c1

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/bragg.html#c1

Davisson and Germer designed and built a vacuum apparatus for the purpose of measuring the energies of electrons scattered from a metal surface. Electrons from a heated filament were accelerated by a voltage and allowed to strike the surface of nickel metal.

The electron beam was directed at the nickel target, which could be rotated to observe angular dependence of the scattered electrons. Their electron detector (called a Faraday box) was mounted on an arc so that it could be rotated to observe electrons at different angles. It was a great surprise to them to find that at certain angles there was a peak in the intensity of the scattered electron beam. This peak indicated wave behavior for the electrons, and could be interpreted by the Bragg law to give values for the lattice spacing in the nickel crystal.

The wave length associated with the above equation agreed with the De-Broglie's prediction. Thus it is confirmed that electron has a wave like nature because only a wave has wave length.

The dual nature of matter is apparent because

each contains both particle quantities (p and E)

and wave properties (λ and f)



Example 37-10: Wavelength of a ball.

Calculate the de Broglie wavelength of a 0.20-kg ball moving with a speed of 15 m/s.



Example 37-11: Wavelength of an electron.

Determine the wavelength of an electron that has been accelerated through a potential difference of 100 V.


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.



The wave nature of electrons is manifested in experiments where an electron beam interacts with the atoms on the surface of a solid. By studying the angular distribution of the diffracted electrons, one can indirectly measure the geometrical arrangement of atoms. Assume that the electrons strike perpendicular to the surface of a solid, and that their energy is low, K = 100 eV, so that they interact only with the surface layer of atoms. If the smallest angle at which a diffraction maximum occurs is at 24°, what is the separation d between the atoms on the surface?


The wavelength of electrons will vary with energy, but is still quite short. This makes electrons useful for imaging – remember that the smallest object that can be resolved is about one wavelength. Electrons used in electron microscopes have wavelengths of about 0.004 nm.

37-8 Electron Microscopes


Transmission electron microscope – the electrons are focused by magnetic coils



Scanning electron microscope – the electron beam is scanned back and forth across the object to be imaged.



It was known that atoms were electrically neutral, but that they could become charged, implying that there were positive and negative charges and that some of them could be removed.

One popular atomic model was the “plum-pudding” model:

37-9 Early Models of the Atom


This model had the atom consisting of a bulk positive charge, with negative electrons buried throughout.

Rutherford did an experiment that showed that the positively charged nucleus must be extremely small compared to the rest of the atom. He scattered alpha particles – helium nuclei – from a metal foil and observed the scattering angle. He found that some of the angles were far larger than the plum-pudding model would allow.



The only way to account for the large angles was to assume that all the positive charge was contained within a tiny volume – now we know

that the radius of the nucleus is 1/10,000 that of the atom.



Therefore, Rutherford’s model of the atom is mostly empty space:



A very thin gas heated in a discharge tube emits light only at characteristic frequencies.

37-10 Atomic Spectra: Key to the Structure of the Atom


An atomic spectrum is a line spectrum – only certain frequencies appear. If white light passes through such a gas, it absorbs at those same frequencies.



The wavelengths of electrons emitted from hydrogen have a regular pattern:

This is called the Balmer series. R is the Rydberg constant:



Other series include the Lyman series:

and the Paschen series:



A portion of the complete spectrum of hydrogen is shown here. The lines cannot be explained by the Rutherford theory.



Bohr proposed that the possible energy states for atomic electrons were quantized – only certain values were possible. Then the spectrum could be explained as transitions from one level to another.

37-11 The Bohr Model


Bohr found that the angular momentum was quantized:


.


An electron is held in orbit by the Coulomb force:



Using the Coulomb force, we can calculate the radii of the orbits:


.


The lowest energy level is called the ground state; the others are excited states.


Example 37-13: Wavelength of a Lyman line.

Use this figure to determine the wavelength of the first Lyman line, the transition from n = 2 to n = 1. In what region of the electromagnetic spectrum does this lie?


Example 37-14: Wavelength of a Balmer line.

Determine the wavelength of light emitted when a hydrogen atom makes a transition from the n = 6 to the n = 2 energy level according to the Bohr model.


Example 37-15: Absorption wavelength.

Use this figure to determine the maximum wavelength that hydrogen in its ground state can absorb. What would be the next smaller wavelength that would work?


Example 37-16: He+ ionization energy.

