Recent content by scigal89

I'd really appreciate any insight on any of this since I've hit a wall. It is about the Fermi gas. --- My teacher did an example in class that didn't make much sense, and I'm trying to understand it. He had us take the real-part of the antiderivative of exp(ik(x-x'))dk, then evaluate it to...

Homework Statement Show that the following function is not square integrable, i.e. that it is not continuous. \int_{-\infty}^{\infty} \left ( e^{ikx} \right )^{2}dx Homework Equations See above. Also: \int \left ( e^{ikx} \right )^{2}dx = -\frac{ie^{2ikx}}{2k} The Attempt at a...

Any ideas? It seems box normalizing is done w/the wavefunction for a free particle to assume a finite box rather than infinite dimensions. Is that basically box normalizing? How is that helpful with the black body?

What is box normalizing and why is it important in quantum mechanical problems such as the black body radiation?

I didn't know that! I never thought of it parametrically before. Interesting... To be honest, though, I'm still not sure about the bounds for my case. I figure since "physically" they have to do with finding an electron at some displacement, it is extended over all space from negative infinity...

That's what I originally thought (over all space for my "functions"). Yet, that made me wonder, for something like the orthonormal Legendre polynomials, which are generated by Gram Schmidt on (-1,1) they are not used merely within those bounds...

Right. But the process involves integrating over some space, which is why I asked about the bounds. I was thinking that given my functions, 0 to 2pi might not be a good choice, but negative pi to pi or negative infinity to infinity are better choices. How do I choose the appropriate interval...

How do you choose the interval? Does it matter conceptually that the Legendre polynomials, for instance, are orthonormalised over (-1,1)? Does that mean that even though they are not orthonormal over all space even though you can obviously graph them over all space? Specifically, I want to...

We're covering probability of the distance for free electrons with parallel spin (long-range oscillations should go to zero) and using that to get a correlation energy. My teacher wants us to elaborate the following 1D case. \int e^{ik(x-X)}dk=\frac{e^{ik(x-X)}}{i(x-X)}\Rightarrow...

Any suggestions?

For Gram-Schmidt I take interval as (0, 2π). Then \phi _{0} = \frac{sin(x)}{x\sqrt {Si(4 \pi)}} \psi _{1} = \left \{ \frac{sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{(2x)}\right \}-\left [ \int_{0}^{2\pi}\left \{ \frac{sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{(2x)}\right \}\frac{sin(x)}{x\sqrt...

In class we worked out the following \int e^{ik(x-X)}dk=\frac{e^{ik(x-X)}}{i(x-X)}\approx \frac{sin[k(x-X)]}{x-X} by taking the real part of the solution. My teacher wants us to graph the following functions \psi_{1} \sim \frac{sin(x)}{x} \psi_{2} \sim...

My teacher worked out the following and said it's a unitary transformation (how?) of exp(ikx). He said we're supposed to find the periodic bounds of integration - but I thought for Fourier transforms the bounds are negative infinity to infinity, so in this case shouldn't it just be the Dirac...