Quantum Mechanics 300 fall 18... · 2018-09-14 · Photoelectric Effect Fall 2018 Prof. Sergio B. Mendes 44 • Electromagnetic radiation interacts with electrons within the metal

Quantum Mechanics

Fall 2018 Prof. Sergio B. Mendes 1

An essential theory to understand properties of matter and light.

• Chemical• Electronic• Magnetic• Thermal• Optical• Etc.

Experimental Basis of Quantum Physics


CHAPTER 3

Topics


• 3.1 Discovery of the X Ray and the Electron• 3.2 Determination of Electron Charge• 3.3 Line Spectra• 3.4 Quantization• 3.5 Blackbody Radiation• 3.6 Photoelectric Effect• 3.7 X-Ray Production• 3.8 Compton Effect• 3.9 Pair Production and Annihilation

3.1 Discovery of the X Rays


X rays were discovered by Wilhelm Röntgen in 1895.

Observed x rays were emitted by cathode rays bombarding glass

Cathode Ray Experiments


In the 1890s scientists and engineers were familiar with “cathode rays”. These rays were generated from one of the metal plates in an evacuated tube across which a large electric potential had been established.

It was known that cathode rays could penetrate matter and their properties were under intense investigation during the 1890s.

It was considered that cathode rays had something to do with atoms.

Observation of X Rays


These rays were unaffected by magnetic fields and penetrated materials more than cathode rays.

Wilhelm Röntgen studied the effects of cathode rays passing through various materials.

He noticed that a phosphorescent screen near the tube glowed during some of these experiments.

He called them x rays and deduced that they were produced by the cathode rays bombarding the glass walls of his vacuum tube.

Röntgen’s X Ray Tube


Röntgen constructed an x-ray tube by allowing cathode rays to impact the glass wall of the tube to produce x rays.

He used x rays to image the bones of a hand on a phosphorescent screen.

First X Ray Diagnostic !!


Discovery of the Electron


Electrons were discovered by J. J. Thomson, 1897.

Observed that cathode rays were charged particles

Thomson’s Experiment


Thomson used an evacuated cathode-ray tube to show that the cathode rays were negatively charged particles (electrons) by deflecting them in electric and magnetic fields.

Analysis of Thomson’s Experiment


𝐹𝐹𝑦𝑦 = 𝑞𝑞 𝐸𝐸 = 𝑚𝑚 𝑎𝑎𝑦𝑦 𝑡𝑡𝑎𝑎𝑎𝑎 𝜃𝜃 =𝑣𝑣𝑦𝑦𝑣𝑣𝑥𝑥

=𝑞𝑞 𝐸𝐸𝑚𝑚

𝑙𝑙𝑣𝑣𝑥𝑥 2

𝑣𝑣𝑦𝑦 = 𝑎𝑎𝑦𝑦 𝑡𝑡 =𝑞𝑞 𝐸𝐸𝑚𝑚

𝑙𝑙𝑣𝑣𝑥𝑥

𝑞𝑞𝑚𝑚

=𝑣𝑣𝑥𝑥 2 𝑡𝑡𝑎𝑎𝑎𝑎 𝜃𝜃

𝐸𝐸 𝑙𝑙

𝐵𝐵 = 0

Magnetic Field turned ON and adjust for:


𝑞𝑞 𝐸𝐸 + 𝑞𝑞 �⃗�𝑣 × 𝐵𝐵 = 0

𝑣𝑣𝑥𝑥 =𝐸𝐸𝐵𝐵

𝑞𝑞𝑚𝑚

=𝑣𝑣𝑥𝑥 2 𝑡𝑡𝑎𝑎𝑎𝑎 𝜃𝜃

𝐸𝐸 𝑙𝑙=𝐸𝐸 𝑡𝑡𝑎𝑎𝑎𝑎 𝜃𝜃𝐵𝐵2 𝑙𝑙1.76 × 1011

𝐶𝐶𝑘𝑘𝑘𝑘

=

3.2 Millikan’s Experiment and Determination of

Electron Charge


Major Steps:


−𝑚𝑚𝑘𝑘 + 𝑏𝑏 𝑣𝑣 = 𝐹𝐹

−𝑚𝑚𝑘𝑘 + 𝑞𝑞 𝐸𝐸 − 𝑏𝑏 𝑣𝑣 = 𝐹𝐹

= 𝑚𝑚𝑑𝑑𝑣𝑣𝑑𝑑𝑡𝑡

= 0

= 𝑚𝑚𝑑𝑑𝑣𝑣𝑑𝑑𝑡𝑡= 0

−𝑚𝑚𝑘𝑘 + 𝑏𝑏 𝑣𝑣𝑑𝑑 = 0

−𝑚𝑚𝑘𝑘 + 𝑞𝑞 𝐸𝐸 − 𝑏𝑏 𝑣𝑣𝑢𝑢 = 0

𝑞𝑞 𝐸𝐸 − 𝑏𝑏 𝑣𝑣𝑢𝑢 − 𝑏𝑏 𝑣𝑣𝑑𝑑 = 0

Charges were multiples of unit “e”


