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Plane Waves Vs. Wave Packets Plane Waves Vs. Wave Packets

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Plane Waves Vs. Wave Packets Plane Waves Vs. Wave Packets 3/29/11 PH300 Modern Physics SP11 Exam 2 is this Thursday, 3/31, in class. HW09 will be due Wednesday (3/30) by 5pm in my mailbox. HW09 solu0ons posted at 5pm on Wednesday “It doesn't maer how beau0ful your Recently: theory is; it doesn't maer how smart 1. Electron diffrac0on and maer waves you are. If it doesn't agree with 2. Wave packets and uncertainty experiment, it's wrong.” – Richard Feynman Today: 1. Finish maer waves/wave func0on 2. Review for Exam 2 Day 19, 3/29: Thursday: Quesons? Maer Waves and Measurement Exam 2 (in class) Review for Exam 2 2 Superposion Superposion If and are solu0ons to a wave equaon, then so is Superposion (linear combinaon) of two waves We can construct a “wave packet” by combining many plane waves of different energies (different k’s). 4 Plane Waves vs. Wave Packets Plane Waves vs. Wave Packets For which type of wave are the posi0on (x) and momentum (p) most well-defined? A) x most well-defined for plane wave, p most well-defined for wave packet. B) p most well-defined for plane wave, x most well-defined for wave packet. C) p most well-defined for plane wave, x equally well-defined for both. Which one looks more like a par0cle? D) x most well-defined for wave packet, p equally well-defined for both. E) p and x are equally well-defined for both. 1 3/29/11 Uncertainty Principle Uncertainty Principle In math: Δx small Δp – only one wavelength In words: The posi0on and momentum of a par0cle cannot both be determined with complete precision. The more precisely one is determined, Δx medium Δp – wave packet made of several waves the less precisely the other is determined. What do (uncertainty in posi0on) and (uncertainty in momentum) mean? Δx large Δp – wave packet made of lots of waves Uncertainty Principle Uncertainty Principle A Realist A Sta.scal Interpretaon: Interpretaon: • Photons are scaered by localized par0cles. •Measurements are performed on an ensemble of similarly prepared systems. • Due to the lens’ resolving power: • Distribu0ons of posi0on and momentum values are obtained. • Uncertain0es in posi0on and momentum are defined in terms of • Due to the size of the lens: the standard deviaon. Uncertainty Principle Ma@er Waves (Summary) • Electrons and other par0cles have wave proper0es A Wave (interference) Interpretaon: • When not being observed, electrons are spread out in space (delocalized waves) • When being observed, electrons are found in one place (localized par0cles) • Par0cles are described by wave func0ons: • Wave packets are constructed from a series of plane waves. (probabilis0c, not determinis0c) • The more spaally localized the wave packet, the less uncertainty • Physically, what we measure is in posion. (probability density for finding a par0cle in a par0cular place • With less uncertainty in posi0on comes a greater uncertainty in at a par0cular 0me) momentum. • Simultaneous knowledge of x & p are constrained by the Uncertainty Principle: 2 3/29/11 PE summary: Summary PE effect: h5p://phet.colorado.edu/simulaons/photoelectric/photoelectric.jnlp 2. Current vs. f: 1. Current vs. Voltage: • Current responds instantaneously to light on/off high intensity à Not just a heang effect! I I low intensity • Color maers: Below a certain frequency: No electrons à 0 U Not just a heang effect! 0 Frequency • Posi0ve voltage does not increase the current (remember the 'pool analogy') 3. Current vs. intensity: or: Inial KE vs. f: • Negave voltages do decrease current à Stopping voltage is equal to ini0al kine0c energy! I • Ini0al kine0c energy depends on color of the light. • Current is propor0onal to the light intensity. KE Initial 0 Intensity 0 Frequency Summary of what we know so far: individual atoms vs. atoms bound in solids 1. If light can kick out electron, then even smallest intensi0es of that light will con0nue to kick out electrons. KE of electrons does not depend on 23 3 Individual atom: Solid metal: ~10 atoms per cm intensity. Specific, discrete energy levels Highest energy electrons are in a 2. Lower frequencies of light means lower ini0al KE of electrons “band” of energy levels: Lots and & KE changes linearly with frequency. lots of energy levels spaced really close together 3. Minimum frequency below which light won’t kick out electrons. (a con0nuum of levels) Energy (Einstein) Need “photon” picture of light to explain observaons: - Light comes in chunks (“par0cle-like”) of