(a) Use the Bohr model to determine the ionization energy of the He+ ion, which has a single electron. (b) Also calculate the maximum wavelength a photon can have to cause ionization.


Conceptual Example 37-17: Hydrogen at 20°C.

Estimate the average kinetic energy of whole hydrogen atoms (not just the electrons) at room temperature, and use the result to explain why nearly all H atoms are in the ground state at room temperature, and hence emit no light.


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 Bohr had proposed.

In addition, it makes more reasonable the fact that the electrons do not radiate, as one would otherwise expect from an accelerating charge.

37-12 de Broglie’s Hypothesis Applied to Atoms


These are circular standing waves for n = 2, 3, and 5.

37-12 de Broglie’s Hypothesis Applied to Atoms


• Planck’s hypothesis: molecular oscillation energies are quantized:

• Light can be considered to consist of photons, each of energy

• Photoelectric effect: incident photons knock electrons out of material.


.


• Compton effect and pair production also support photon theory.

• Wave–particle duality – both light and matter have both wave and particle properties.

• Wavelength of an object:


.


• Principle of complementarity: both wave and particle properties are necessary for complete understanding.

• Rutherford showed that atom has tiny nucleus.

• Line spectra are explained by electrons having only certain specific orbits.

• Ground state has the lowest energy; the others are called excited states.


Chapter 38Quantum Mechanics

• Quantum Mechanics – A New Theory

• The Wave Function and Its Interpretation; the Double-Slit Experiment

• The Heisenberg Uncertainty Principle

• Philosophic Implications; Probability versus Determinism

• The Schrödinger Equation in One Dimension – Time-Independent Form

• Time-Dependent Schrödinger Equation

• Free Particles; Plane Waves and Wave Packets

• Particle in an Infinitely Deep Square Well Potential (a Rigid Box)

• Finite Potential Well

• Tunneling Through a Barrier

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.

Quantum Mechanics – A New Theory

Some characteristics of ideal particle and ideal wave.

An ideal particle has an zero size and an essential feature of a particle is that it is localized in space.

An ideal wave has a single frequency and is infinitely long . Therefore, an ideal wave is unlocalized in space.

A localized entity can be built from infinitely long waves as follows:

Imagine drawing one wave along the x-axis.

Now, draw a second wave of the same amplitude but a different frequency.

As a result of superposition of these two waves, beats exist as the wave alternately in phase and out of phase.

We have already introduced some localization by superposing the two waves.

Imagine more and more waves are added.

The result is shown as below:

The small region of constructive interference is called wave packet. This localized region of space is different from all other regions.

We can identify the wave packet as a particle because it has the localized nature of a particle.

The location of the wave packet corresponds to the particle’s position.

From the wave equation, v =fλ = ω/k ( for individual particle)

Factor in the bracket

Δω/ Δk

For many waves,

d ω/dk= vg

Vg = dE/dp

• Exploring the possibility of the envelope of combined waves as particle moving with speed v ( << c), the energy of the particle is equivalent of kinetic energy K = ½ mv2

½ mv2 = p2/2m

Find dE/dp = vg

An electromagnetic wave has oscillating electric and magnetic fields. What is oscillating in a matter wave?

This role is played by the wave function, Ψ. The square 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.

The Wave Function and Its Interpretation; the Double-Slit

Experiment

For 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 Wave Function and Its Interpretation; the Double-Slit

Experiment

Quantum mechanics tells us there are limits to measurement – not because of the limits of our instruments, but inherently.

This is due to wave-particle duality, and to interaction between the observing equipment and the object being observed.

The Heisenberg Uncertainty Principle

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 – it could transfer anywhere from none to all of its momentum.

In addition, the position can only be measured to about one wavelength of the photon.


Combining, we find the combination of uncertainties:

This is called the Heisenberg uncertainty principle.

It tells us that the position and momentum cannot simultaneously be measured with precision.


This relation can also be written as a relation between the uncertainty in time and the uncertainty in energy:

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.


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 do.

Philosophic Implications; Probability versus Determinism

The Schrödinger Equation in OneDimension—Time-Independent Form

The Schrödinger equation cannot be derived, just as Newton’s laws cannot. However, we know that it must describe a traveling wave, and that energy must be conserved.

Therefore, the wave function will take the form:

where


Since energy is conserved, we know:

This suggests a form for the Schrödinger equation, which experiment shows to be correct:


Since the solution to the Schrödinger equation is supposed to represent a single particle, the total probability of finding that particle anywhere in space should equal 1:

When this is true, the wave function is normalized.

