Differences

This shows you the differences between two versions of the page.

--- classes:2009:fall:phys4101.001:lec_notes_0921 [2009/09/21 22:02] – 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 6: / Line 6: @@
 **next lecture note: [[lec_notes_0923]]**\\
-Please try to include the following
-  * main points understood, and expand them - what is your understanding of what the points were.
-    * expand these points by including many of the details the class discussed.
-  * main points which are not clear.  - describe what you have understood and what the remain questions surrounding the point(s).
-    * Other classmates can step in and clarify the points, and expand them.
-  * How the main points fit with the big picture of QM.  Or what is not clear about how today's points fit in in a big picture.
-  * wonderful tricks which were used in the lecture.
 === 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
+  *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.
 == Raising/Lowering Operators ==
+As derived and discussed in the last lecture as the mechanism for the Algebraic Method of solution to the Harmonic Oscillator, the ladder operators are:
- With these, the Hamiltonian can be written as
+<math> a_\pm = \frac{1}{\sqrt{2 \hbar m \omega}}(\mp i p + m \omega x)</math>
+for the raising and lowering operator, respectively.
+Notice that the term outside the parentheses makes the ladder operators dimensionless quantities: <math>\frac{1}{\sqrt{(kg*m^2/s)(kg)(1/s)}}(kg*m/s)=dimensionless</math>
+  * It is often helpful and preferable to structure factors in DEs and operations such as the ladder operators as dimensionless quantities.
+A few more examples are:
+<math>\xi=x/x_0</math> is natural length scale and
+<math> E/E_0</math> is a natural energy scale for <math> E_0 = \hbar \omega </math> for the Harmonic Oscillator.
+Recall that we can write momentum as <math> p=-i\hbar\partial_x </math>
+=> then the Hamiltonian is <math> H=\hbar\omega(\partial_\xi^2+ \xi^2) </math>
+and the ladder operators become <math> a_\pm= \mp \partial_\xi+\xi </math>
+Rewritten with the basic ladder operators, The Schrodinger equation is:  <math> \hbar\omega(a_+a_- + 1/2)\psi_n=E_n \psi_n = \hbar \omega(n+1/2) \psi_n</math>
+and canceling, <math>a_+a_-\psi_n=n\psi_n</math>
+  * Even though the ladder operators seem very complicated, using characteristics like their relation to the Hamiltonian and commutation, they prove to be very useful and probably easier and simpler than the Analytical approach to the Harmonic Oscillator.
+The Ladder Operators allow us to write the Hamiltonian as
  <math> H=\hbar\omega(a_+a_- + 1/2) </math>.
- Which makes the Schrodinger equation
+and the Schrodinger equation as
 <math> \hbar\omega(a_\mp a_\pm \pm (1/2))\psi=E\psi </math>
+== Expectation of Momentum and Momentum Squared ==
+It was asked in lecture today how to compute the expectation value of the momentum and momentum^2. Yuichi showed us that
+<math> <p>_\Psi_n = \int_{-\infty}^{\infty}\Psi^*(x,t)(-i \hbar \partial_x)\Psi(x,t)dx</math>
+which yields <math>\Psi_0=(\frac{m\omega}{\pi\hbar})^{1/4}e^{-\xi^2/2}</math> for n = 0
+and <math>\Psi_n=sqrt{2/L}sin(n \pi x / L)</math> for <math>n=0=\Psi_0(x)==0</math>
+=> <math> \Psi_n(x) = sqrt{2/L}sin(\frac{n+1}{L} \pi x) </math> for <math> n >= 0 </math>.
+ *Starting at n=0 or n=1 is purely a matter of convention. Equations like
+<math>E_n=\hbar\omega(n+1/2)</math> for <math>n>=0</math>
+and <math>E_n=\hbar\omega(n-1/2)</math> for <math>n>=1</math>
+are purely based on notational preferences and chosen conventions, and are identical physically.
+Now the expectation value of momentum is given by
+<math> <p>_\Psi_n = \int_{-\infty}^{\infty}\Psi_n^*(-i sqrt{\frac{m \hbar \omega}{2}} (a_- - a_+))\Psi_n dx</math>
+<math> = (-i sqrt{\frac{m \hbar \omega}{2}}) \int_{-\infty}^{\infty}(\Psi_n^*(a_-)\Psi_n-\Psi_n^*(a_+)\Psi_n) dx</math>
+=> which is proportional to <math> \int_{-\infty}^{\infty} \Psi_n^*\Psi_{n-1} dx - \int_{-\infty}^{\infty}\Psi_n^*\Psi_{n+1} dx = 0 </math>
+-which is not a surprising result; for a harmonic oscillator, the average momentum is zero because the particle spends an equal amount of time having positive momentum as an equal but opposite momentum going the other direction.
+The expectation of the momentum squared is given by:
+<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)

User Tools

Differences

Page Tools