PHYS 172: Modern Mechanics Fall 2009

Transcription

1 PHYS 172: Modern Mechanics Fall 2009 Lecture 14 Energy Quantization Read

2 Reading Question: Ch. 7, Secs 1-5 A simple model for the hydrogen atom treats the electron as a particle in circular orbit about a massive proton. This is similar to orbital motion of the Space Shuttle about the Earth. Choose the INCORRECT statement: A. Both the electron and the Shuttle are in free fall (accelerating towards the center). B. You can always change K+U a little to change the radius a little. C. There are no centrifugal forces pushing out on the electron or the Shuttle. D. Both systems have a potential energy that goes as U~ -1/r.

3 Quantization Classical Physics: quantities are continuous. Quantum Physics: Some quantities are limited to a discrete set of values. Example: charge, Q = N. e

4 Energy Levels are quantized for many microscopic systems:

5 Energy of a photon Photon energy and wavelength: Ephoton = h! = light hc " light Planck s constant: h = J. s Planck: light is emitted by quanta (1900) Einstein: light consists of quanta! (1905) Visible light Electromagnetic spectrum E = 3.1 ev ν = E = 1.8 ev ν = Wavelength nm

6 Experimental fact: Atoms have QUANTIZED energy levels. Light from these lamps comes only in a few narrow colors, which means the photons in those lines have a fixed, well-defined energy. The photon energy is the difference between two quantized energy levels. If atomic energy levels were not quantized, the light would be white (all energies).

7 Quantized energy levels in a hydrogen atom 13.6 ev E N =!, where N is a nonzero positive integer (1, 2, 3...) N 2 1 ev=1.6x10-19 J

8 Absorption and Emission Spectra Quantized atomic energy levels can be inferred from the specific wave lengths of light seen in absorption and emission. Photon energy and wavelength: E photon = hc! light -34 Planck's constant h = 6.6x10 joule! second h h = = # 2! " x10 joule second

9 Absorption & Emission Absorption Emission

10 Temperature Effects Emission & absorption can happen between any pair of states, in principle. At low temperature, only the lowest level will have any electrons in it. This is called the ground state.

11 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

12 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

13 Clicker Question 3 A collection of these atoms is kept very cold, so that all are in the ground state. Light consisting of photons with a range of energies from 1 to 7.5 ev passes through this collection of objects. What photon energies will be absorbed from the light beam ( dark lines )? A) 2 ev, 5 ev, 9 ev B) 3 ev, 4 ev C) 0.5 ev, 3 ev, 4 ev D) 4 ev, 7 ev E) 3 ev, 4 ev, 7 ev

14 Effect of temperature Hydrogen atom: Boltzmann constant: k= J/K Population of level: ~ exp (!E / kt ) Temperature, K Energy of the level above the ground state, E N E 1 ground state Population of levels for visible light transition at room temperature: E ~ 2 ev, T=300K! 33 (! E kt ) = ~ exp / 10 Sun, 6000K: ~0.02

15 Energy conversion: light and matter Absorption: photon is absorbed electron jumps to higher level Spontaneous emission: photon is emitted electron jumps to lower level Stimulated emission: external photon causes electron jump to lower level a photon is emitted the original photon is not absorbed! Makes laser work!

16 Laser L ight A mplification by S timulated E mission of R adiation Laser media Requirement: inverted population, more atoms must be in excited state E than in state E. First laser: 1960, Theodore Maiman (Hughes Laboratory, California)

17 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...

18 Quantized Vibrational Energy Levels far away from equilibrium, atomic bond doesn t behave as quantum spring (levels not evenly spaced) Nearly uniform spacing: equilibrium!e = h" 0 = h k s m

19 Quantized Vibrational Energy Levels Classical harmonic oscillator: E = 1 2 mv kx2 = 1 ka 2 2 max Any value of A is allowed any E is possible. Quantum harmonic oscillator: E N = Nh! 0 + E 0 where N = 0, 1, 2,... Only certain values of E are possible. Note that levels are evenly spaced: 0E!"=h! 0 = k s m

20 Clicker Question 4 Two atoms joined by a chemical bond can be modeled as two masses connected 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 to the second excited vibrational state? A) ev B) ev C) 0.05 ev D) 0.10 ev E) 0.20 ev

21 Clicker Question 5 Molecule A: 2 atoms of mass M A Molecule B: 2 atoms of mass 4xM 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

22 Clicker Question 6 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