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Phys 172 Modern Mechanics Summer 2010

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Phys 172 Modern Mechanics Summer 2010 Phys 172 Modern Mechanics Summer 2010 r r Δ=ΔpFtsys net ΔEWsys=+ surr Q r r Δ=ΔLsysτ net t Lecture 14 – Energy Quantization Read:Ch 8 Reading Quiz 1 An electron volt (eV) is a measure of: A) Electricity 1 eV = 1.6 x 10-19 J B) Force C) Energy D) Momentum E) None of the above Spectroscopy Spectrum of “white” light is essentially continuous. Sppygectrum of hydrogen gas is clearly discrete. What’s going on here? Light and Energy “cooler” “hotter” Differen t co lors o f lig ht → different ph ot on energi es h Epcc== the wavelength λ determines photon photon color of light λ photon Planck’s constant: h = 6.6x10-34 J·s Energy Quantization in Atoms Consider a hydrogen atom ((p1 proton and 1 electron) It turns out that the electron may only assume certain orbits. N=1 Then U + Kelectron can be only certain values. N=2 N=3 Bohr Model of the Atom Energy Quantization in Atoms −13.6 eV EKU≡ += Nee N 2 N =1, 2 , 3, etc electronic energy levels of hydrogen atom (no other atom has these levels!) CLICKER QUESTION 1 Suppose that these are the quantized energy levels (K+U) for A) 9 eV an atom. Initially the atom is in its B) 6 eV ground state[ ] (symbolized by a dot). C) 5 eV An electron with kinetic energy D) 3 eV 6 eV collides with the atom and E) 2 eV excites it. What is the remaining kinetic energy of the electron? Only possible excitation: -9 eV → -5 eV. Not enouggyh K in electron for any other excitation. System = atom + electron: ΔEatom + ΔEelectron = W + Q = 0 → ΔE =- 4eV4 eV ΔEatom =[(= [(-5eV)5 eV) – (-9eV)]=49 eV)] = 4 electron Kf,electron = 2 eV (no change in rest energies, etc.) Quantum Mechanics … In this course we won’t touch most of quantum mechanics. It’s a very interesting story, however . Emission and Absorption of Photons emitted photon absorbed photon How Do We Determine Energy Levels? We look at light emitted from some gas of atoms , and we see photons with energies 1 eV,,,,,, 2 eV, 3 eV, 6 eV, 8 eV, 9 eV Play with the numbers for a while. The following energy levels are consistent with this data: -10 eV, -9 eV, -7 eV, -1 eV (or -11, -10, -8, -2 etc.) CLICKER QUESTION 2 Suppose that these are the quantized energy A) 2, 5, and 9 eV levels (K+U) for an atom. B) 3, 4, and 7 eV If the atom is excited to C) 3 or 7 eV the second excited state D) 5 or 9 eV (marked by a dot), what E) 2 eV are the possible energies of ppghotons it might emit? Possible atomic transitions: •-2 → -9 gives ΔEatom =-7 eV which gives Ephoton = 7 eV OR •-2 → -5 gives Ephoton = 3 eV, followed by -5→-9 gives Ephoton = 4 eV CLICKER QUESTION 3 Light consisting of photons A) 2 eV, 5 eV, 9 eV with a range of energies from B) 3 eV, 4 eV 1 to 7.5 eV passes through C) 0.5 eV, 3 eV, 4 eV this collection of objects. D) 4 eV, 7 eV E) 3 eV, 4 eV, 7 eV A collection of these What photon energies will be atoms is kept very absorbed from the light beam cold, so that all are in (“dark lines”)? the ground state. NOTE: Excited states fall back to the ground state so quickly that we’ll never see “double transitions” like -9 → -5→ -2. Joseph von Fraunhofer Solar Spectrum Quantizing Two Interacting Atoms U for two atoms If atoms don’t move too far from eqq,uilibrium, U looks like Uspring. Thus, energy levels should correspond to a quantized spring . Quantized Vibrational Energy Levels Classical harmonic oscillator: 11122 2 E =+=222mv ks kAmax AlfAilldAny value of A is allowed → any EiE is poss ible. Quantum harmonic oscillator: ENN = hω00+ E where N = 0120, 1, 2, . k ω = s 0 m Only certain values of E are possible. Note that levels are evenly spaced: ΔE = hω0 Quantized Vibrational Energy Levels far away from equilib r ium, a tom ic bon d doesn’t behave as quantum spring (levels not evenly spaced) Nearly uniform spacing: k Δ=E ω = s hh0 m equilibrium CLICKER QUESTION 4 Pb: ks ~ 5 N/m Al: ks ~ 16 N/m Which vibrational energy level diagram represents Pb, and which is Al? A) A is Pb and B is Al B) A is Al and B is Pb C) A is both Pb and Al D) B is both Pb and Al k ks,Al > ks,Pb s ω > ω ΔE ==hhω0 0,Al 0,Pb m mAl < mPb ΔEAl > ΔEPb CLICKER QUESTION 5 (if time) Two atoms joined by a chemical bond How much energy is required to can be modeled as two masses raise the molecule from its first connectdted bby a spring. excite d sta te to the second excited In one such molecule, it takes vibrational state? 0.05 eV to raise the molecule from its A) 0.0125 eV vibrational ground state to the first B) 0.025 eV excited vibrational energy state. C) 0.05 eV D) 0. 10 eV E) 0.20 eV CLICKER QUESTION 6 (if time) Molecule A: 2 atoms of mass MA Which molecule has vibrational Molecule B: 2 atoms of mass 4MA energy levels spaced closer together? Stiffness of interatomic bond is A) Molecule A approximately the same for both. B) Molecule B C) the spacing is the same k Δ=E ω = s m⇑→ΔE⇓ hh0 m CLICKER QUESTION 7 (if time) Suppose the atoms in diatomic Which molecule has vibrational molecules C and D had approximately energy levels spaced closer the same masses, but . together? Stiffness of bond in C is 3 times as large C) Molecule C as stiffness of bond in D. D) Molecule D E) the spacing is the same k Δ=E ω = s k⇑→ΔE ⇑ hh0 m.
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