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--- classes:2009:fall:phys4101.001:lec_notes_0921 [2009/09/21 23:19] – x500_maxwe120
+++ classes:2009:fall:phys4101.001:lec_notes_0921 [2009/09/23 22:45] (current) – yk
@@ Line 1: / Line 1: @@
-===== Sept 21 (Mon) =====
+===== Sept 21 (Mon) Raising/lowering operators: where they came from and their applications =====
 ** Responsible party:  Zeno, Blackbox **
@@ Line 9: / Line 9: @@
 === Main Points ===
-*Special Functions
-*Raising/Lowering Operators
+  *Special Functions
-*Structuring DEs
-*Expectation of Momentum / Momentum Squared
+  *Raising/Lowering Operators
-*Orthogonality of Hermitian Operators
-*Introduction to the Analytical Method (next lecture)
+  *Structuring DEs
+  *Expectation of Momentum / Momentum Squared
+  *Orthogonality of Hermitian Operators
+  *Introduction to the Analytical Method (next lecture)
 == 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.
@@ Line 83: / Line 89: @@
 <math> <p^2>_\Psi_n = \int_{-\infty}^{\infty}\Psi_n^*(-i sqrt{\frac{m \hbar \omega}{2}} (a_- - a_+))^2\Psi_n dx</math>
+<math> = -\frac{m \hbar \omega}{2} \int_{-\infty}^{\infty}\Psi_n^*(a_- - a_+)^2 \Psi_n dx</math>
+<math> = -\frac{m \hbar \omega}{2} \int_{-\infty}^{\infty}\Psi_n^*(a_-^2 - a_-a_+ - a_+a_- + a_+^2) \Psi_n dx</math>
+-where the first and last terms of the expansion raise and lower the n-value twice, canceling. The middle two cancel because __For all Hermitian Operators, their eigenvectors are orthogonal.__ Here's a link that explains Hermitian Operators in more depth:
+[[http://mathworld.wolfram.com/HermitianOperator.html]]
+(*I'm still a little unsure about this last part, so any clarifications would be appreciated)

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