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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.
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.
Angular frequency for (deep) water waves is <math>\omega = \sqrt{\frac{g }{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>.
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?
How do we evaluate an integral of which has a solution containing the erf error function?
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.”
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.
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?