𝑞𝑞 = 𝑎𝑎 𝑒𝑒 =𝑏𝑏 𝑣𝑣𝑑𝑑 + 𝑣𝑣𝑢𝑢

𝐸𝐸

𝑒𝑒 = 1.6 × 10−19 𝐶𝐶

𝑚𝑚 = 9.1 × 10−31 𝑘𝑘𝑘𝑘

3.3 Line Spectra


Color Separation by a Diffraction Grating


𝑑𝑑 𝑠𝑠𝑠𝑠𝑎𝑎 𝜃𝜃𝑚𝑚 = 𝑚𝑚 𝜆𝜆 grating equation

0 1 2𝑚𝑚 =

𝑚𝑚 = 1

Emission and Absorption Experiments


Emission Spectra of a Few Atoms


• Identification of Chemical Elements

• Composition of Materials

Balmer’s series for the Hydrogen atom, 1885


𝜆𝜆 = 364.56 𝑎𝑎𝑚𝑚𝑘𝑘2

𝑘𝑘2 − 4

𝑘𝑘 = 3, 4, 5, …

A simple rearragenment:


1𝜆𝜆

= 𝑅𝑅𝐻𝐻1

22−

1𝑘𝑘2

𝜆𝜆 = 364.56 𝑎𝑎𝑚𝑚𝑘𝑘2

𝑘𝑘2 − 4

𝑅𝑅𝐻𝐻 ≡4

364.56 𝑎𝑎𝑚𝑚

𝑘𝑘 = 3, 4, 5, …

=1.096776 × 107

𝑚𝑚

Rydberg equation, 1890


1𝜆𝜆

= 𝑅𝑅𝐻𝐻1𝑎𝑎2

−1𝑘𝑘2

𝑘𝑘 > 𝑎𝑎

3.4 Quantization


Discrete quantities !!

• Electric charge

• Discrete spectral lines

• Atomic mass

• ?? (many more to come)

3.5 Blackbody Radiation


Matter & Electromagnetic Radiation in Thermal Equilibrium


• Constant temperature

• Absorption and emission are balanced

• Continuous spectral distribution

𝑻𝑻

Experimental Spectral Distribution


Empirical Laws:


Wien’s Law:

Stefan-Boltzmann Law:

𝜆𝜆𝑚𝑚𝑚𝑚𝑥𝑥 𝑇𝑇 ≅ 2.898 × 10−3 𝑚𝑚 𝐾𝐾

𝑅𝑅 𝑇𝑇 = �0

∞𝛪𝛪 𝜆𝜆,𝑇𝑇 𝑑𝑑𝜆𝜆 = 𝜖𝜖 𝜎𝜎 𝑇𝑇4

𝜎𝜎 = 5.6705 × 10−8 W/(m2 K4)

𝜖𝜖 = 0 , 1

Classical Calculation of Blackbody Radiation


𝑢𝑢 𝑓𝑓 = 𝑁𝑁 𝑓𝑓 �𝐸𝐸

𝛪𝛪 𝑓𝑓,𝑇𝑇 = 𝑐𝑐 𝑢𝑢 𝑓𝑓

∆𝐴𝐴

𝑐𝑐 ∆𝑡𝑡

𝑢𝑢 𝑓𝑓

𝑐𝑐 ∆𝑡𝑡

𝑐𝑐 ∆𝑡𝑡14

Power per-unit-area per-unit-frequency:

Appendix 1

Density of Modes (standing waves) of Electromagnetic Radiation


𝑁𝑁 𝑓𝑓 = 8 𝜋𝜋𝑓𝑓2

𝑐𝑐3

𝑁𝑁 𝑓𝑓 = ? ?

