Wavelength of 1 ev electron

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1 HW8: M Chap 15: Question B, Exercises 2, 6 M Chap 16: Question B, Exercises 1 M Chap 17: Questions C, D From Last Time Essay topic and paragraph due Friday, Mar. 24 Light waves are particles and matter particles are waves! Electromagnetic radiation (e.g. light) made up of photon PARTICLES with energy E=hf and momentum p=e/c=h/ λ Matter particles are waves with wavelength λ=h/p. Wavelength of 1 ev electron For an electron, " = rest energy 1 ev electron, 1240 ev # nm 2 $ MeV 10 ev electron 100 ev electron constant 1 E kinetic λ=1.23 nm λ=0.39 nm λ=0.12 nm = 1.23 ev 1/ 2 # nm E kinetic kinetic energy Wed. Mar. 22, 2006 Phy107 Lect 23 1 Wed. Mar. 22, 2006 Phy107 Lect 23 2 Question A 10 ev electron has a wavelength of ~ 0.4 nm. What is the wavelength of a 40 ev electron? Can this be correct? If electrons are waves, they should demonstrate wave-like effects e.g. Interference, diffraction A. 0.2 nm B. 0.4 nm C. 0.8 nm A 25 ev electron has wavelength 0.25 nm, similar to atomic spacings in crystals Wavelength = constant rest energy Kinetic energy Wed. Mar. 22, 2006 Phy107 Lect 23 3 Wed. Mar. 22, 2006 Phy107 Lect 23 4 Wave reflection from crystal Reflection from next plane Reflection from top plane Constructive & Destructive Interference Interference arises when waves change their phase relationship. Can vary phase relationship of two waves by changing physical location of speaker. in-phase 1/2 λ phase diff side view Interference of waves reflecting from different atomic layers in the crystal. Difference in path length ~ spacing between atoms Wed. Mar. 22, 2006 Phy107 Lect 23 5 Constructive Destructive Wed. Mar. 22, 2006 Phy107 Lect

2 X-ray diffraction Molecular structure Diffraction spot arrangement indicates atomic arrangement Used to determine atomic arrangements of complex molecules. e.g. DNA Davisson-Germer experiment Diffraction of electrons from a nickel single crystal. Established that electrons are waves Bright spot: constructive Davisson: Nobel Prize 1937 X-ray diffraction pattern 54 ev electrons (λ=0.17nm) Wed. Mar. 22, 2006 Phy107 Lect 23 7 Wed. Mar. 22, 2006 Phy107 Lect 23 8 Suppose an electron is a wave Analogy with sound Here is a wave: " = h p Sound wave also has the same characteristics But we can often locate sound waves E.g. echoes bounce from walls. Can make a sound pulse λ where is the electron? Wave extends infinitely far in +x and -x direction x Example: Hand clap: duration ~ 0.01 seconds Speed of sound = 340 m/s Spatial extent of sound pulse = 3.4 meters. 3.4 meter long hand clap travels past you at 340 m/s Wed. Mar. 22, 2006 Phy107 Lect 23 9 Wed. Mar. 22, 2006 Phy107 Lect Beat frequency: spatial localization What does a sound particle look like? One example is a beat frequency between two notes Two sound waves of almost same wavelength added. Making a particle out of waves 440 Hz Hz 440 Hz Hz Hz Constructive Large amplitude Destructive Small amplitude Constructive Large amplitude Wed. Mar. 22, 2006 Phy107 Lect Hz Hz Hz Hz Hz Wed. Mar. 22, 2006 Phy107 Lect

3 Spatial extent of localized sound wave J Δx = spatial spread of wave packet Spatial extent decreases as the spread in included wavelengths increases. Wed. Mar. 22, 2006 Phy107 Lect Δx Same occurs for a matter wave Construct a localized particle by adding together waves with slightly different wavelengths. Since de Broglie says λ = h /p, each of these components has slightly different momentum. We say that there is some uncertainty in the momentum And still don t know exact location of the particle! Wave still is spread over Δx ( uncertainty in position) Can reduce Δx, but at the cost of increasing the spread in wavelength (giving a spread in momentum). Wed. Mar. 22, 2006 Phy107 Lect Interpreting For sound, we would just say that the sound pulse is centered at some position, but has a spread. Can t do that for a quantum-mechanical particle. Many measurements indicate that the electron is indeed a point particle. Interpretation is that the magnitude of electron wavepulse at some point in space determines the probability of finding the electron at that point. Wed. Mar. 22, 2006 Phy107 Lect J Heisenberg Uncertainty Principle Using Δx = position uncertainty Δp = momentum uncertainty Heisenberg showed that the product ( Δx ) ( Δp ) is always greater than ( h / 4π ) Often write this as where h = h 2"! ("x)("p) ~ h/2 is pronounced h-bar Planck s constant Wed. Mar. 22, 2006 Phy107 Lect 23 16! Thinking about uncertainty ("x)("p) ~ h/2 For a classical particle, p=mv, so an uncertainty in momentum corresponds to an uncertainty in velocity.! ("x)("v) ~ h /2m This says that the uncertainty is small for massive objects, but becomes important for very light objects, such as electrons.! Large, massive objects don t show effects of quantum mechanics. Uncertainty principle question Suppose an electron is inside a box 1 nm in width. There is some uncertainty in the momentum of the electron. We then squeeze the box to make it 0.5 nm. What happens to the momentum? A. Momentum becomes more uncertain B. Momentum becomes less uncertain C. Momentum uncertainty unchanged Wed. Mar. 22, 2006 Phy107 Lect Wed. Mar. 22, 2006 Phy107 Lect

