Contemporary Physics: Photoelectric Effect, Wave-Particle Duality, and Quantum Mechanics, Study notes of Physics

Download Contemporary Physics: Photoelectric Effect, Wave-Particle Duality, and Quantum Mechanics and more Study notes Physics in PDF only on Docsity! 1 Quantum Mechanics . Highlights of Contemporary Physics – first module Outline: – Physics around 1900: clear boundaries and the clockwork universe. – Catastrophes: e.g., photoelectric effect and electron diffraction; blurring boundaries and “infinitely wrong” predictions. – New paradigms: quantization and matter waves; particle- wave duality; uncertainty relations. – Paradoxes: Schrödinger’s cat and Bell’s inequality. Why Einstein and many others remain unhappy. The “speakable and unspeakable in quantum mechanics”: What is reality? Catastrophic Experiments • Blackbody radiation (~1860; h “born” in 1901) • Photoelectric effect (~1899; explained in 1905) • Atomic spectra, Rutherford scattering, & atomic models (1880-1911; “explained” in 1913) • Electron Diffraction (~1929; theory came first) . M Brief Review Review and more… Blackbody radiation • Classical physics predicts that, in any “oven” with any temperature T, the total energy in there is infinity × ½kT. • In reality, it is finite, of course… • Moreover, energy associated with each wavelength are miniscule for those λ much shorter than a characteristic λmax … • “Hotter is Shorter!” - λmax ∝ 1/T • Planck found simple formula, with new constant h, that “fits” all known results. Photoelectric effect • Shine light of frequency f and intensity I on metal • Electrons may be ejected (“dug out of the material”). • The e’s that get out have max (kinetic) energy Emax . • Classical physics predicts that, once I gets beyond a certain threshold, Emax increase with I (and who cares what color, i.e., f the beam has!) • Reality is: Emax increases with f rather than I !!! • Summary: Emax = hf – W or hν – W • Einstein’s proposal: Light comes in specific “lumps” (quanta), each carrying energy hf or hν f I V Photoelectric effect has many characteristics of waves, but to explain this effect, we must admit that it is also particle like! Atomic spectra • Light emitted from or absorbed by atoms are associated with distinct set of colors. (“DNA of element!”) • Those from Hydrogen are simplest and fit to Rydberg’s formula: • Rutherford scattering showed that atoms are put together like planets around the sun: very light e’s running around very heavy nucleus (lb vs. tons!). • Classical physics predicts that “moving” charged particles radiate (and so consistent with light emission from atoms), but the e’s should spiral into the nucleus in nanosec’s!! ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= 22 111 mn R Hλ 2 de Broglie waves Louis de Broglie asked: If light waves can behave like particles… …can particles have wave characteristics? and proposed in his PhD thesis (1924) de Broglie waves Louis de Broglie asked: If light waves can behave like particles… …can particles have wave characteristics? and proposed in his PhD thesis (1924) A particle has wavelength: λ = h / p particle’s momentum (like mv!!) . de Broglie waves If we take seriously λ = h / p …what are the consequences? de Broglie waves If we take seriously λ = h / p …what are the consequences? Particles would wiggle and can interfere! (“subtract” as well as “add”) HUH??? This we can test!! Interference and Diffraction a brief reminder • Double slit (Young 1803) • Multiple slits and diffraction in real life • Bragg and X-ray crystallography • … then onto Electron diffraction Nice URLs: http://en.wikipedia.org/wiki/Diffraction http://en.wikipedia.org/wiki/Bragg_diffraction http://www.itp.uni-hannover.de/ ~zawischa/ITP/multibeam.html Nice Applets: http://www.falstad.com/mathphysics.html Thomas Young's sketch of his “two-slit” experiment, presented to the Royal Society in 1803: Light (single wavelength) falls (from left) on barrier with two narrow slits: A and B. Waves from these interfere, “subtracting” at C, D, E, and F. 5 Electron Diffraction Electron Diffraction .de Broglie gives you e-microscopes, TEM, LEED, … google “electron diffraction”!! Bohr’s model of the atom • Remember the puzzle about planetary models? … no specific orbit means continuous radiation at “all” wavelengths! • Bohr exploited deBroglie’s