Number of modes per-unit-volume per-unit-frequency

1D Standing Waves


𝑓𝑓 𝑥𝑥 = 𝐴𝐴 sin(𝑘𝑘 𝑥𝑥)

𝑘𝑘 𝐿𝐿 = 𝑚𝑚 𝜋𝜋 ∆𝑚𝑚 = ∆𝑘𝑘𝐿𝐿𝜋𝜋

𝑥𝑥

𝑓𝑓 𝑥𝑥

𝑚𝑚 = 1, 2, 3, …

3D Standing Waves


𝑓𝑓 𝑥𝑥 = 𝐴𝐴 sin(𝑘𝑘𝑥𝑥 𝑥𝑥) sin 𝑘𝑘𝑦𝑦 𝑦𝑦 sin(𝑘𝑘𝑧𝑧 𝑧𝑧)

𝑘𝑘𝑥𝑥 𝐿𝐿𝑥𝑥 = 𝑚𝑚𝑥𝑥 𝜋𝜋 ∆𝑚𝑚𝑥𝑥 = ∆𝑘𝑘𝑥𝑥𝐿𝐿𝑥𝑥𝜋𝜋

𝑘𝑘𝑦𝑦 𝐿𝐿𝑦𝑦 = 𝑚𝑚𝑦𝑦 𝜋𝜋 ∆𝑚𝑚𝑦𝑦 = ∆𝑘𝑘𝑦𝑦𝐿𝐿𝑦𝑦𝜋𝜋

𝑘𝑘𝑧𝑧 𝐿𝐿𝑧𝑧 = 𝑚𝑚𝑧𝑧 𝜋𝜋 ∆𝑚𝑚𝑧𝑧 = ∆𝑘𝑘𝑧𝑧𝐿𝐿𝑧𝑧𝜋𝜋

∆𝑚𝑚𝑥𝑥 ∆𝑚𝑚𝑦𝑦 ∆𝑚𝑚𝑧𝑧 = ∆𝑘𝑘𝑥𝑥 ∆𝑘𝑘𝑦𝑦 ∆𝑘𝑘𝑧𝑧𝑉𝑉𝜋𝜋3

𝑀𝑀 =18

4 𝜋𝜋3𝑘𝑘3

𝑉𝑉𝜋𝜋3

𝑑𝑑𝑀𝑀𝑉𝑉

=1

2 𝜋𝜋2𝑘𝑘2 𝑑𝑑𝑘𝑘 = 4 𝜋𝜋 𝑓𝑓2

𝑐𝑐3 𝑑𝑑𝑓𝑓

𝑘𝑘 = 2 𝜋𝜋𝑓𝑓𝑐𝑐


𝑁𝑁 𝑓𝑓 = 2𝑑𝑑𝑀𝑀𝑉𝑉 𝑑𝑑𝑓𝑓 = 2 × 4 𝜋𝜋

𝑓𝑓2

𝑐𝑐3 = 8 𝜋𝜋𝑓𝑓2

𝑐𝑐3

𝑑𝑑𝑀𝑀𝑉𝑉

= 4 𝜋𝜋𝑓𝑓2

𝑐𝑐3𝑑𝑑𝑓𝑓

Average Energy for Each Mode of the Electromagnetic Radiation


�𝐸𝐸 = ? ?

�𝐸𝐸 =∫0∞𝐸𝐸 𝑝𝑝 𝐸𝐸 𝑑𝑑𝐸𝐸

∫0∞𝑝𝑝 𝐸𝐸 𝑑𝑑𝐸𝐸

𝑝𝑝 𝐸𝐸 = 𝑒𝑒−𝐸𝐸𝑘𝑘𝑘𝑘

= 𝑘𝑘 𝑇𝑇=∫0∞𝐸𝐸 𝑒𝑒−

𝐸𝐸𝑘𝑘𝑘𝑘𝑑𝑑𝐸𝐸

∫0∞ 𝑒𝑒−

𝐸𝐸𝑘𝑘𝑘𝑘 𝑑𝑑𝐸𝐸

�𝐸𝐸 = 𝑘𝑘 𝑇𝑇

Appendix 2

Bringing All the Pieces Together:




𝑐𝑐3

𝛪𝛪 𝑓𝑓,𝑇𝑇 =14𝑐𝑐 𝑢𝑢 𝑓𝑓 = 2 𝜋𝜋 𝑓𝑓2

𝑐𝑐2𝑘𝑘 𝑇𝑇



�𝐸𝐸 = 𝑘𝑘 𝑇𝑇


𝛪𝛪 𝑓𝑓,𝑇𝑇 = 2 𝜋𝜋 𝑓𝑓2


𝛪𝛪 𝜆𝜆,𝑇𝑇 𝑑𝑑𝜆𝜆 = 𝛪𝛪 𝑓𝑓,𝑇𝑇 𝑑𝑑𝑓𝑓

𝜆𝜆 𝑓𝑓 = 𝑐𝑐

𝛪𝛪 𝜆𝜆,𝑇𝑇 = 2 𝜋𝜋 𝑐𝑐𝜆𝜆4𝑘𝑘 𝑇𝑇

𝑓𝑓 𝑑𝑑𝜆𝜆 + 𝜆𝜆 𝑑𝑑𝑓𝑓 = 0𝑑𝑑𝑓𝑓𝑑𝑑𝜆𝜆

=𝑓𝑓𝜆𝜆

=𝑐𝑐𝜆𝜆2

Rayleigh-Jeans Formula

Theory and Experiment


𝛪𝛪 𝜆𝜆,𝑇𝑇 = 2 𝜋𝜋 𝑐𝑐𝜆𝜆4𝑘𝑘 𝑇𝑇

Rayleigh-Jeans

Planck’s Assumption


�𝐸𝐸 = ? ?

