PHYS 172: Modern Mechanics Lecture 14 – Energy Quantization Read Summer 2012

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

Quantum means quantized Answers come in whole numbers Example: The number of unopened Coke cans in your refrigerator is quantized.

Quantum Waves are Quantized There are discrete vibrational modes (normal modes) 1D: One Dimension Violin string, jumprope 2D: Two Dimensions Modes of a drumhead, coffee sloshing in your mug 3D: Three Dimensions Electron Waves around Atomic Nuclei! Higher Frequency = Higher Energy

Photons One PHOTON = One packet of light Photons come in discrete particles, or packets of energy. And yet it's still a wave: = Wavelength (crest to crest) speed of light [m/s] frequency [1/s] wavelength [m] Number of wavelengths which go by per second

Photons Visible light Electromagnetic spectrum Wavelength nm E = 3.1 eV ν = 7.5x10 14 s -1 E = 1.8 eV ν = 4.2x10 14 s -1 Photons come in discrete particles, or packets of energy. One PHOTON = One packet of light

Atoms and Light Absorb a Photon Adding a photon increases the energy of the atom 1S 2S 3S

1S 3S Atoms and Light Release a Photon Releasing a photon decreases the energy of the atom 2S

Atoms and Light QUANTUM MECHANICS says each ELEMENT (type of atom) can only have specific, QUANTIZED energies. Each atomic transition has a CHARACTERISTIC COLOR Photon Energy = Frequency = Color

The Sun Dark lines correspond to specific atomic transitions, such as “1s to 2s in Hydrogen”, or “1s to 2p in Helium”. Dark lines correspond to specific atomic transitions, such as “1s to 2s in Hydrogen”, or “1s to 2p in Helium”.

Hydrogen atom: electron energy 1S 2S 3S

Emission spectra Energy of emitted photon: Line spectrum – light is emitted at fixed frequencies Hydrogen atom: 1S 2S 3S

Absorption spectra Energy of absorbed photon: Line spectrum – absorption at fixed frequencies Different atoms – different energies Atomic spectra – signature of element Example: He was discovered on Sun first Hydrogen atom: 1S 2S 3S

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 Clicker Question

Effect of temperature Population of level: Energy of the level above the “ground state”, E N – E 1 Temperature, K Boltzmann constant: k=1.4× J/K Population of levels for visible light transition (E = 2eV): At Room Temperature, 300K: On the Sun, 6000K:

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!

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.

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... E r Spring (harmonic oscillator)

Quantizing two interacting atoms Any value of A is allowed and any E is possible. Classical harmonic oscillator: Quantum harmonic oscillator: U = (1/2)k s s EE  EE  Energy levels: N EN  0 / s km  J s 2 h    X 0 s k m  E  equidistant spacing ground state  h

Yes, Tiny Harmonic Oscillators are Quantized Quantum harmonic oscillator: U = (1/2)k s s EE  EE  Energy levels: N EN  Js 2 h    ｴ 0 s k m  E  equidistant spacing ground state WEB DEMO: 

Quantized vibrational energy levels 00 N ENE  U = (1/2)k s x 2 Larger resonance frequency – larger level separation Anharmonic oscillator: Not an equidistant spacing of levels

Home study: Rotational energy levels (8.5,page 338) Nuclear & Hadronic energy levels (8.6) Comparison of energy level spacing (8.7)

Laser Ruby: aluminum oxide crystal (sapphire) where some Al were replaced by Cr