Free Particles; Plane Waves and Wave Packets

Free particle: no force, so U = 0. The Schrödinger equation becomes the equation for a simple harmonic oscillator, with solutions:

Since U = 0,

where

Free electron.

An electron with energy E = 6.3 eV is in free space (where U = 0).

Find (a) the wavelength λ (in nm) and

(b) the wave function for the electron (assuming B =

0).

The solution for a free particle is a plane wave, as shown in part (a) of the figure; more realistic is a wave packet, as shown in part (b). The wave packet has both a range of momenta and a finite uncertainty in width.

One of the few geometries where the Schrödinger equation can be solved exactly is the infinitely deep square well. As is shown, this potential is zero from the origin to a distance , and is infinite elsewhere.

The solution for the region between the walls is that of a free particle:

Requiring that ψ = 0 at x = 0 and x = gives B = 0 and k = nπ/ . This means that the energy is limited to the values:

Particle in an Infinitely Deep SquareWell Potential (a Rigid Box)

These plots show the energy levels, wave function, and probability distribution for several values of n.

Electron in an infinite potential well.

(a) Calculate the three lowest energy levels for an electron trapped in an infinitely deep square well potential of width = 1.00 x 10-10 m (about the diameter of a hydrogen atom in its ground state). (b) If a photon were emitted when the electron jumps from the n = 2 state to the n = 1 state, what would its wavelength be?

Calculating a normalization constant .

Show that the normalization constant A for all wave functions describing a particle in an infinite potential well of width has a value of:

2 / .A

Probability near center of rigid box.

An electron is in an infinitely deep square well potential of width = 1.0 x 10-10 m. If the electron is in the ground state, what is the probability of finding it in a region dx = 1.0 x 10-12

m of width at the center of the well (at x = 0.5 x 10-10 m)?

Probability of e- in ¼ of box.

Determine the probability of finding an electron in the left quarter of a rigid box—i.e., between one wall at x = 0 and position x = /4. Assume the electron is in the ground state.

Most likely and average positions.

Two quantities that we often want to know are the most likely position of the particle and the average position of the particle. Consider the electron in the box of width = 1.00 x 10-10 m in the first excited state n = 2. (a) What is its most likely position? (b) What is its average position?

A confined bacterium.

A tiny bacterium with a mass of about 10-14 kg is confined between two rigid walls 0.1 mm apart. (a) Estimate its minimum speed. (b) If instead, its speed is about 1 mm in 100 s, estimate the quantum number of its state.

A finite potential well has a potential of zero between x = 0 and x = , but outside that range the potential is a constant U0.

The potential outside the well is no longer zero; it falls off exponentially.

The wave function in the well is different than that outside it; we require that both the wave function and its first derivative be equal at the position of each wall (so it is continuous and smooth), and that the wave function be normalized.

These graphs show the wave functions and probability distributions for the first three energy states.

Figure 38-13a goes here. Figure 38-13b goes here.

Since the wave function does not go to zero immediately upon encountering a finite barrier, there is some probability of finding the particle represented by the wave function on the other side of the barrier. This is called tunneling.

Figure 38-15 goes here.

The probability that a particle tunnels through a barrier can be expressed as a transmission coefficient, T, and a reflection coefficient, R (where T + R = 1). If T is small,

The smaller E is with respect to U0, the smaller the probability that the particle will tunnel through the barrier.

Barrier penetration.

A 50-eV electron approaches a square barrier 70 eV high and (a) 1.0 nm thick, (b) 0.10 nm thick. What is the probability that the electron will tunnel through?

Alpha decay is a tunneling process; this is why alpha decay lifetimes are so variable.

Scanning tunneling microscopes image the surface of a material by moving so as to keep the tunneling current constant. In doing so, they map an image of the surface.

Figure 38-18 goes here.

• Quantum mechanics is the basic theory at the atomic level; it is statistical rather than deterministic

• Heisenberg uncertainty principle:

• Particles are represented by wave functions.

• The absolute square of the wave function represents the probability of finding particle in a given location.

• Wave function satisfies Schrödinger equation:

Equation 38-5 goes here.

• A particle in an infinite potential well has quantized energy levels:

Equation 38-13 goes here.

• In a finite well, probability extends into classically forbidden areas.

• Particles can tunnel through barriers of finite height and width.