�𝐸𝐸 =∑𝑛𝑛=0∞ 𝐸𝐸𝑛𝑛 𝑝𝑝 𝐸𝐸𝑛𝑛∑𝑛𝑛=0∞ 𝑝𝑝 𝐸𝐸𝑛𝑛

𝑝𝑝 𝐸𝐸 = 𝑒𝑒−𝐸𝐸𝑘𝑘𝑘𝑘

�𝐸𝐸 =∫0∞𝐸𝐸 𝑝𝑝 𝐸𝐸 𝑑𝑑𝐸𝐸

∫0∞𝑝𝑝 𝐸𝐸 𝑑𝑑𝐸𝐸

𝐸𝐸𝑛𝑛 = 𝑎𝑎 ℎ 𝑓𝑓

�𝐸𝐸 =ℎ 𝑓𝑓

𝑒𝑒ℎ 𝑓𝑓𝑘𝑘𝑘𝑘 − 1

Appendix 3

Planck Theory of Blackbody Radiation




𝑐𝑐3

𝛪𝛪 𝑓𝑓,𝑇𝑇 =14𝑐𝑐 𝑢𝑢 𝑓𝑓 =

2 𝜋𝜋 ℎ 𝑓𝑓3

𝑐𝑐2 𝑒𝑒ℎ 𝑓𝑓𝑘𝑘𝑘𝑘 − 1


𝑐𝑐3ℎ 𝑓𝑓


�𝐸𝐸 =ℎ 𝑓𝑓


Comparison:


𝛪𝛪 𝑓𝑓,𝑇𝑇 = 2 𝜋𝜋 𝑓𝑓2


𝛪𝛪 𝑓𝑓,𝑇𝑇 =2 𝜋𝜋 ℎ 𝑓𝑓3


Rayleigh-Jeans

Planck

ℎ = 6.6261 × 10−34 𝐽𝐽 𝑠𝑠

Frequency and Wavelength Descriptions:


𝛪𝛪 𝑓𝑓,𝑇𝑇 =2 𝜋𝜋 ℎ 𝑓𝑓3


𝛪𝛪 𝜆𝜆,𝑇𝑇 =2 𝜋𝜋 ℎ 𝑐𝑐2

𝜆𝜆5 𝑒𝑒ℎ 𝑐𝑐𝜆𝜆𝑘𝑘𝑘𝑘 − 1

𝛪𝛪 𝜆𝜆,𝑇𝑇 𝑑𝑑𝜆𝜆 = 𝛪𝛪 𝑓𝑓,𝑇𝑇 𝑑𝑑𝑓𝑓

𝑑𝑑𝑓𝑓𝑑𝑑𝜆𝜆

=𝑓𝑓𝜆𝜆

=𝑐𝑐𝜆𝜆2

𝜆𝜆 𝑓𝑓 = 𝑐𝑐

Wien’s Law from Planck’s Theory


𝑑𝑑𝛪𝛪 𝜆𝜆,𝑇𝑇𝑑𝑑𝜆𝜆

= 0



ℎ 𝑐𝑐𝜆𝜆𝑚𝑚𝑚𝑚𝑥𝑥 𝑘𝑘 𝑇𝑇

≅ 4.966 𝜆𝜆𝑚𝑚𝑚𝑚𝑥𝑥 𝑘𝑘 𝑇𝑇 ≅ℎ 𝑐𝑐

4.966 𝑘𝑘= 2.898 × 10−3 𝑚𝑚 𝐾𝐾

Appendix 4

Stefan-Boltzmann Law from Planck’s Theory




𝑅𝑅 𝑇𝑇 = �0

∞𝛪𝛪 𝜆𝜆,𝑇𝑇 𝑑𝑑𝜆𝜆 =

2 𝜋𝜋5 𝑘𝑘4

15 ℎ3 𝑐𝑐2𝑇𝑇4

𝜎𝜎 =2 𝜋𝜋5 𝑘𝑘4

15 ℎ3 𝑐𝑐2= 5.6705 × 10−8 W/(m2 K4)

PhET on Blackbody Radiation


https://phet.colorado.edu/sims/blackbody-spectrum/blackbody-spectrum_en.html

3.6 Methods of Electron Emission:


Thermionic emission: Application of heat allows electrons to gain enough energy to escape.

