1. Chapter 27
Quantum Physics

2. Need for Quantum Physics
Problems remained that classical mechanics couldn’t explain:
Blackbody radiation – electromagnetic radiation emitted by a heated object
Photoelectric effect – emission of electrons by an illuminated metal
Spectral lines – emission of sharp spectral lines by gas atoms in an electric discharge tube

3. Blackbody Radiation
Josef Stefan
1835 – 1893
An object at any temperature emits electromagnetic radiation, sometimes called thermal radiation
Stefan’s Law describes the total power radiated
The spectrum of the radiation depends on the temperature and properties of the object
As the temperature increases, the total amount of energy increases and the peak of the distribution shifts to shorter wavelengths

4. Wien’s Displacement Law
The wavelength of the peak of the blackbody distribution was found to follow Wein’s displacement law
λmax T = x 10-2 m • K
λmax – wavelength at which the curve’s peak
T – absolute temperature of the object emitting the radiation
Wilhelm Carl Werner Otto Fritz Franz Wien
1864 – 1928

5. The Ultraviolet Catastrophe
Classical theory did not match the experimental data
At long wavelengths, the match is good
At short wavelengths, classical theory predicted infinite energy
At short wavelengths, experiment showed no or little energy
This contradiction is called the ultraviolet catastrophe

6. Max Karl Ernst Ludwig Planck
Planck’s Resolution
Planck hypothesized that the blackbody radiation was produced by resonators, submicroscopic charged oscillators
The resonators could only have discrete energies
En = n h ƒ
n – quantum number, ƒ – frequency of vibration, h –Planck’s constant, x J s
Key point is quantized energy states
Max Karl Ernst Ludwig Planck
1858 – 1947

7. Albert Einstein
1879 – 1955
Photoelectric Effect
Photoelectric effect (first discovered by Hertz): when light is incident on certain metallic surfaces, electrons are emitted from the surface –photoelectrons
The successful explanation of the effect was given by Einstein
Electrons collected at C (maintained at a positive potential) and passing through the ammeter are a current in the circuit

8. Photoelectric Effect
The current increases with intensity, but reaches a saturation level for large DV’s
No current flows for voltages less than or equal to –DVs, the stopping potential, independent of the radiation intensity
The maximum kinetic energy of the photoelectrons is related to the stopping potential: KEmax = e DVs

9. Features Not Explained by Classical Physics
No electrons are emitted if the incident light frequency is below some cutoff frequency that is characteristic of the material being illuminated
The maximum kinetic energy of the photoelectrons is independent of the light intensity
The maximum kinetic energy of the photoelectrons increases with increasing light frequency
Electrons are emitted from the surface almost instantaneously, even at low intensities

10. Einstein’s Explanation
KEmax = hƒ – f
Einstein’s Explanation
Einstein extended Planck’s idea of quantization to electromagnetic radiation
A tiny packet of light energy – a photon – is emitted when a quantized oscillator jumps from one energy level to the next lower one
The photon’s energy is E = hƒ
Each photon can give all its energy to an electron in the metal
The maximum kinetic energy of the liberated photoelectron is KEmax = hƒ – f
f is called the work function of the metal

11. Einstein’s Explanation
KEmax = hƒ – f
Einstein’s Explanation
The effect is not observed below a certain cutoff frequency since the photon energy must be greater than or equal to the work function
Without this, electrons are not emitted, regardless of the intensity of the light
The maximum KE depends only on the frequency and the work function, not on the intensity
The maximum KE increases with increasing frequency
The effect is instantaneous since there is a one-to-one interaction between the photon and the electron

12. Verification of Einstein’s Theory
KEmax = hƒ – f
Verification of Einstein’s Theory
Experimental observations of a linear relationship between KE and frequency confirm Einstein’s theory
The x-intercept is the cutoff frequency
The cutoff wavelength is related to the work function
Wavelengths greater than lC incident on a material with a work function f don’t result in the emission of photoelectrons
c = λƒ

13. Chapter 27 Problem 16
An isolated copper sphere of radius 5.00 cm, initially uncharged, is illuminated by ultraviolet light of wavelength 200 nm. What charge will the photoelectric effect induce on the sphere? The work function for copper is 4.70 eV.

