1st and 2 Midterm Exam - University of Rochesterbadolato/PHY_123/Resources_files/04_04_2012.pdf1st...

1st and 2nd Midterm Exam

Average 83.1 73.4

Stand. Dev. 12.0 12.8

Max. 99 95

Min. 40 14

Chapter 37 Early Quantum Theory and

Models of the Atom

•  37-1 Planck’s Quantum Hypothesis; Blackbody Radiation

•  37-2 Photon Theory of Light and the Photoelectric Effect

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

•  37-4 Compton Effect

•  37-5 Photon Interactions; Pair Production

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

•  37- 7 Wave Nature of Matter

Units of Chapter 37

At the end of the XXI century several relevant phenomena in nature could not be explained by the classical theories of mechanics, electrodynamics, and thermodynamics. Only the introduction of two revolutionary new theories

Theory of Relativity and the Quantum Mechanics

allowed us to describe nature in a more satisfactory way, determining a truly paradigm shift.

Chapter 37 Early Quantum Theory and Models

of the Atom

The construction of the first nuclear reactor in Chicago in 1942 and the development of nuclear technology could not have happened without a proper understanding of the quantum properties of particles and nuclei. However, the real breakthrough for a wide recognition of the relevance of quantum effects in technology occurred with the invention of the transistor in 1948 and the ensuing rapid development of semiconductor electronics.

Chapter 37 Early Quantum Theory and Models

of the Atom

Electronic devices like transistors rely heavily on the quantum mechanical. Today the rapid developments of spintronics, photonics, and nanotechnology provide continuing testimony to the technological relevance of quantum mechanics.

Blackbody (or Thermal) Radiation One of the phenomena that was unexplained by classical theories was the spectrum of light emitted by hot objects.

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.

37-1 Planck’s Quantum Hypothesis; Blackbody Radiation

This plot 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:


λmaxT = 2.90 ×10−3 m ⋅K

Applying Wien's Law to humans, one finds that the peak wavelength (λmax) of light emitted by a person is This is why thermal imaging devices designed for humans are most sensitive to 7–14 µm wavelength.

λmax =2.90 ×10−3 mK

305 K≅ 9.5 µm

Example:


Much of a person's energy is radiated away in the form of infrared energy. Some mater ia ls a re transparent to infrared light, while opaque to visible light (note the plastic bag).

O t h e r m a t e r i a l s a r e transparent to visible light, while opaque or reflective to the infrared (note the man's glasses).


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


In 1900 the German physicist Max Planck, proposed a formula (Planck’s radiation formula) that nicely fit the data


I(�, T ) =2⇡hc2��5

ehc/�kBT � 1

Planck made a new and radical assumption:

The energy emitted by charged oscillators in matter is limited to a set of discrete, integer multiples of a fundamental unit of energy, E, proportional to the oscillation frequency f :

E = n h f n =1,2,3,...h = 6.6 ×10−34 J ⋅s

The constant h is now called Planck’s constant.


Planck’s theory of radiation is therefore a quantum theory, because energy is not a continuous quantity as was believed for centuries; rather it is quantized, that is exist only in discrete amounts.

E = hf , 2hf , 3hf , ...


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

E = hf

Since light comes from a radiating source, when light is emitted by a charged oscillator, the oscillator energy of nhf must decrease by an amount nhf. Thus to conserve energy, the light must be emitted in packets, or quanta, each with energy

E = hf

The photon is massless, has no electric charge, and does not decay spontaneously in empty space. A photon has two possible polarization states and is described by exactly three continuous parameters: the components of its wave vector, which determine its wavelength λ and its direction of propagation.


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.


Photoelectric Effect

photons

Photo-generated electrons

Photoelectric Effect

A diagram illustrating the emission of electrons from a metal plate, requiring energy gained from an incoming photon to be more than the work function of the material.

When light shines on a metal surface, electrons are found to be emitted from the surface. This effect is called photoelectric effect.

Ca: 2.87 eV —› 432 nm Pt: 5.12-5.93 eV —› 220 nm

Na: 2.36 eV —› 526 nm

Kmax = eV0


Maximum Electron Kinetic Energy (Kmax) vs. Light Frequency

According to the wave theory of light:

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

2.  Frequency would not matter.


If light is described as 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.