Secondary emission: The electron gains enough energy by transfer from another high-speed particle that strikes the material from outside.

Field emission: A strong external electric field pulls the electron out of the material.

Photoelectric effect: Incident light (electromagnetic radiation) shining on the material transfers energy to the electrons, allowing them to escape.

Photoelectric Effect


• Electromagnetic radiation interacts with electrons within the metal and can increase the electrons kinetic energy.

• Eventually electrons get enough extra kinetic energy that allow them to escape from the metal.

• We call the ejected electrons photoelectrons.

PhET on Photoelectric Effect


https://phet.colorado.edu/en/simulation/legacy/photoelectric

Same frequency, different light intensities


For a given frequency, the retarding potential energy is the same regardless of

the light intensity

Same light intensity,different frequencies


For a given light intensity, the retarding potential energy required to stop the

current increases with the light frequency.

Retarding potential energy versus frequency


The retarding potential energy required to stop the current increases linearly with

the light frequency.

Some Conclusions from Experiments:


• The kinetic energies of the photoelectrons are independent of the light intensity.

• The maximum kinetic energy of the photoelectrons, for a given emitting material, depends only on the frequency of the light.

• The smaller the work function of the emitter material, the smaller is the threshold frequency of the light that can eject photoelectrons.

• The photoelectrons are emitted almost instantly following illumination of the photocathode, independent of the intensity of the light.

• When the photoelectrons are produced, however, their number is proportional to the intensity of light.

A Few Puzzles for Classical Physics:


• Why the maximum kinetic energy of the photoelectrons does not depend on the light intensity ?

• Why there is a threshold frequency for the photocurrent ?

• Why the photoelectrons are ejected almost immediately after illumination, regardless of the light intensity ?

• Why the maximum kinetic energy depends on the light frequency ?

Einstein’s Article of 1905 on “Creation and Conversion of Light”



• Einstein suggested that electromagnetic radiation is quantized into particles called photons.

• Each photon has the energy quantum 𝐸𝐸 = ℎ 𝑓𝑓, where 𝑓𝑓 is the frequency of the light and ℎ is Planck’s constant.

• The photon travels at the speed of light 𝑐𝑐 in a vacuum, and its wavelength is given by 𝜆𝜆 = ⁄𝑐𝑐 𝑓𝑓

From those assumptions:


• Conservation of energy yields:

where 𝜙𝜙 is the work function of the metal.

• The retarding potentials measured in the photoelectric effect are the opposing potentials needed to stop the most energetic electrons.

ℎ 𝑓𝑓 − 𝜙𝜙 =12𝑚𝑚 𝑣𝑣2

𝑒𝑒 𝑉𝑉𝑜𝑜 =12𝑚𝑚 𝑣𝑣𝑚𝑚𝑚𝑚𝑥𝑥 2


• The kinetic energy of the electron does not depend on the light intensity at all, but only on the light frequency and the work function of the material.

• The stopping potential is linearly proportional to the light frequency, with a slope ℎ, the same constant found by Planck.

12𝑚𝑚 𝑣𝑣2 = ℎ 𝑓𝑓 − 𝜙𝜙

12𝑚𝑚 𝑣𝑣𝑚𝑚𝑚𝑚𝑥𝑥

2 = ℎ 𝑓𝑓 − 𝜙𝜙𝑒𝑒 𝑉𝑉𝑜𝑜 = = ℎ 𝑓𝑓 − 𝑓𝑓𝑜𝑜

Work Function of Materials


Millikan on Photoelectric Effect:


Millikan published data in 1916 for the photoelectric effect in which he shone light of varying frequency on a sodium electrode and measured the maximum kinetic energies of the photoelectrons.

He found that no photoelectrons were emitted below a frequency of 4.39 X 1014 Hz (or longer than a wavelength of 683 nm).

The results were independent of light intensity, and the slope of a straight line drawn through the data produced a value of Planck’s constant in excellent agreement with Planck’s theory.

Even though Millikan admitted his own data were sufficient proof of Einstein’s photo-electric effect equation, Millikan was not convinced of the photon concept for light and its role in quantum theory.

The photoelectric effect can be described by 𝛾𝛾 + 𝑒𝑒𝑒 → 𝑒𝑒.


Can the reverse process 𝑒𝑒 → 𝑒𝑒𝑒 + 𝛾𝛾occur ?

The short answer is yes.

