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 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 Josef Stefan 1835 – 1893

4. Wien’s Displacement Law The wavelength of the peak of the blackbody distribution was found to follow Wein’s displacement law λ max T = 0.2898 x 10 -2 m K λ max – 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. Planck’s Resolution Planck hypothesized that the blackbody radiation was produced by resonators, submicroscopic charged oscillators The resonators could only have discrete energies E n = n h ƒ n – quantum number, ƒ – frequency of vibration, h – Planck’s constant, 6.626 x 10 -34 J s Key point is quantized energy states Max Karl Ernst Ludwig Planck 1858 – 1947

7. 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 Albert Einstein 1879 – 1955

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

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 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 KE max = hƒ –   is called the work function of the metal KE max = hƒ – 

11. 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 KE max = hƒ – 

12. 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 C incident on a material with a work function  don’t result in the emission of photoelectrons KE max = hƒ –  c = λ ƒ

13. Chapter 27 Problem 12 Lithium, beryllium, and mercury have work functions of 2.30 eV, 3.90 eV, and 4.50 eV, respectively. Light with a wavelength of 4.00 × 10 2 nm is incident on each of these metals. (a) Which of these metals emit photoelectrons in response to the light? Why? (b) Find the maximum kinetic energy for the photoelectrons in each case.

14. 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/m e c = 0.002 43 nm (very small compared to visible light) is called the Compton wavelength Arthur Holly Compton 1892 – 1962

26. Chapter 27 Problem 22 X-rays are scattered from a target at an angle of 55.0° with the direction of the incident beam. Find the wavelength shift of the scattered x-rays.

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. Dual Nature of Light Experiments can be devised that will display either the wave nature or the particle nature of light – in some experiments light acts as a wave and in others it acts as a particle Nature prevents testing both qualities at the same time “Particles” of light – photons Each photon has a particular energy E = h ƒ Each photon encompasses both natures of light – interacts like a particle and has a frequency like a wave

29. 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

30. 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

31. 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)

32. Chapter 27 Problem 30 (a) An electron has a kinetic energy of 3.00 eV. Find its wavelength. (b) A photon has energy 3.00 eV. Find its wavelength.

33. Erwin Rudolf Josef Alexander Schrödinger 1892 – 1987 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 Ψ 2 at some location at a given time is proportional to the probability of finding the particle at that location at that time

34. Erwin Rudolf Josef Alexander Schrödinger 1892 – 1987 The Wave Function

35. 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

36. 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 Δp x, 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

37. 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  t Werner Karl Heisenberg 1901 – 1976

38. 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

39. The Uncertainty Principle

40. Chapter 27 Problem 38 (a) Show that the kinetic energy of a nonrelativistic particle can be written in terms of its momentum as KE = p 2 /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.

41. Answers to Even Numbered Problems Chapter 27: Problem 10 (a) 288 nm (b) 1.04 × 10 15 Hz (c) 1.19 eV

42. Answers to Even Numbered Problems Chapter 27: Problem 16 (a) 8.29 × 10 −11 m (b) 1.24 × 10 −11 m

43. Answers to Even Numbered Problems Chapter 27: Problem 20 m = 2, q = 25.8° m = 3, q = 40.8° m = 4, q = 60.8°

44. Answers to Even Numbered Problems Chapter 27: Problem 46 0.14 keV