energy (“photon”) Energy - a photon interacts only with single electron - Photon energy depends on frequency of light … Inside for lower frequencies, photon energy not enough to free an electron Outside metal metal 16 Metal A Metal B Metal A Metal B Ephot Ephot Ephot Ephot work func0on Φ work func0on Φ Most loosely bound e Energy Energy Inside Outside Inside Outside Inside Outside Inside Outside metal metal metal metal metal metal metal metal In each case, the blue photon ejects the red electron. Consider the In each case, the blue photon ejects the red electron. Consider the following statements: following statements: I. The work functions are the same in both cases. I. The work functions are the same in both cases. II. The KE of the ejected electrons are the same in both cases. II. The KE of the ejected electrons are the same in both cases. A. I=True, II=True A. I=True, II=True Work func0on = amount of energy needed to B. I=True, II=False kick out most loosely bound electron out of the C. I=False, II=True B. I=True, II=False metal. D. I=False, II=False C. I=False, II=True D. I=False, II=False 17 18 3 3/29/11 Metal A Metal B Ephot Ephot E=hc/λ… work func0on Φ work func0on Φ Most loosely A. …is true for both photons and electrons. bound e Energy B. …is true for photons but not electrons. Inside Outside Inside Outside C. …is true for electrons but not photons. metal metal metal metal D. …is not true for photons or electrons. In each case, the blue photon ejects the red electron. Consider the following statements: I. The work functions are the same in both cases. c = speed of light! II. The KE of the ejected electrons are the same in both cases. E = hf is always true but f = c/λ only applies to light, A. I=True, II=True B. I=True, II=False Conservaon of energy so E = hf ≠ hc/λ for electrons. KE of ejected = Photon – Energy needed C. I=False, II=True D. I=False, II=False electron energy to extract electron from metal. 19 20 Balmer had a mathemacal formula to describe hydrogen spectrum, but no mechanism for why it Hydrogen atom- worked. Balmer series Hydrogen energy levels 410.3 486.1 656.3 nm 434.0 Balmer (1885) no0ced wavelengths followed a progression 410.3 486.1 Balmer’s formula656.3 nm 434.0 where n = 3,4,5, 6, …. As n gets larger, what happens to wavelengths of emi5ed light? a. gets larger and larger without limit b. gets larger and larger, but approaches a limit where m=1,2,3 c. gets smaller and smaller without limit and where n = m+1, m+2 m=1, n=2 21 d. get smaller and smaller, but approaches a limit 22 If electron going around in li5le orbits, important implicaons from classical physics If calculate angular momentum of atom L = mvr, for orbit, get L=nh/2π. v Basic connec0ons between Angular momentum only comes in units of h/2π = ħ - r, v, and energy! ( “quan0zed” in units of “h-bar” ) Fcent r F = ma= Fcent = ? (quick memory check) a) -mv Bohr Model: + b) -mv2/r Success! What it can do: c) -v2/r2 predict energy levels fairly precisely (although not fine d) -mvr splings discovered short 0me later) for any one electron e) don’t remember learning atom. anything related to this Problems: It cannot: 2 Ans b) Fcent = -mv /r explain anything about jumps between levels, atomic 2 Equate to Coulomb force, = kq+ q-/r , life0mes, why some transi0ons stronger than others, atoms 2 2 2 mv /r =ke /r with mul0ple electrons. mv2 = ke2/r 23 24 4 3/29/11 −13.6eV E = Photon energy n 5ev 2 3ev n 2ev Compare the energy of the photon given off when 0 100 200 300 400 500 600 700 800 nm the electron goes from the n=2 level of H to the What energy levels for electrons are consistent with this ground level (n=1) with the energy difference spectrum for “Finkelnolium”? between the n=4 level and the n=2 level. Electron Energy levels: E4 E3 E /E = ???? E2 2-1 4-2 A B C D E a. 2, b. 4 c. ½, d. ¼ , e. 3/16 E1 0eV 0eV 0eV 0eV -2eV -2eV -3eV 5eV -5eV -5eV -5eV -5eV -7eV 3eV -7eV 2eV -8eV 25 0eV -10eV -10eV 26 Electron energy levels = PE + KE Since PE = 0 at infinity (e.g. electron escaping In discharge lamps, one electron bashes into an atom. Photon energy from atom). 5ev 3ev 2ev A positive total energy would mean that 10 V KE>PE and electron would leave atom! -2 eV 0 100 200 300 400 500 600 700 800 900 1000 nm 1 -3 eV What energy levels for electrons are consistent 2 with this spectrum for “Finkelnolium”? 3 -6 eV At 0eV, electron has escaped atom. (So no transi0ons from 0 eV back down) Electron Energy levels: -10eV A B C D E If atoms fixed at these points in tube, what will you see from each atom? 0eV 0eV 0eV 0eV A.
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