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classes:2009:fall:phys4101.001:lec_notes_0921 [2009/09/21 23:29] x500_maxwe120classes:2009:fall:phys4101.001:lec_notes_0921 [2009/09/23 22:45] (current) yk
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-===== Sept 21 (Mon) =====+===== Sept 21 (Mon) Raising/lowering operators: where they came from and their applications =====
 ** Responsible party:  Zeno, Blackbox **  ** Responsible party:  Zeno, Blackbox ** 
  
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 == Special Functions == == Special Functions ==
  
-One of the first points made today was that special functions such as the Bessel Functions, Legendre Polynomials and Hermite Polynomials keep emerging as solutions to various classes of differential equations. The important aspect of this fact is that when one function of a class is a solution, usually other functions in that class and sometimes linear combinations of the functions are solutions as well. One of the most familiar examples is the types of differential equations we commonly see as physics students: <math> y'' + k^2 y = 0</math>, <math> y'' +b y' k^2 y = 0</math>, etc., which have exponential solutions. Similar DEs have other exponential solutions, and using Euler's Formulae we can express these as trigonometric functions; all being a class of exponential functions that form solutions to these types of differential equations. This concept pertains to Quantum Mechanics with the exponential and trig solutions we've used thus far, and with the Hermite Polynomials in the Analytical solution for the wave functions of the Harmonic Oscillator (see pg 56), which we'll cover Wednesday.+One of the first points made today was that special functions such as the Bessel Functions, Legendre Polynomials and Hermite Polynomials keep emerging as solutions to various classes of differential equations. The important aspect of this fact is that when one function of a class is a solution, usually other functions in that class and sometimes linear combinations of the functions are solutions as well. One of the most familiar examples is the types of differential equations we commonly see as physics students: <math> y'' + k^2 y = 0</math>, <math> y'' +b y' +k^2 y = 0</math>, etc., which have exponential solutions. Similar DEs have other exponential solutions, and using Euler's Formulae we can express these as trigonometric functions; all being a class of exponential functions that form solutions to these types of differential equations. This concept pertains to Quantum Mechanics with the exponential and trig solutions we've used thus far, and with the Hermite Polynomials in the Analytical solution for the wave functions of the Harmonic Oscillator (see pg 56), which we'll cover Wednesday.
  
  
classes/2009/fall/phys4101.001/lec_notes_0921.1253593742.txt.gz · Last modified: 2009/09/21 23:29 by x500_maxwe120