3.7 X-Ray Production


𝑒𝑒 𝑉𝑉𝑜𝑜 = 𝐸𝐸𝑖𝑖

𝐸𝐸𝑓𝑓 + ℎ 𝑓𝑓 = 𝐸𝐸𝑖𝑖

𝐸𝐸𝑓𝑓 + ℎ𝑐𝑐𝜆𝜆

= 𝐸𝐸𝑖𝑖0, 𝑒𝑒 𝑉𝑉𝑜𝑜 = 𝐸𝐸𝑓𝑓

Bremsstrahlung, from the German word for “braking radiation”

𝑒𝑒 → 𝑒𝑒𝑒 + 𝛾𝛾

Three Different Materials


The relative intensity of x rays produced in an x-ray tube is shown for an accelerating voltage of 35 kV.

𝐸𝐸𝑓𝑓 + ℎ𝑐𝑐𝜆𝜆

= 𝐸𝐸𝑖𝑖𝐸𝐸𝑓𝑓 = 0, 𝑒𝑒 𝑉𝑉𝑜𝑜

𝐸𝐸𝑖𝑖 = 𝑒𝑒 𝑉𝑉𝑜𝑜

Notice that 𝜆𝜆𝑚𝑚𝑖𝑖𝑛𝑛 is the same for all three targets.

𝜆𝜆 =ℎ 𝑐𝑐

𝐸𝐸𝑖𝑖 − 𝐸𝐸𝑓𝑓𝜆𝜆𝑚𝑚𝑖𝑖𝑛𝑛 =

ℎ 𝑐𝑐𝑒𝑒 𝑉𝑉𝑜𝑜𝜆𝜆𝑚𝑚𝑖𝑖𝑛𝑛 = 0.0354 𝑎𝑎𝑚𝑚

3.8 Compton Effect


(Classical) Light-Matter Interactionand Scattered Light


• Electric field in the light wave drives the oscillation of electrons present in matter.

• Frequency of oscillation coincides with the frequency of the incident light.

• Re-emitted (scattered) light has the same frequency as the incident light.

• Thomson radiation

Compton Effect, 1923


𝛾𝛾 + 𝑒𝑒 → 𝛾𝛾𝑒 + 𝑒𝑒𝑒

• Conservation of electric charge

• Conservation of energy

• Conservation of linear momentum

• Instead of the collective interaction between light wave & electrons, Compton considered photon-electron interaction

• Described the phenomenon as single-photon & single-electron collision

Linear Momentum for Photons:


𝐸𝐸 = ℎ 𝑓𝑓 = 𝑝𝑝 𝑐𝑐

𝐸𝐸2 − 𝑝𝑝2 𝑐𝑐2 = 𝑚𝑚2 𝑐𝑐4

𝐸𝐸 = 𝑝𝑝 𝑐𝑐

𝑚𝑚 = 0

𝑝𝑝 = ℎ𝑓𝑓𝑐𝑐 =

ℎ𝜆𝜆

𝑝𝑝 =ℎ𝜆𝜆


Conservation of Energy:


ℎ 𝑓𝑓 + 𝐸𝐸𝑖𝑖 = ℎ 𝑓𝑓𝑒 + 𝐸𝐸𝑓𝑓

𝐸𝐸𝑖𝑖 = 𝑚𝑚 𝑐𝑐2

𝐸𝐸𝑓𝑓2 = 𝑚𝑚 𝑐𝑐2 2 + 𝑝𝑝𝑒𝑒2𝑐𝑐2

ℎ 𝑓𝑓 = ℎ𝑐𝑐𝜆𝜆

ℎ 𝑓𝑓′ = ℎ𝑐𝑐𝜆𝜆𝑒

ℎ𝑐𝑐𝜆𝜆

+ 𝑚𝑚 𝑐𝑐2 = ℎ𝑐𝑐𝜆𝜆𝑒

+ 𝑚𝑚 𝑐𝑐2 2 + 𝑝𝑝𝑒𝑒2𝑐𝑐2

Conservation of Linear Momentum:


ℎ𝜆𝜆

=ℎ𝜆𝜆𝑒𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 + 𝑝𝑝𝑒𝑒 𝑐𝑐𝑐𝑐𝑠𝑠 𝜙𝜙𝑥𝑥:

0 =ℎ𝜆𝜆𝑒𝑠𝑠𝑠𝑠𝑎𝑎 𝜃𝜃 − 𝑝𝑝𝑒𝑒 𝑠𝑠𝑠𝑠𝑎𝑎 𝜙𝜙𝑦𝑦:

From linear momentum equations:


ℎ𝜆𝜆

=ℎ𝜆𝜆𝑒𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 + 𝑝𝑝𝑒𝑒 𝑐𝑐𝑐𝑐𝑠𝑠 𝜙𝜙

0 =ℎ𝜆𝜆𝑒𝑠𝑠𝑠𝑠𝑎𝑎 𝜃𝜃 − 𝑝𝑝𝑒𝑒 𝑠𝑠𝑠𝑠𝑎𝑎 𝜙𝜙

ℎ𝜆𝜆

2

+ℎ𝜆𝜆𝑒

2

− 2ℎ𝜆𝜆

ℎ𝜆𝜆𝑒

𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 = 𝑝𝑝𝑒𝑒2

From energy equation:


