Chapter 28

1. Chapter 28 Quantum Theory April 8th, 2013

2. Quantum Theory Two things are really different at the atomic and sub-atomic scale: • Wave-particle duality • Particles are waves, waves are particles • Energies are not continuous but discrete. They vary by discrete increments called quanta • e. g. electron binding energies

3. demos • Young’s double-slit experiment demonstrates the wave-nature of light • Wave-particle duality: electron are waves, they diffract like x-ray photons (1956 electron diffraction instrument)

4. Quantum Regime • Macroscopic-world explanations fail at the atomic-scale • Newtonian mechanics • Maxwell’s equations describing electromagnetism • The atomic-scale world is referred to as the quantum regime • Quantum refers to a very small increment, or parcel, or packet of energy • The discovery and development of quantum theory began in the late 1800s and continued during the early 1900s

5. Waves vs. Particles • In the world of Newton and Maxwell, energy can be carried by particles and waves • Waves produce an interference pattern when passed through a double slit • Classical particles (bullets) will pass through one of the slits and no interference pattern will be formed

6. Particles and Waves, Classical • Waves exhibit interference; particles do not • Particles often deliver their energy in discrete amounts • The energy carried or delivered by a wave is not discrete • The energy carried by a wave is described by its intensity • The amount of energy absorbed depends on the intensity and the absorption time

7. Interference with Electrons • The separation between waves and particles is not found in the quantum regime • Electrons are used in a double slit experiment • The blue lines show the probability of the electrons striking particular locations

8. Interference with Electrons, cont. • The probability curve of the electrons has the same form as the variation of light intensity in the double-slit interference experiment • The experiment shows that electrons undergo constructive interference at certain locations on the screen • At other locations, the electrons undergo destructive interference • The probability for an electron to reach those location is very small or zero • The experiment also shows aspects of particle-like behavior since the electrons arrive one at a time at the screen

9. Particles, Waves, Quanta • All objects, including light and electrons, can exhibit interference • All objects, including light and electrons, carry energy in discrete amounts • These discrete parcels of energy are called quanta

10. Work function • In the 1880s, Hertz discovered the work function and the photoelectric effect • If V is the electric potential at which electrons begin to jump across the vacuum gap, the work function is Wc = eV • The work function, Wc is the minimum energy required to remove a single electron from a piece of metal • This energy can be delivered either as electric potential or by shining light on the metal Wc = eV

11. Work Function, cont. • A metal contains electrons that are free to move around within the metal • The electrons are still bound to the metal and need energy to be removed from the metal • This energy is the work function • The value of the work function is different for different metals

12. Work Functions Wc of several metals

13. Photoelectric Effect • Another way to extract electrons from a metal is by shining light onto it • Light striking a metal is absorbed by the electrons • If an electron absorbs an amount of light energy greater than Wc, it is ejected off the metal • This is called the photoelectric effect

14. Photoelectric Effect, cont. • No electrons are emitted unless the light’s frequency is greater than a critical value ƒc, the intensity of light does not matter • When the frequency is above ƒc, the kinetic energy of the emitted electrons varies linearly with the frequency, not the intensity of light • These results could not be explained with the classical wave theory of light

15. Photoelectric Effect, Problems • Trying to explain the photoelectric effect with the classical wave theory of light presented two difficulties: • Experiments showed that the critical frequency is independent of the intensity of the light • Classically, the energy is proportional to the intensity • It should always be possible to eject electrons by increasing the intensity to a sufficiently high value • Yet it is observable that below the critical frequency, there are no ejected electrons no matter how great the light intensity • The kinetic energy of an ejected electron is independent of the light intensity • Classical theory predicts that increasing the intensity will cause the ejected electrons to have a greater kinetic energy • Yet experiments show that the electron kinetic energy depends on the frequency of light, not at all on its intensity