idea: • If electrons are also wavelike, then the orbit circumference had better be an integer multiple of its wavelength (lest it will destructively interfere with itself!). Bohr’s model of the atom • Relate radius r to momentum p through Newtonian physics. • Use de Broglie’s formula to get to the wavelengths. • Integral wavelengths must fit into orbit circumference: 2πr. • Result is discrete, allowed orbits! • …with discrete, allowed energies! o • Use de Broglie’s formula to get to the wavelengths • • Result is discrete, allowed orbits Warning: This picture isn’t kosher… Bohr was lucky!!! Bohr’s model of the atom detour for the experts… …won’t be on test! • K/r2 = mv2/r = p2/mr • mK/r = h2/λ2 λ2 = (h2/mK) r • 2πr = nλ • λ = n (h2/ 2πmK) • 2πr = n2 (h2/ 2πmK) • E ~ p2 = h2/λ2 ~ 1/n2 • Relate r to p = mv • Use de Broglie’s to get λ • Integral wavelengths must fit orbit circumference: 2πr. • Result is discrete, allowed orbits! • …with discrete, allowed energies! Bohr’s model of the atom • … discrete, allowed energies: • n = 1, 2, … • Light emission is regarded as e’s “jumping” from one orbit to another, losing energy to photon. • So, these energies can only be of the form • So, with Einstein’s E=hf =hc/λ, we can understand Light absorption is regarded as e’s “jumping” from one orbit to another, gaining energy from a photon. 6 Bohr’s model of the atom • … discrete, allowed energies: • n = 1, 2, … • Light emission is regarded as e’s “jumping” from one orbit to another, losing energy to photon. • So, these energies can only be of the form • So, with Einstein’s E=hf =hc/λ, we can understand ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= 22 111 mn R Hλ . . amplitude . . wavelength . . frequency . . energy . . momentum c (universal constant)v (you choose!)velocity 0m (fixed by particle)mass ×.×medium . ×. ×localized Light waves Particles characteristic …classical boundaries now gone ! Summary New paradigms: Light waves can be particle-like Particles can be wave-like Quantum Mechanics . Highlights of Contemporary Physics – first module Outline: – Physics around 1900: clear boundaries and the clockwork universe. – Catastrophes: e.g., photoelectric effect and electron diffraction; blurring boundaries and “infinitely wrong” predictions. – New paradigms: quantization and matter waves; particle- wave duality; uncertainty relations. – Paradoxes: Schrödinger’s cat and Bell’s inequality. Why Einstein and many others remain unhappy. The “speakable and unspeakable in quantum mechanics”: What is reality? Particle ~ Wave Duality • Particles behave like waves - sometimes • Waves behave like particles - sometimes double talk ?!?!?!?!? Waves behave like particles - sometimes • Only for EM waves (like light) … …NOT water, slinky, etc. • Photoelectric effect • Low intensity photos are “grainy” 7 5.5: Waves or Particles? Dimming the light in Young’s two-slit experiment results in single photons at the screen. Since photons are particles, each can only go through one slit, so, at such low intensities, their distribution should become the single-slit pattern. Each photon actually goes through both slits! Waves behave like particles - sometimes • Only for EM waves (like light) … …NOT water, slinky, etc. • Photoelectric effect • Low intensity photos are “grainy” • … key through probability normalization (later…) …as well as: E = hf Particles behave like waves - sometimes • Atomic orbits • Electron diffraction • Electron “optics” – e.g., microscopes • Other “beams” in particle accelerators • … key through deBroglie: Particle ~ Wave Duality • Particles behave like waves - sometimes • Waves behave like particles - sometimes “Sometimes” means how you look at it !! Schmittmann: PARVE or WACLE 10 Fun things about Bohr… The Order of the Elephant is the highest order of Denmark. The order is of ancient origin, but was instituted in its current form in 1693 by King Christian V. The order of the Elephant has one class: Knight of the Elephant (Ridder af Elefantordenen). Two other prominent figures appointed to this order around Bohr’s time: Dwight D. Eisenhower (1945) and Winston Churchill (1950). A more “illustrious” appointment was Nicolae Ceauşescu, President of Romania (Note: Awarded on the November 1980 state visit to Denmark but revoked by the Queen on December 23, 1989 for the first time in history. The insignia have been returned to Denmark and Ceauşescu's name has been deleted from the official rolls.) http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Complementarity/CompCopen.html Fun things about Bohr… Where can you find these Elephants? Frederiksborg Palace Fun things about Bohr… Where can you find these Elephants? http://en.wikipedia.org/wiki/Frederiksborg_Palace Frederiksborg Palace Chapel Fun things about Bohr… Particle ~ Wave Duality • If particles behave like waves… …what’s wiggling??? • Also, why did you say Bohr’s model wasn’t kosher? (What’s orbit plane?)