ℎ𝑐𝑐𝜆𝜆

+ 𝑚𝑚 𝑐𝑐2 = ℎ𝑐𝑐𝜆𝜆𝑒

+ 𝑚𝑚 𝑐𝑐2 2 + 𝑝𝑝𝑒𝑒2𝑐𝑐2

ℎ𝑐𝑐𝜆𝜆− ℎ

𝑐𝑐𝜆𝜆𝑒

2+ 2 ℎ

𝑐𝑐𝜆𝜆− ℎ

𝑐𝑐𝜆𝜆𝑒

𝑚𝑚 𝑐𝑐2 = 𝑝𝑝𝑒𝑒2𝑐𝑐2

ℎ𝜆𝜆−ℎ𝜆𝜆𝑒

2

+ 2ℎ𝜆𝜆−ℎ𝜆𝜆𝑒

𝑚𝑚 𝑐𝑐 = 𝑝𝑝𝑒𝑒2

Combining the two equations:


ℎ𝜆𝜆

2

+ℎ𝜆𝜆𝑒

2

− 2ℎ𝜆𝜆

ℎ𝜆𝜆𝑒

𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 = 𝑝𝑝𝑒𝑒2

ℎ𝜆𝜆−ℎ𝜆𝜆𝑒

2


𝑚𝑚 𝑐𝑐 = 𝑝𝑝𝑒𝑒2

−2ℎ𝜆𝜆

ℎ𝜆𝜆𝑒

𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 = −2ℎ𝜆𝜆

ℎ𝜆𝜆𝑒


𝑚𝑚 𝑐𝑐

ℎ𝜆𝜆

ℎ𝜆𝜆𝑒

1 − 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 =ℎ𝜆𝜆−ℎ𝜆𝜆𝑒

𝑚𝑚 𝑐𝑐

Compton Equation:


ℎ𝑚𝑚 𝑐𝑐

1 − 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 = 𝜆𝜆𝑒 − 𝜆𝜆

ℎ𝜆𝜆

ℎ𝜆𝜆𝑒

1 − 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 =ℎ𝜆𝜆−ℎ𝜆𝜆𝑒

𝑚𝑚 𝑐𝑐

• Compton wavelength: ℎ𝑚𝑚𝑒𝑒 𝑐𝑐

= 0.00243 𝑎𝑎𝑚𝑚

• Compton equation

• Independent of wavelength

• Usually observed with x rays

Experimental Results by Compton:


Compton’s original data:

(a) the primary x-ray beam from Mo unscattered

(b) the scattered spectrum from carbon at 135°showing both the modified and unmodified wave.


3.9 Pair Production𝛾𝛾 → 𝑒𝑒− + 𝑒𝑒+

Conservation of electric charge

• Cannot satisfy simultaneously conservation of energy and linear momentum

Conservation of electric charge

Conservation of energy

Conservation of linear momentum

ℎ 𝑓𝑓 > 2 𝑚𝑚𝑒𝑒 𝑐𝑐21.022 𝑀𝑀𝑒𝑒𝑉𝑉

Positron-Electron Annihilation


𝑒𝑒− + 𝑒𝑒+ → 𝛾𝛾 + 𝛾𝛾𝑒 Conservation of electric charge

Conservation of energy

Conservation of linear momentum

2 𝑚𝑚𝑒𝑒 𝑐𝑐2 ≅ ℎ 𝑓𝑓1 + ℎ 𝑓𝑓2

Annihilation of a positronium atom (consisting of an electron and positron) producing two photons.

0 ≅ℎ 𝑓𝑓1𝑐𝑐

−ℎ 𝑓𝑓2𝑐𝑐

ℎ 𝑓𝑓 = 𝑚𝑚𝑒𝑒 𝑐𝑐2 ≅ 0.511 𝑀𝑀𝑒𝑒𝑉𝑉

Positron Emission Tomography (PET)

Fall 2018 75

A useful medical diagnostic tool to study the path and location of a positron-emitting radiopharmaceutical in the human body.

Appropriate radiopharmaceuticals are chosen to concentrate by physiological processes in the region to be examined.

The positron travels only a few millimeters before annihilation, which produces two

photons that can be detected to give the positron position.

