classes:2009:fall:phys4101.001:q_a_0928 [School of Physics and Astronomy Wiki]

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===== Sept 28 (Mon) SHO wrapping up, Free particle and wave packet=====
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====Schrodinger's Dog 09/24/2009 11:03am====
Ok, so I asked this question a while back, but didn't get a answer to it, but found it with help from Yuichi. So my questions was:

"How does Griffths goes from <math>a_{j+2} \approx \frac{2}{j}a_{j} </math> to <math>a_j \approx \frac{C}{(j/2)!}</math>, where C is some constant, on the bottom page 53 and top of page 54?"

Well, what Griffth simply did was a large about substitution. Griffths basically just used the formula <math>a_{j+2} \approx \frac{2}{j}a_{j} </math> , and looked at vaules for j-2, j-4,...etc, so we get <math>a_{j} \approx \frac{2}{j-2}a_{j-2} </math> , <math>a_{j-4} \approx \frac{2}{j-4}a_{j-4} </math>, etc. We then subsitute <math>a_{j} \approx \frac{2}{j-2}a_{j-2} \approx \frac{2}{j-2}...\frac{2}{1}a_{1} </math>. <math> a_{1}=C</math>, since it is the first term. Using factorials allows us to obtain a closed expression for <math>\frac{2}{j-2} ...\frac{2}{1}=\frac{1}{(j/2)!}</math>.  Combining the fact that <math>a_{1}=C </math>, using the factorial expression, and disregarding <math>a_{j+2} </math> term, we find that <math>a_j \approx \frac{C}{(j/2)!}</math>. 

Sorry if this is messy, but this it I guess. Back to MXP lab :P. 

====liux0756 09/25/2009 9:22am====
In page 64, water waves are mentioned, however, I cannot understand the description in the textbook why the group velocity is one half of phase velocity, for water.

===chavez 9:35 9/25===
Angular frequency for (deep) water waves is <math>\omega = \sqrt{\frac{g k}{2}}</math> and the phase velocity is given by <math>v_{phase}=\frac{\omega}{k}=\sqrt{\frac{g}{2k}}</math>. The group velocity is given by <math>v_{group}=\frac{\delta\omega}{\delta k}=\frac{1}{2}\sqrt{\frac{g}{2k}}=\frac{v_{phase}}{2}</math>.

=== Can 9:45 9/27 ===
I didn't see the water wave problem on page 64, it is talking about the analytical methods.
Anyway, a question for chavez, if <math>\omega = \sqrt{\frac{g }{2}}</math>, then shouldn't <math>v_{phase}=\frac{\omega}{k}=\sqrt{\frac{g}{2k^2}}</math>, is there a typo in the expression for the frequency somewhere?

=== chavex 10:14 9/27 ===
Ah thanks for catching my mistake. It should have been <math>\omega = \sqrt{\frac{g k}{2}}</math>. I'll edit that into my original post.



====Dagny 9/25 3:40pm ====
How do we evaluate an integral of which has a solution containing the erf error function?

=== Anaximenes - 16:05 - 09/25/09 ===
I assume you're referring to the integral of <math>e^{-cx^2}</math> or something similar?  I would try using a definite integral that doesn't contain the erf in the solution.  Wikipedia seems to be a good source for that; search for "list of integrals of exponential functions."

=== Schrodinger's Dog - 11:40 - 09/27/09 ===
You can't evaluate it, you can numerically, unless of course you have the trivial case where the error function has the limits 0 to <math> \infty </math>, where you get the error function being 1. For very large upper limits and very low upper limits, I suppose you can assume the value of 1 for large upper limits and use the Trapezoid rule to approixmate the integral in very low limits. You could even write a rough program evaluating the integral using the Trapezoid rule, which I think would give you a sufficient solution. 

=== Blackbox - 23:49 - 09/27/09 ===
Yes, I had the same problem when I tried to calculate the expectation value in homework #1 and #3. For example, some of numerical integration can be solved by substituting "x^2" into "t" but some others are not easily done by integration rules. Should we memorize some specific integration results? 

=== Esquire -10:54 09/30/09 ===
Here is a very nice site which discusses just about everything you need to know about the properties of the error function.

[[http://functions.wolfram.com/GammaBetaErf/Erf/introductions/ProbabilityIntegrals/05/]]

==== Daniel Faraday 9/27 11am ====
Homework Question: on the question that's on the discussion sheet about energy scales in eV, what do we use for the size of the well for the neutron in the nucleus?

===Schrodinger's Dog 9/27 2:07am===
Well, I would guess the the rest mass of a neutron, but I was wondering what problem you are talking about? I didn't see this in our homework. 

===Pluto 4ever 9/27 10:31pm===
If you are referring to the second half of the discussion problem then you just have to use the one half nanometer scale as it says in the problem.

====Hydra   9/27 11:00pm ====
Can somebody show me why the gaussian wave packet has the minimum uncertainty? It makes sense intuitively, I just want to see quantitatively.

===Schrodinger's Dog 9/27 3:03am===
Find <math>\sigma_x</math> and <math>\sigma_p</math>, multiply them together and you should get minimum uncertainty (i.e. <math>\frac{\hbar}{2}</math>).

//**Yuichi**// You can check out section 3.5.2.



==== time to move on ====


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