14. Wilhelm Conrad Röntgen
X-Rays
X-rays (discovered and named by Roentgen): electromagnetic radiation with short – typically about 0.1 nm – wavelengths
X-rays have the ability to penetrate most materials with relative ease
X-rays are produced when high-speed electrons are suddenly slowed down
Wilhelm Conrad Röntgen
1845 – 1923

15. Production of X-rays
X-rays can be produced by electrons striking a metal target
A current in the filament causes electrons to be emitted
These freed electrons are accelerated toward a dense metal target (the target is held at a higher potential than the filament)

16. X-ray Spectrum The x-ray spectrum has two distinct components
1) Bremsstrahlung: a continuous broad spectrum, which depends on voltage applied to the tube
2) The sharp, intense lines, which depend on the nature of the target material

17. Bremsstrahlung
An electron passes near a target nucleus and is deflected from its path by its attraction to the nucleus
This produces an acceleration of the electron and hence emission of electromagnetic radiation
If the electron loses all of its energy in the collision, the initial energy of the electron is completely transformed into a photon
The wavelength then is

18. Bremsstrahlung Not all radiation produced is at this wavelength
Many electrons undergo more than one collision before being stopped
This results in the continuous spectrum produced

19. Diffraction of X-rays by Crystals
For diffraction to occur, the spacing between the lines must be approximately equal to the wavelength of the radiation to be measured
The regular array of atoms in a crystal can act as a three-dimensional grating for diffracting X-rays

20. Diffraction of X-rays by Crystals
A beam of X-rays is incident on the crystal
The diffracted radiation is very intense in the directions that correspond to constructive interference from waves reflected from the layers of the crystal

21. Diffraction of X-rays by Crystals
The diffraction pattern is detected by photographic film
The array of spots is called a Laue pattern
The crystal structure is determined by analyzing the positions and intensities of the various spots
Max Theodore Felix
von Laue
1879 – 1960

22. Bragg’s Law
The beam reflected from the lower surface travels farther than the one reflected from the upper surface
If the path difference equals some integral multiple of the wavelength, constructive interference occurs
Bragg’s Law gives the conditions for constructive interference
2 d sin θ = m λ
m = 1, 2, 3…
Sir William Lawrence
Bragg
1890 – 1971

23. The Compton Effect
Compton directed a beam of x-rays toward a block of graphite and found that the scattered x-rays had a slightly longer wavelength (lower energy) that the incident x-rays
The change in wavelength (energy) – the Compton shift – depends on the angle at which the x-rays are scattered
Arthur Holly Compton
1892 – 1962

24. The Compton Effect
Compton assumed the photons acted like other particles in collisions with electrons
Energy and momentum were conserved
The shift in wavelength is given by
Arthur Holly Compton
1892 – 1962

25. The Compton Effect
The Compton shift depends on the scattering angle and not on the wavelength
h/mec = nm (very small compared to visible light) is called the Compton wavelength
Arthur Holly Compton
1892 – 1962

26. Chapter 27 Problem 33
A 0.45-nm x-ray photon is deflected through a 23° angle after scattering from a free electron. (a) What is the kinetic energy of the recoiling electron? (b) What is its speed?