Appendix 1


𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 =∫02 𝜋𝜋 𝑑𝑑𝜑𝜑 ∫0

𝜋𝜋/2 𝑑𝑑𝜃𝜃 𝑠𝑠𝑠𝑠𝑎𝑎 𝜃𝜃 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃

∫02 𝜋𝜋 𝑑𝑑𝜑𝜑 ∫0

𝜋𝜋/2 𝑑𝑑𝜃𝜃 𝑠𝑠𝑠𝑠𝑎𝑎 𝜃𝜃= �

0

𝜋𝜋/2𝑑𝑑𝜃𝜃

𝑠𝑠𝑠𝑠𝑎𝑎 2 𝜃𝜃2

=−𝑐𝑐𝑐𝑐𝑠𝑠 2 𝜃𝜃

4 0

𝜋𝜋/2

=12

Appendix 2


∫0∞𝐸𝐸 𝑒𝑒−

𝐸𝐸𝑘𝑘 𝑘𝑘 𝑑𝑑𝐸𝐸

∫0∞ 𝑒𝑒−


=𝐸𝐸 −𝑘𝑘 𝑇𝑇 𝑒𝑒−

𝐸𝐸𝑘𝑘 𝑘𝑘 𝐸𝐸=0

𝐸𝐸=∞− ∫0

∞ −𝑘𝑘 𝑇𝑇 𝑒𝑒−𝐸𝐸𝑘𝑘 𝑘𝑘 𝑑𝑑𝐸𝐸

∫0∞ 𝑒𝑒−


= 𝑘𝑘 𝑇𝑇

Appendix 3


�𝐸𝐸 =∑𝑛𝑛=0∞ 𝐸𝐸𝑛𝑛 𝑝𝑝 𝐸𝐸𝑛𝑛∑𝑛𝑛=0∞ 𝑝𝑝 𝐸𝐸𝑛𝑛

=∑𝑛𝑛=0∞ 𝑎𝑎 ℎ 𝑓𝑓𝑒𝑒−

𝑛𝑛 ℎ 𝑓𝑓𝑘𝑘𝑘𝑘

∑𝑛𝑛=0∞ 𝑒𝑒−𝑛𝑛 ℎ 𝑓𝑓𝑘𝑘𝑘𝑘

= 𝑘𝑘𝑇𝑇∑𝑛𝑛=0∞ 𝑎𝑎𝑛𝑛 𝑒𝑒−𝑛𝑛𝛼𝛼

∑𝑛𝑛=0∞ 𝑒𝑒−𝑛𝑛𝛼𝛼= 𝑘𝑘𝑇𝑇 −𝑛𝑛

𝑑𝑑 𝑙𝑙𝑎𝑎 ∑𝑛𝑛=0∞ 𝑒𝑒−𝑛𝑛𝛼𝛼

𝑑𝑑𝑛𝑛

= 𝑘𝑘𝑇𝑇 −𝑛𝑛𝑑𝑑 𝑙𝑙𝑎𝑎 1

1 − 𝑒𝑒−𝛼𝛼𝑑𝑑𝑛𝑛

= 𝑘𝑘𝑇𝑇 𝑛𝑛𝑑𝑑 𝑙𝑙𝑎𝑎 1 − 𝑒𝑒−𝛼𝛼

𝑑𝑑𝑛𝑛

= 𝑘𝑘𝑇𝑇 𝑛𝑛𝑒𝑒−𝛼𝛼

1 − 𝑒𝑒−𝛼𝛼=

ℎ 𝑓𝑓


𝑛𝑛 ≡ℎ 𝑓𝑓𝑘𝑘𝑇𝑇

Appendix 4


−5


−

ℎ 𝑐𝑐𝑘𝑘 𝑇𝑇 − 1

𝜆𝜆2 𝑒𝑒ℎ 𝑐𝑐𝜆𝜆 𝑘𝑘 𝑘𝑘

𝜆𝜆5 𝑒𝑒ℎ 𝑐𝑐𝜆𝜆 𝑘𝑘 𝑘𝑘 − 1

= 0

−5 +ℎ 𝑐𝑐𝑘𝑘 𝑇𝑇 𝑒𝑒

ℎ 𝑐𝑐𝜆𝜆 𝑘𝑘 𝑘𝑘

𝜆𝜆 𝑒𝑒ℎ 𝑐𝑐𝜆𝜆 𝑘𝑘 𝑘𝑘 − 1

= 0

𝑑𝑑𝛪𝛪 𝜆𝜆,𝑇𝑇𝑑𝑑𝜆𝜆

= 0𝛪𝛪 𝜆𝜆,𝑇𝑇 =

2 𝜋𝜋 ℎ 𝑐𝑐2


Quantum Mechanics 300 fall 18... · 2018-09-14 · Photoelectric Effect Fall 2018 Prof. Sergio B. Mendes 44 • Electromagnetic radiation interacts with electrons within the metal

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