Phys 172 Modern Mechanics Summer 2010

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1 Phys 172 Modern Mechanics Summer 2010 r r Δ p = F Δt sys net Δ E = W + Q sys sys net surr r r Δ L = τ Δt Lecture 14 Energy Quantization Read:Ch 8

2 Reading Quiz 1 An electron volt (ev) is a measure of: A) Electricity B) Force C) Energy D) Momentum E) None of the above 1 ev = 1.6 x J

3 Spectroscopy Spectrum of white light is essentially continuous. Spectrum of hydrogen gas is clearly discrete. What s going on here?

4 Light and Energy cooler hotter Different colors of light different photon energies h Ephoton = pphoton c = c λ photon the wavelength λ determines color of light Planck s constant: h = 6.6x10-34 J s

5 Energy Quantization in Atoms Consider a hydrogen atom (1 proton and 1 electron) It turns out that the electron may only assume certain orbits. Then U + K electron can be only certain values. N=1 N=2 N=3 Bohr Model of the Atom

6 Energy Quantization in Atoms 13.6 ev E N K e + U e = 2 N N = 1, 2, 3, etc electronic energy levels of hydrogen atom (no other atom has these levels!)

7 CLICKER QUESTION 1 Suppose that these are the quantized energy levels (K+U) for an atom. Initially the atom is in its ground state [ ] (symbolized by a dot). An electron with kinetic energy 6 ev collides with the atom and excites it. What is the remaining kinetic energy of the electron? A) 9 ev B) 6 ev C) 5 ev D) 3 ev E) 2 ev Only possible excitation: -9 ev -5 ev. Not enough K in electron for any other excitation. System = atom + electron: ΔE atom + ΔE electron = W + Q = 0 ΔE atom =[(5eV) [(-5 (-9eV)]=4 ΔE electron =-4eV K f,electron = 2 ev (no change in rest energies, etc.)

8 Quantum Mechanics In this course we won t touch most of quantum mechanics. It s a very interesting story, however...

9 Emission and Absorption of Photons emitted photon absorbed photon

10 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.)

11 CLICKER QUESTION 2 Suppose that these are the quantized energy levels (K+U) for an atom. If the atom is excited to the second excited state (marked by a dot), what are the possible energies of photons it might emit? A) 2, 5, and 9 ev B) 3, 4, and 7 ev C) 3 or 7 ev D) 5 or 9 ev E) 2 ev Possible atomic transitions: -2-9 gives ΔE atom =-7 ev which gives E photon = 7 ev OR -2-5 gives E photon = 3 ev, followed by -5-9 gives E photon = 4 ev

12 CLICKER QUESTION 3 A collection of these atoms is kept very cold, so that all are in the ground state. 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 What photon energies will be absorbed from the light beam ( dark lines )? NOTE: Excited states fall back to the ground state so quickly that we ll never see double transitions like

13 Joseph von Fraunhofer Solar Spectrum

14 Quantizing Two Interacting Atoms U for two atoms If atoms don t move too far from equilibrium, U looks like U spring. Thus, energy levels should correspond to a quantized spring...

15 Quantized Vibrational Energy Levels Classical harmonic oscillator: E = mv + ks = ka max Any value of fai is allowed any Ei is possible. Quantum harmonic oscillator: E = Nhω + E N 0 0 where N = 0, 1, 2,... ω = 0 k s m Only certain values of E are possible. Δ E = hω Note that levels are evenly spaced: 0

16 Quantized Vibrational Energy Levels far away from equilibrium, i atomic bond doesn t behave as quantum spring (levels not evenly spaced) equilibrium Nearly uniform spacing: Δ E = hω = 0 k h s m

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

17 CLICKER QUESTION 4 Pb: k s ~ 5 N/m Al: k s ~ 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 Δ E = hω = h 0 k s m k s,al > k s,pb m Al < m Pb ω 0,Al > ω 0,Pb ΔE Al > ΔE Pb

CLICKER QUESTION 5 (if time) Two atoms joined by a chemical bond can be modeled as two masses connected td by a spring. In one such molecule, it takes 0.

18 CLICKER QUESTION 5 (if time) Two atoms joined by a chemical bond can be modeled as two masses connected td by a spring. In one such molecule, it takes 0.05 ev to raise the molecule from its vibrational ground state to the first excited vibrational energy state. How much energy is required to raise the molecule from its first excited state t to the second excited vibrational state? A) ev B) ev C) 0.05 ev D) 0.10 ev E) 0.20 ev

19 CLICKER QUESTION 6 (if time) Molecule A: 2 atoms of mass M A Molecule B: 2 atoms of mass 4M A Stiffness of interatomic bond is approximately the same for both. Which molecule has vibrational energy levels spaced closer together? A) Molecule A B) Molecule B C) the spacing is the same k Δ E = hω s 0 = h m ΔE m

20 CLICKER QUESTION 7 (if time) Suppose the atoms in diatomic molecules C and D had approximately the same masses, but... Stiffness of bond in C is 3 times as large as stiffness of bond in D. Which molecule has vibrational energy levels spaced closer together? C) Molecule C D) Molecule D E) the spacing is the same k Δ E = hω s 0 = h k ΔE m

PHYS 172: Modern Mechanics Fall 2009

PHYS 172: Modern Mechanics Fall 2009 Lecture 14 Energy Quantization Read 7.1 7.9 Reading Question: Ch. 7, Secs 1-5 A simple model for the hydrogen atom treats the electron as a particle in circular orbit