( ) = ! in nm. E( in ev) ! (in nm) 1240 E (in ev) = Quantum Mechanics. The photon model. First example of quantization.

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1 Quantum Mechanics Light has a particle nature. This is most clearly shown by the photoelectric effect. Particles have a wave nature. All of the wave phenomena we have seen apply to particles as well. Quantum principles are well understood and well accepted. But they are pretty weird. 1 The photon model 1240 E (in ev) =! (in nm) ( ) = 1240 E( in ev)! in nm First example of quantization. 2 Creating X rays If an electron is accelerated through a 5.0 kv potential difference, what is the maximum photon energy of the resulting x ray? What is the wavelength? One electron. One photon. 3

2 Photon Production A particular species of bioluminescent copepod (a small marine crustacean, typically a few mm in length) emits blue light at a peak wavelength of 490 nm. In a typical flash lasting 2.4 s, the copepod emits 1.4 x photons. What power does this correspond to? What is the intensity at a distance of 10 m? E photon = hc! h = 6.62 "10 #34 J $s P =!E!t I = P source 4"r 2 4 Can You See a Single Photon? At the wavelength corresponding to the maximum sensitivity of the human eye, 510 nm, the limit of sensitivity of the darkadapted eye has been shown to be correspond to a 100 ms flash of light of total energy 240 ev. (Weaker flashes of light may be detected, but not reliably.) a) What is the energy of a single photon at this wavelength? b) How many photons does the flash contain? c) If 60% of the incident light is lost to reflection and absorption by tissues of the eye, how many photons reach the retina? The light from the flash covers well over 500 rod cells. d) So, can you see a single photon? The star Sirius is much hotter than the sun, with a peak wavelength of 290 nm compared to the sun s 500 nm. It is also larger, with a diameter 1.7 times that of the sun. By what factor does the energy emitted by Sirius exceed that of the sun? 55. The photon energies used in different types of medical x-ray Ratios. 6

3 72. A light-emitting diode (LED) connected to a 3.0 V power supply emits 440 nm blue light. The current in the LED is 10 ma, and the LED is 60% efficient at converting electric power input into light power output. How many photons per second does the LED emit? 73. A 1000 khz AM radio station broadcasts with a power of What you had to pay What you get 7 Quantum Concept #1: EM Waves have a particle nature 8 The Photoelectric Effect Light Window I Cathode Anode A Ammeter DV I 0 f 0 f There is a threshold frequency. Above it, electrons are emitted. Below it, not so much. 9

4 Just Checking In the photoelectric effect experiment, why does red light not cause the emission of an electron though blue light can? The photons of red light don t have sufficient energy to eject an electron. The electric field of the red light oscillates too slowly to eject an electron. Red light contains fewer photons than blue, not enough to eject electrons. The red light doesn t penetrate far enough into the metal electrode. 10 The Photoelectric Effect Light Window I Intense light Cathode Anode A Ammeter 2V stop 0 Weak light DV DV I Changing the accelerating voltage changes the current. But only within certain limits. 11 Just Checking In the photoelectric effect experiment, increasing the accelerating voltage from 3.0 V to 5.0 V does not increase the current. How can we explain this result? The resistance of the tube changes as well. The electrons are already at their maximum speed. 3.0 V makes all the electrons reach the anode, so increasing voltage causes no change. Increasing the voltage doesn t change the electron kinetic energy. 12

5 The Work Function How much it costs to release an electron. This varies with the electrode. TABLE 28.1 The work functions for some metals Element E 0 (ev) Potassium 2.30 Sodium 2.75 Aluminum 4.28 Tungsten 4.55 Copper 4.65 Iron 4.70 Gold Think About It. Light Window Cathode Anode A Ammeter DV I 5.0 ev photons strike an electrode with work function 3.0 ev. a. What is the kinetic energy of emitted electrons? b. What potential is needed to reduce the current to zero? 14 Just Checking. Monochromatic light shines on the cathode in a photoelectric effect experiment, causing the emission of electrons. If the intensity of the light stays the same but the frequency of the light is increased, the emitted electrons will be moving at a higher speed. both A and B are true. there will be more electrons emitted. neither A nor B are true. 15