4 Using quantum mechanics Planetary model of atom Quantum mechanics makes astonishingly accurate predictions of the physical world Can apply to atoms, molecules, solids. An early success was in understanding Structure of atoms Interaction of electromagnetic radiation with atoms Positive charge is concentrated in the center of the atom ( nucleus ) Atom has zero net charge: Positive charge in nucleus cancels negative electron charges. Electrons orbit the nucleus like planets orbit the sun (Attractive) Coulomb force plays role of gravity nucleus electrons Wed. Mar. 22, 2006 Phy107 Lect Wed. Mar. 22, 2006 Phy107 Lect Difference between atoms No net charge to atom number of orbiting negative electrons same as number of positive protons in nucleus Different elements have different number of orbiting electrons Hydrogen: 1 electron Helium: 2 electrons Copper: 29 electrons Uranium: 92 electrons! Organized into periodic table of elements Elements in same column have similar chemical properties Wed. Mar. 22, 2006 Phy107 Lect Wed. Mar. 22, 2006 Phy107 Lect Planetary model and radiation Circular motion of orbiting electrons causes them to emit electromagnetic radiation with frequency equal to orbital frequency. Same mechanism by which radio waves are emitted by electrons in a radio transmitting antenna. In an atom, the emitted electromagnetic wave carries away energy from the electron. Electron predicted to continually lose energy. The electron would eventually spiral into the nucleus However most atoms are stable! Atoms and photons Experimentally, atoms do emit electromagnetic radiation, but not just any radiation! In fact, each atom has its own fingerprint of different light frequencies that it emits. 400 nm 500 nm 600 nm 700 nm Hydrogen Mercury Wavelength (nm) Wed. Mar. 22, 2006 Phy107 Lect Wed. Mar. 22, 2006 Phy107 Lect

5 Hydrogen emission spectrum Hydrogen is simplest atom One electron orbiting around one proton. The Balmer Series of emission lines empirically given by Hydrogen emission This says hydrogen emits only photons of a particular wavelength, frequency Photon energy = hf, so this means a particular energy. n = 4, λ = nm Hydrogen n = 3, λ = nm Conservation of energy: Energy carried away by photon is lost by the orbiting electron. Wed. Mar. 22, 2006 Phy107 Lect Wed. Mar. 22, 2006 Phy107 Lect The Bohr hydrogen atom Retained planetary picture: one electron orbits around one proton Only certain orbits are stable Radiation emitted only when electron jumps from one stable orbit to another. Here, the emitted photon has an energy of E initial -E final Stable orbit #2 Stable orbit #1 E initial E final Wed. Mar. 22, 2006 Phy107 Lect Photon Energy levels Instead of drawing orbits, we can just indicate the energy an electron would have if it were in that orbit. Zero energy E 3 = " ev E 2 = " ev E 1 = " ev Wed. Mar. 22, 2006 Phy107 Lect Energy axis Photon emitted hf=e 2 -E 1 Emitting and absorbing light Photon is emitted when electron drops from one quantum state to another Zero energy E 3 = " ev E 2 = " ev E 1 = " ev Photon absorbed hf=e 2 -E 1 E 2 = " ev Wed. Mar. 22, 2006 Phy107 Lect E 3 = " ev E 1 = " ev Absorbing a photon of correct energy makes electron jump to higher quantum state. Photon emission question An electron can jump between the allowed quantum states (energy levels) in a hydrogen atom. The lowest three energy levels of an electron in a hydrogen atom are ev, -3.4 ev, -1.5 ev. These are part of the sequence E n = -13.6/n 2 ev. Which of the following photons could be emitted by the hydrogen atom? A ev B. 3.4 ev C. 1.7 ev The energy carried away by the photon must be given up by the electron. The electron can give up energy by dropping to a lower energy state. So possible photon energies correspond to differences between electron orbital energies. The 10.2 ev photon is emitted when the electron jumps from the -3.4 ev state to the ev state, losing 10.2 ev of energy. Wed. Mar. 22, 2006 Phy107 Lect

6 Energy conservation for Bohr atom Each orbit has a specific energy E n =-13.6/n 2 Photon emitted when electron jumps from high energy to low energy orbit. E i E f = h f Photon absorption induces electron jump from low to high energy orbit. E f E i = h f Agrees with experiment! Wed. Mar. 22, 2006 Phy107 Lect Example: the Balmer series All transitions terminate at the level Each energy level has energy E n =-13.6 / n 2 ev E.g. to transition Emitted photon has energy ## E photon = " 13.6 & # % ( " " 13.6 && % % ( $ $ 3 2 ' $ 2 2 ( =1.89 ev '' Emitted wavelength E photon = hf = hc ", " = hc 1240 ev # nm = = 656 nm E photon 1.89 ev Wed. Mar. 22, 2006 Phy107 Lect Spectral Question Compare the wavelength of a photon produced from a transition from to with that of a photon produced from a transition to. A. λ 31 < λ 21 B. λ 31 = λ 21 C. λ 31 > λ 21 E 31 > E 21 so λ 31 < λ 21 Wed. Mar. 22, 2006 Phy107 Lect

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