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This 11 page Class Notes was uploaded by Ms. Bryce Wisoky on Wednesday October 21, 2015. The Class Notes belongs to PHY 300 at Syracuse University taught by Staff in Fall. Since its upload, it has received 25 views. For similar materials see /class/225632/phy-300-syracuse-university in Physics 2 at Syracuse University.
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Date Created: 10/21/15
Lec9 o Intro to Quantum Mechanics o Numerical solution of Schddinger s equa tion Whywhen quantum Newton s laws give a very accurate descrip tion of the behavior of everyday objectsmotions But they fail miserably to describe atoms This was a crisis for physics at turn of cen tury u Eg Laws of EM Newton mechanics pre dicts atoms should be unstable Electrons classically have any energy but see only discrete energies Energy of electromagnetic waves in vac uum infinite o Photoelectric effect light waves like par ticles o Recent expts eg electron diffraction Resolution o Radical Took many physicists about 20 years to discover a Discovered twice Schrddinger Heisenberg 1926 o Arguably become the most well tested and accurate scientific theory QED Basics Discard notion that microscopic objects like electrons can a well defined position veloc ity etc Not a practical issue but one of principle Instead think of them as being described by a wavefunction lxt Like a usual wave in sense that electron is not localized like a classical particle But this is a probability wave l1lt2 yields probability of finding particle at 1215 Dynamics replace Newton s laws simple ordinary differential equations by Schrodnger s equation partial differential equation 4 Points to note R2 82w aw VW 39h 2m 8122 Z at o Equation looks like a funny wave equation but only first order in t o Involves square root of minus 1 i In gen eral lJ is complex Hence need lJ2 for positive real probability a New fundamental constant introduced h 105 x 10 34 Js Planck s constant Stationary states a Put xi gbxeiEtT Plug into equation Find time independent SchrOdinger equa tion 0 Such a wavefunction describes the allowed state of say electron in an atom with E being its energy 0 Try to solve schematically In general only certain energies allowed Allowed energies Rewrite equation d2gb 2m gb R2 o For a bound state need V gt E at large 12 o In which case this equation develops expo nential solutions gb N 65 e mc with H2 o For small 1 V lt E and generate oscillatory solutions gb N sin m cos m Allowed energies II Need ffomdxgb2x 1 Probability Thus need to choose correct solution as 1 gt loo Inside will get oscillations Must smoothly match at boundary Requires that E be very carefully chosen In general discrete set of possible E39s How to solve numerically Rewrite equations dcb g P dp 2mVI E R2 W Consider case V x Va Can show 012 is either even or odd function In former case choose gb0 1 and pO 0 Choose some E and integrate equations using Eulerleapfrog Look at gbx for large x If it is growing will not be able to impose 2 1 Choose another E and try again Shooting method Summary c Find only discrete set of E work So both wavefunction and E are output from cal culation o E0 gt me Particle cannot be stationary at minimum of potential Quantum fluc tuations o E increases with number of oscillations 10
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