6 Just Checking. Monochromatic light shines on the cathode in a photoelectric effect experiment, causing the emission of electrons. If the frequency of the light stays the same but the intensity of the light is increased, the emitted electrons will be moving at a higher speed. both A and B are true. there will be more electrons emitted. neither A nor B are true. 16 The Details. Light of wavelength 400 nm illuminates a potassium electrode (work function 2.3 ev). a. What is the photon energy? b. What is the energy of the emitted electron? c. What is the stopping potential? Window! Light Cathode Anode Ammeter A!!! DV! I 17 Metal surfaces on spacecraft in bright sunlight develop a net electric charge. Do they develop a negative or a positive charge? Explain. What s The Fizics? 18

7 Diffraction and Interference 19 Diffraction Diffraction and Interference 20 Double Slit Interference Pattern Viewing screen Incident laser beam Longer wavelength means bigger spacing. 21

8 Grating Interference Pattern Screen y y 2 m 5 2 Grating y 1 m 5 1 u2 u 1 0 2y 1 m 5 0 m 5 1 Dr between these paths is exactly 2l (m 5 2). L 2y m Appearance of screen 22 Particles have a Wave Nature! = h p = h mv De Broglie wavelength for a moving particle 23 Particle or Wave? m! Localized. Smeared out. Wavelength of a squirrel running at 3 m/s: 1x10-33 m 24

9 Particle or Wave? In a television set, an electron is accelerated by a voltage of 150 V. a. What is the kinetic energy of the electron? b. What is the speed of the electron? c. What is the De Broglie wavelength? Does this matter? Size of a hydrogen atom Orbitals 0.1 nm 25 Looking Deeper Electron microscope view of pigment molecule. 26 Quantum Concept #3: The wave nature of particles leads to quantization. 27

10 Particles have a wave nature. So... Particle: L m v Wave: L...the possible states are quantized. 28 The Crux of the Quantum Biscuit Photons have a particle nature. Their energy is quantized. It comes in chunks of a particular size. Particles have a wave nature. Confining them restricts them to certain energy states. The energy of a confined particle is quantized. It is restricted to certain values. 29 The wave nature of particles leads to quantized energy levels for electrons in atoms. Only certain transitions are possible. Energy 160 ev n 5 4 Energy 160 ev n ev 40 ev 10 ev 0 n 5 3 n 5 2 n 5 1 Ground state Energy levels for a particle in a 0.10-nm-long box 90 ev 40 ev 10 ev 0 DE system 5 E 3 2 E ev DE system 5 E 1 2 E ev n 5 3 n 5 2 n 5 1 Possible transitions for a system with these energy levels E n = 1! hn $ 2m " # 2L % & 2 = h2 8mL 2 n2 n = 1,2, 3,

11 What is the maximum photon energy that could be emitted by the quantum system with the energy level diagram shown below? The minimum? 31 The Details. Light of wavelength 400 nm illuminates a potassium electrode (work function 2.3 ev). a. What is the photon energy? b. What is the energy of the emitted electron? c. What is the stopping potential? Window! Light Cathode Anode Ammeter A!!! DV! I 32 Ocean water is most transparent at wavelengths of 470 nm, so bioluminescent creatures emit light at approximately this wavelength. Firefly squid use ATP to provide the energy for this reaction. Metabolizing one molecule of ATP releases 0.32 ev. How many molecules of ATP must be metabolized to produce one photon of blue light at 470 nm? 33