16. Photoelectric Effect, solution: Photons • Einstein proposed that light carries energy in discrete quanta, now called photons • Each photon carries a quantum of energy Ephoton = hƒ • h is a constant of nature called Planck’s constant • h = 6.626 x 10-34 J ∙ s • A beam of light should be thought of as a collection of photons • Each photon has an energy dependent on its frequency • If the intensity of monochromatic light is increased, the number of photons increases, but the energycarried by each photon does not change • Vending machine analogy

17. Photoelectric Effect, solution: Photons • The introduction of photons accounts for all the problems with the classical explanation • The absorption of light by an electron is just like a collision between two particles, a photon and an electron • The photon carries an energy that is absorbed by the electron • If this energy is less than the work function, the electron is not able to escape from the metal • The energy of a single photon depends on frequency but not on the light intensity

18. Photoelectric Effect, solution: Photons • The kinetic energy of the ejected electrons depends on light frequency but not intensity • The critical frequency corresponds to photons whose energy is equal to the work function h ƒc = Wc • This electron is ejected with 0 kinetic energy • If the photon has a higher frequency, the difference goes into kinetic energy of the ejected electron KEelectron = h ƒ - h ƒc = h ƒ - Wc • This linear relationship is what was observed experimentally

19. Photoelectric EffectNobel prizeAlbert Einstein 1921 (his discovery was in 1905) KEelectron = h ƒ - h ƒc = h ƒ– Wc With his explanation of the photoelectric effect, Einstein introduced the idea that light is made of particles, now called photons,and that their energies are quantized. We now say that: a photon is a quantum of light

20. Momentum of a Photon • A light wave with energy E also carries a certain momentum • Particles of light called photonscarry a discrete amount of both energy and momentum • Photons have two properties that are different from classical particles • Photons do not have any mass • Photons exhibit interference effects

21. Blackbody Radiation • Blackbody radiation is emitted over a range of wavelengths • To the eye, the color of the cavity is determined by the wavelength at which the radiation intensity is largest

22. Blackbody Radiation, Classical • The blackbody intensity curve has the same shape for a wide variety of objects • Electromagnetic waves form standing waves as they reflect back and forth inside the oven’s cavity • The frequencies of the standing waves follow the pattern ƒn = n ƒ where n = 1, 2, 3, … • There is no limit to the value of n, so the frequency can be infinitely large • But as the frequency increases, so does the energy • Classical theory predicts that the blackbody intensity should become infinite as the frequency approaches infinity. This is nonsense!

23. Blackbody Radiation and Quanta • The disagreement between the classical predictions and experimental observations was called the “ultraviolet catastrophe” • Planck proposed solving the problem by assuming the energy in a blackbody cavity must come in discrete quanta • Each parcel would have energy E = h ƒn • His theory fit the experimental results, but gave no reason why it worked • Planck’s work is generally considered to be the beginning of quantum theory

24. Particle-Wave Nature of Light • Some phenomena can only be understood in terms of the particle nature of light • Photoelectric effect • Blackbody radiation • Light also has wave properties at the same time • Interference • Light has both wave-like and particle-like properties

25. Wave-like Duality • The notion that the properties of both classical waves and classical particles are present at the same time is also called wave-particle duality and it is essential for understanding the micro-scale world • All particles at all scales are capable of wave-like properties, as first proposed by Louis de Broglie • De Broglie suggested that if a particle has a momentum p, its wavelength is • Even baseballs, although those would have a wavelength, albeit a very small one (e.g. 10-34m)!

26. Electrons are waves! • To test de Broglie’s hypothesis, an experiment was designed by Davisson-Germer to observe interference of electrons • The experiment showed conclusively that electrons have wavelike properties: the diffraction pattern they form is identical to that obtained by x-ray photons. • The calculated wavelength was in good agreement with de Broglie’s theory

27. Wavelengths of Macroscopic Particles • From de Broglie’s equation and using the classical expression for kinetic energy • As the mass of the particle (object) increases, its wavelength decreases • In principle, you could observe interference with baseballs • Has not yet been observed

28. Problem 28.36 An electron and a neutron have the same wavelength. What is the ration of (a) their kinetic energies and (b) their momenta? Assume the speeds are low enough that you can ignore relativity.