27. Photons and Electromagnetic Waves
Light (as well as all other electromagnetic radiation) has a dual nature. It exhibits both wave and particle characteristics
The photoelectric effect and Compton scattering offer evidence for the particle nature of light – when light and matter interact, light behaves as if it were composed of particles
On the other hand, interference and diffraction offer evidence of the wave nature of light

28. Wave Properties of Particles
In 1924, Louis de Broglie postulated that because photons have wave and particle characteristics, perhaps all forms of matter have both properties
Furthermore, the frequency and wavelength of matter waves can be determined
The de Broglie wavelength of a particle is
The frequency of matter waves is
Louis de Broglie
1892 – 1987

29. Wave Properties of Particles
The de Broglie equations show the dual nature of matter
Each contains matter concepts (energy and momentum) and wave concepts (wavelength and frequency)
The de Broglie wavelength of a particle is
The frequency of matter waves is
Louis de Broglie
1892 – 1987

30. The Davisson-Germer Experiment
Davisson and Germer scattered low-energy electrons from a nickel target and followed this with extensive diffraction measurements from various materials
The wavelength of the electrons calculated from the diffraction data agreed with the expected de Broglie wavelength
This confirmed the wave nature of
electrons
Other experimenters confirmed the
wave nature of other particles
Clinton Joseph Davisson (1881 – 1958) and Lester Halbert Germer (1896 – 1971)

31. Chapter 27 Problem 40
A monoenergetic beam of electrons is incident on a single slit of width nm. A diffraction pattern is formed on a screen 20.0 cm from the slit. If the distance between successive minima of the diffraction pattern is 2.10 cm, what is the energy of the incident electrons?

32. Erwin Rudolf Josef Alexander Schrödinger
The Wave Function
In 1926 Schrödinger proposed a wave equation that describes the manner in which matter waves change in space and time
Schrödinger’s wave equation is a key element in quantum mechanics
Schrödinger’s wave equation is generally solved for the wave function, Ψ, which depends on the
particle’s position and the time
The value of Ψ2 at some location at
a given time is proportional to the
probability of finding the particle
at that location at that time
Erwin Rudolf Josef Alexander Schrödinger
1892 – 1987

33. The Uncertainty Principle
When measurements are made, the experimenter is always faced with experimental uncertainties in the measurements
Classical mechanics offers no fundamental barrier to ultimate refinements in measurements and would allow for measurements with arbitrarily small uncertainties
Quantum mechanics predicts that a barrier to measurements with ultimately small uncertainties does exist

34. The Uncertainty Principle
In 1927 Heisenberg introduced the uncertainty principle: If a measurement of position of a particle is made with precision Δx and a simultaneous measurement of linear momentum is made with precision Δpx, then the product of the two uncertainties can never be smaller than h/4
Mathematically,
It is physically impossible to measure
simultaneously the exact position and the
exact linear momentum of a particle
Werner Karl Heisenberg
1901 – 1976

35. The Uncertainty Principle
Another form of the principle deals with energy and time:
Energy of a particle can not be measured with complete precision in a short interval of time Dt
Werner Karl Heisenberg
1901 – 1976

36. The Uncertainty Principle
A thought experiment for viewing an electron with a powerful microscope
In order to see the electron, at least one photon must bounce off it
During this interaction, momentum is transferred from the photon to the electron
Therefore, the light
that allows you to
accurately locate
the electron changes
the momentum of the electron

37. The Uncertainty Principle

38. Chapter 27 Problem 46
(a) Show that the kinetic energy of a nonrelativistic particle can be written in terms of its momentum as KE = p2/2m. (b) Use the results of (a) to find the minimum kinetic energy of a proton confined within a nucleus having a diameter of 1.0 × 10−15 m.

39. Answers to Even Numbered Problems
Chapter 27:
Problem 12
5.4 eV

40. Answers to Even Numbered Problems
Chapter 27:
Problem 20
(a) 8.29 × 10−11 m
(b) 1.24 × 10−11 m

41. Answers to Even Numbered Problems
Chapter 27:
Problem 24
6.7°

42. Answers to Even Numbered Problems
Chapter 27:
Problem 28
1.8 keV, 9.7 × 10−25 kg⋅m/s

43. Answers to Even Numbered Problems
Chapter 27:
Problem 34
1.98 × 10−11 m
1.98 × 10−14 m

44. Answers to Even Numbered Problems
Chapter 27:
Problem 52
191 MeV