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classes:2008:fall:phys4101.001:chapter2 [2008/09/18 22:23] – ykclasses:2008:fall:phys4101.001:chapter2 [2008/09/19 08:56] (current) – yk
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 Feel free to write anything related to what you have learned in Chapter 2 or wanted to learn but did not learn enough. Feel free to write anything related to what you have learned in Chapter 2 or wanted to learn but did not learn enough.
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 +**If you want to get credits for your entries, please sign your name at the end of your contributions.**  I can in principle investigate who did what, but that's pretty tedious.  If there is any accusation of wrongful conducts, I will use it to investigate who did what in this note, but that's the only time I will use this tool.
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 +If you need some help in learning how to edit wiki or math equations, please go to the bottom of this wiki page.
  
 ===== Section 2.1 Stationary States ===== ===== Section 2.1 Stationary States =====
 This section covers how to solve the Schroedinger Eqn by separating //x// and //t// using the technique called separation of variables.  This is possible because the potential energy //V(x)// is usually a function of only //x//, and not //t//. This section covers how to solve the Schroedinger Eqn by separating //x// and //t// using the technique called separation of variables.  This is possible because the potential energy //V(x)// is usually a function of only //x//, and not //t//.
  
-Using the Schroedinger Eqn, boundary condition(s) and normalization condition, we will find still a number of (sometime infinite) solutions each of which is expressed as a product of a function of //x// and a function of //t//.  The latter takes a form of <math>e^{-i\omega t}</math>, where <math>\omega = \frac{E}{\strike h}</math>. +Using the Schroedinger Eqn, boundary condition(s) and normalization condition, we will find still a number of (sometime infinite) solutions each of which is expressed as a product of a function of //x// and a function of //t//.  The latter takes a form of <math>e^{-i\omega t}</math>, where <math>\omega = \frac{E}{\strike h}</math>.  The spatial part depends on the potential energy involved in the problem, so at this point, we cannot write it explicitly.  We will be spending the rest of the class figuring out this solution for samples of potentials. 
 ===== Section 2.2 The Infinite Square Well ===== ===== Section 2.2 The Infinite Square Well =====
 The infinite square well is the special case of a potential well where the potential inside the well is 0 and the potential outside is infinite.  There is zero probability for a particle to be outside the well.  Solutions for ψ: The infinite square well is the special case of a potential well where the potential inside the well is 0 and the potential outside is infinite.  There is zero probability for a particle to be outside the well.  Solutions for ψ:
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 The General Solution:  Ψ(x,t)=∑c(n)*sqrt(2/L)*sin(nπx/L)*e^[(-i*n^2*π^2*hbar*t/(2*m*L^2)] The General Solution:  Ψ(x,t)=∑c(n)*sqrt(2/L)*sin(nπx/L)*e^[(-i*n^2*π^2*hbar*t/(2*m*L^2)]
-<math>\psi(x,t)=\sum_{n=1}^{\infty}c_n\sqrt{2/L}\sin{\frac{n\pi x}{L}}\exp{-i\frac{n^2\pi^2*\hbar}{2 m L^2}t}</math>+<math>\psi(x,t)=\sum_{n=1}^{\infty}c_n\sqrt{2/L}\sin{\frac{n\pi x}{L}}\exp{-i\frac{n^2\pi^2\hbar}{2 m L^2}t}</math>
  
 The <math>c_n</math>'s are just coefficients of the <math>\psi_n</math>'s as they are sumed, so that some of them are weighted more than others.  In class, we discussed how <math>|c_n|^2</math> can be thought of as a "probability" that the particle is in state //n//.  Really, the wavefunction describing the particle is composed of a little bit of many states, and the coefficients (the <math>c_n</math>'s) dictate how much state each contributes to the general solution of the particle. The <math>c_n</math>'s are just coefficients of the <math>\psi_n</math>'s as they are sumed, so that some of them are weighted more than others.  In class, we discussed how <math>|c_n|^2</math> can be thought of as a "probability" that the particle is in state //n//.  Really, the wavefunction describing the particle is composed of a little bit of many states, and the coefficients (the <math>c_n</math>'s) dictate how much state each contributes to the general solution of the particle.
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-===== quick wiki primer =====+---- 
 + 
 +==== math equation primer ==== 
 +First of all, math expressions must be preceded by <code><math> and followed by </math></code> 
 + 
 +Here are some expressions that you may find useful. 
 + 
 +\sum_{n=a}^{b}: sum n = a to b <math>\sum_{n=a}^{b}</math>\\ 
 +\int_{a}^{b}{f(x) dx}: integrate f(x) from x = a to b <math>\int_{a}^{b}{f(x) dx}</math>\\ 
 +\partial: partial derivative, <math>\partial</math> \\ 
 +\frac{a}{b}: fraction a/b <math>\frac{a}{b}</math>\\ 
 +\hbar: hbar <math>\hbar</math>\\ 
 +\infty: infinity symbol <math>\infty</math>\\ 
 +\pi: Greek letter pi <math>\pi</math>\\ 
 +\omega: Greek letter omega <math>\omega</math>\\ 
 +"A_a" or "A_{abc}": subscript "a" or "abc" <math>A_{abc}</math>\\ 
 +"A^a" or "A^{abc}": superscript "a" or "abc" <math>A^{abc}</math> 
 + 
 + 
 + 
 + ===== quick wiki primer =====
  
  
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 === H4 === === H4 ===
 == H5 == == H5 ==
- 
-==== math equation primer ==== 
-\sum_{n=a}^{b} sum n = a to b \\ 
-\int_{x=a}^{b} integrate x = a to b \\ 
-\frac{a}{b} fraction a/b \\ 
-\hbar hbar\\ 
-\infty infinity symbol\\ 
-\pi Greek letter pi\\ 
-\omega Greek letter omega\\ 
-"_a" or "_{abc}" subscript "a" or "abc"\\ 
-"^a" or "^{abc}" superscript "a" or "abc" 
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classes/2008/fall/phys4101.001/chapter2.1221794624.txt.gz · Last modified: 2008/09/18 22:23 by yk

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