… 11 Particle ~ Wave Duality Schrödinger formulated the answer in terms of PROBABILITY AMPLITUDE Ψ(x,t) • like any wave, this “psi” function has an amplitude, goes + and –, and … • has a phase so that it can interfere with itself! • ordinarily composed of many wavelengths/frequencies, so as to … • make up any “waveform” you’d want (Fourier Transform, for the experts) Particle ~ Wave Duality Schrödinger formulated the answer in terms of PROBABILITY AMPLITUDE Ψ(x,t) • like any wave, this “psi” function has an amplitude, goes + and –, and … • has a phase so that it can interfere with itself! • ordinarily composed of many wavelengths/frequencies, so as to … • make up any “waveform” you’d want (Fourier Transform, for the experts) Particle ~ Wave Duality …the square of which is PROBABILITY DENSITY P(x,t) = | Ψ(x,t) | 2 • unlike “psi” function, this P is always positive , and … • the “area under its curve” over any region of space is the probability for finding the quanton (particle or wave!!) in that region • … e.g.,… Particle ~ Wave Duality …the square of which is PROBABILITY DENSITY Probability to find “it” between x =1 and x=2 at time t is ∫ P(x,t) dx 2 1 Particle ~ Wave Duality …the square of which is PROBABILITY DENSITY P(x,t) = | Ψ(x,t) | 2 • unlike “psi” function, this P is always positive , and … • the “area under its curve” over any region of space is the probability for finding the quanton (particle or wave!!) in that region • “Normalization” i.e., the probability to find it anywhere must be ONE Particle ~ Wave Duality …the square of which is PROBABILITY DENSITY Probability to find “it” anywhere is ∫ P(x,t) dx = 1 overall x Remember I said the key to particle like properties for waves is normalization? This is IT !!! Even photons come one at a time. 12 Particle ~ Wave Duality Schrödinger formulated the answer in terms of PROBABILITY AMPLITUDE Ψ(x,t) • like any wave, this “psi” function has an amplitude, goes + and –, and … • has a phase so that it can interfere with itself! • ordinarily composed of many wavelengths/frequencies, so as to … • make up any “waveform” you’d want • BIG question was: how does it EVOLVE in time? http://www.physics.gatech.edu/gcuo/lectures/ MP14 and MP15 6.1: The Schrödinger Wave Equation The Schrödinger wave equation in its time-dependent form for a particle of energy E moving in a potential V in one dimension is: where i is the square root of -1. The Schrodinger Equation is THE fundamental equation of Quantum Mechanics. where V = V(x,t) 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite- Potential Well 6.6 Simple Harmonic Oscillator 6.7 Barriers and Tunneling CHAPTER 6 Quantum Mechanics II Erwin Schrödinger (1887-1961) A careful analysis of the process of observation in atomic physics has shown that the subatomic particles have no meaning as isolated entities, but can only be understood as interconnections between the preparation of an experiment and the subsequent measurement. - Erwin Schrödinger The Copenhagen Interpretation Bohr’s interpretation of the wave function consisted of three principles: Heisenberg’s uncertainty principle Bohr’s complementarity principle Born’s statistical interpretation, based on probabilities determined by the wave function Together these three concepts form a logical interpretation of the physical meaning of quantum theory. In the Copenhagen interpretation, physics describes only the results of measurements. …Next Tuesday: Paradoxes! • What does all this have to do with Heisenberg’s Uncertainty Principle? • Why is Einstein unhappy? • What’s EPR paradox (quantum entanglement) ? • Loose ends … Schrödinger’s cat and Wigner’s friend Test next THURSDAY..(9/10) Last year’s test (and solutions) is on Scholar.