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


E2 = p2c2 + m2c4

Because a photon always travels at the speed of light, it is truly a relativistic particle. The total energy of a relativistic particle can be written as (Eq 36-13):

where k is the wave vector (and the wave number is |k| = 2π/λ), ω= 2πf is the angular frequency, and h the Planck’s constant.

For photons m = 0, therefore the energy and momentum of a photon depend only on its frequency (f) (or equivalently its wavelength):

E = hf = h2π

ω =hcλ

E = pc ⇒ p = Ec=hfc

=hλ=h2π

k p = h2πk

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


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 37-6: 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.


Example 37-7: Photosynthesis.

In photosynthesis, pigments such as chlorophyll in plants capture the energy of sunlight to change CO2 to useful carbohydrate. About nine photons are needed to transform one molecule of CO2 to carbohydrate and O2. Assuming light of wavelength λ = 670 nm (chlorophyll absorbs most strongly in the range 650 nm to 700 nm), how efficient is the photosynthetic process? The reverse chemical reaction releases an energy of 4.9 eV/molecule of CO2.

′λ − λ =hmec

(1− cosφ)

In 1923, Compton observed that the wavelength of scattered X-rays from different materials was slightly longer than the incident ones, and that the wavelength depended on the scattering angle:

37-4 Compton Effect

Compton Effect

′λ − λ =hmec

(1− cosφ)

Radiation scattered from a target such

as graphite

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 c a r b o n . W h a t w i l l b e t h e wavelengths of X-rays scattered at (a) 0°, (b) 90°, and (c) 180°?

Compton Effect Pair Production

Photoelectric Effect Atom absorption/emission

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.


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.


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

LIGHT

is a particle

Photoelectric effect, Compton effect

Some experiments indicate that light behaves like a wave, others like a stream of particles.

is a wave

Interference (and diffraction) experiments


LIGHT

is a particle

Photoelectric effect, Compton effect

is a wave

Interference (and diffraction) experiments

we must accept the dual wave–particle nature of light


The principle of complementarity states that both the wave and particle aspects of light are fundamental to its nature.

Indeed, waves and particles are just our interpretations of how light behaves.

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

Great Books

Wave−Particle Duality

wave particle

pE c

⎛

⎝⎜

⎞

⎠⎟

k

ω c

⎛

⎝⎜

⎞

⎠⎟=

h2π

h = 6.6 ×10−34 J ⋅s

Wave−Particle Duality

wave particle

pE c

⎛

⎝⎜

⎞

⎠⎟

k

ω c

⎛

⎝⎜

⎞

⎠⎟=

h2π

p = h2πk → λ = h

pEc= h2π

ωc

→ E = hf

wave particle

pE c

⎛

⎝⎜

⎞

⎠⎟

k

ω c

⎛

⎝⎜

⎞

⎠⎟=

h2π

The Wave Nature of Matter

λ =hp

If waves can behave like particle, then particles can behave like waves

De Broglie wavelength

Electron diffraction

Photon diffraction

37-7 Wave Nature of Matter

Similarly to X-ray diffraction, atom in crystals can be used to diffract electrons:

In 1927, two American physicists C. J. Davisson and L. H. Germer performed the crucial experiment that confirmed the

de Broglie hypothesis.

Diffraction pattern of electrons scattered from Al foil.



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.


λ =hp=

hmv

=(6.6 ×10−34 J s)(0.2 kg) (15 m/s)

= 2.2 ×10−34 m

The de Broglie wavelength of an ordinary object is too small to be detected.

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.

The properties of waves, such as interference and diffraction, are significant only when the size of objects or slits is not much larger than the wavelength. If the mass is really small, the wavelength can be large enough to be measured.



Example 37-11: Wavelength of an electron.

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


Example 37-11: Wavelength of an electron.

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

λ =hp=

hmv

=(6.6 ×10−34 J s)

(9.1×10−31 kg) (5.9 ×10−6 m/s)= 0.12 ×10−9 m

Similarly to X-ray diffraction, atom in crystals can be used to diffract electrons

1st and 2 Midterm Exam - University of Rochesterbadolato/PHY_123/Resources_files/04_04_2012.pdf1st...

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