12 In a photoelectric effect experiment, light of wavelength 620 nm shines on a cathode with a work function of 1.8 ev. What is the speed of the emitted electron? What anode voltage will stop current in the tube? 34 Electrons are accelerated from rest through an 8000 V potential difference. By what factor would their de Broglie wavelength increase if they were instead accelerated through a 2000 V potential? Electron moving more slowly: Wavelength is longer. K =!U e K = 1 2 mv 2! = h p = h mv Ratio reasoning. 35 The wave nature of particles leads to quantization. L m v L E n = 1 2m! hn $ " # 2L % & 2 = h2 8mL 2 n2 n = 1,2, 3,4... Allowed energies for particle in a box 36

13 The wave nature of particles leads to quantized energy levels for electrons in atoms. Only certain transitions are possible. Energy 160 ev n 5 4 Energy 160 ev n ev 40 ev 10 ev 0 n 5 3 n 5 2 n 5 1 Ground state 90 ev 40 ev 10 ev 0 DE system 5 E 3 2 E ev DE system 5 E 1 2 E ev n 5 3 n 5 2 n 5 1 Energy levels for a particle in a 0.19-nm-long box Possible transitions for a system with these energy levels 37 What energy photons could be emitted by the quantum system sketched below? 38 Electrons of the bonds along the chain of carbon atoms in this dye molecule are shared among the atoms in the chain, but are repelled by the nitrogen-containing rings at the end of the chain. What is the longest wavelength of visible light this molecule will absorb? 0.85 nm 39

14 If the length of the chain is increased, how will this affect the wavelength of the light absorbed by the dye? Ratio reasoning. E = 1! hn $ n 2m # " 2L & % 2 = h2 8mLm L 2 8mL 2 3,4... L n2 n = 1,2,3, 40 Changing Scale The diameter of a typical atomic nucleus is about 10 fm. (1 fm is 1x10-15 m.) What is the kinetic energy, in MeV, of a proton with a de Broglie wavelength of 10 fm? 41 Heisenberg uncertainty principle x p p x h 4p!x 42

15 Uncertainty If I know where you are, I don t know where you are going. "x large!v " 1!x "x small 43 Electrons & Atoms An electron is associated with a particular atom. This limits it to an uncertainty in position of about 1 nm it s somewhere within this range. What uncertainty in speed does this imply? 44 Beaming Someone... 45

16 A spherical virus has a diameter of 50 nm. It is contained inside a long, narrow cell of length m. What uncertainty does this imply for the velocity of the virus along the length of the cell? Assume the virus has a density equal to that of water. 46 Heisenberg uncertainty principle!e!t " h 2# E = mc 2 47 But a quantum compass is different. Quantum mechanics limits the proton to two possible energies... Energy E 2 5 1mB... which correspond to two possible orientations, aligned with or opposite the magnetic field. B r 0 E 1 5 2mB µ proton = 1.41!10 "26 J/T What is the photon energy corresponding to a spin flip for a proton in a 1.0 T magnetic field? What frequency does this correspond to? What type of EM wave is this? 48

17 Changing field, changing frequency. Quantum mechanics limits the proton to two possible energies... Energy E 2 5 1mB... which correspond to two possible orientations, aligned with or opposite the magnetic field. B r 0 E 1 5 2mB µ proton = 1.41!10 "26 J/T If you increase the field from 1.0 T to 2.0 T, how does this change the frequency of the rf (radiofrequency) wave necessary to cause a spin flip? 49 Quantum Weirdness: Non-locality Two places at one time Which slit did the electron go through? Where is the electron? 50 Quantum Weirdness: Superposition Many things in the same place 51

18 Quantum Weirdness: Mixed States Alive and dead cats Schrödinger s Cat 52 Fluorescence A range of wavelengths can excite electrons to the upper band. The electrons fall to the lower edge of the upper band. The electrons then jump to the lower band, emitting photons. Would you expect the absorbed or the emitted light to have a longer wavelength? 53 Absorption band Emission band Relative intensity ns Wavelength (nm) 54