Recent content by euphoricrhino

I can totally see this is the argument behind such rationale. But some dots are still missing for me. The uniqueness theorem's proof depends on the green identity per Jackson section 1.9. In summary, that proof showed that if there are two solutions of poisson equation ##\Phi_1## and ##\Phi_2##...

Hi wise folks, I am working through Jackson problems, and have just encountered problem 5.17: It is pretty straightforward to show that the given image current distributions will satisfy the boundary conditions (both tangent and normal) at the ##z=0## plane. But my question is actually: "why...

This is a straightforward application of the result of problem 3.17, once you have the Green function, then simple application of equation 1.44 gives you the potential in the volume.

Hello, While reading Sakurai (scattering theory/Eikonal approximation section), I encountered a referenced integral ## \int_0^\infty J_1(x)^2\frac{dx}{x}=1/2 ## I also see this integral from a few places (wolfram, DLMF, etc), so I tried to prove this from various angles (recurrence relations...

Now I think I finally come up with a proof for this claim using Wigner-Wickart theorem. The proof is simple but "mathematical", so some physical insights are still appreciated. Here's the proof - Consider the vector operator ##\mathbf{J}##, and transform it into "spherical tensor of rank-1"...

Thinking about it a little more, I think I'm still not understanding the analogy (despite my earlier claim that I did). Take the example where we add ##|l=1,0\rangle## with ##|l=1,0\rangle##, and we are asking why the resulting angular momentum cannot be ##|l=1,0\rangle##. By the analogy, these...

Thanks a lot for the elaboration. I think I'm forming the mental picture now. You are suggesting to visualize any state ##|j,m\rangle## as a 3-D vector of length ##\sqrt{j(j+1)}## whose ##z##-component is ##m##, and claim CG coefficient to be non-zero only if the 3 vectors involved can form a...

Thanks for the reply, but I'm sorry I don't really understand this analogy. Where I got lost is the part when you say "all 3 vectors lie in the xy plane", which 3 vectors are you referring to (it seems you are referring to the ##|m_1=1,m_2=-1\rangle,|m_1=0,m_2=0\rangle,|m_1=-1,m_2=1\rangle## as...

From calculating a few CG-coefficient tables, it occurred to me that when we add two angular mometa ##j_1=l## and ##j_2=1## (with ##l## whole integer), the resulting ##|j=l,m=0\rangle## state always has zero C-G coefficient with ##|j_1=l,j_2=1;m_1=0,m_2=0\rangle## component, i.e., ##\langle...

After some thoughts, I think I misunderstood the statement. Degeneracy in ##l## means two different ##l## values ending up at the same energy eigenvalue. So the two cases are indeed what the text claims to be :)

Hello, I'm hoping someone can help me understand a statement in Sakurai Modern Quantum Mechanics (3rd edition). In particular, in the section that describes free particle in infinite spherical well (page 198, section 3.7.2), after the text has shown that for a given ##l## value, the energy...

Great, thanks a lot. I missed the integration by part trick. This is awesome!

In Sakurai Modern Quantum Mechanics, I came across a statement which says probabiliy flux integrated over all space is just the mean momentum (eq 2.192 below). I was wondering if anybody can help me explain how this is obtained. I can see that ##i\hbar\nabla## is taken as the ##\mathbf{p}##...

I finally figured it out, it's actually quite simple, but all the symbols there have been distracting. The determinant relation $$ \epsilon^{ijk\cdots n}R^{ip}R^{jq}R^{kr}\cdots R^{ns}=\epsilon^{pqr\cdots s} $$ can be viewed as an inner product relation $$ v^nR^{ns}=\epsilon^{pqr\cdots s} $$...

Thanks for the reply! However I must be missing something really obvious, I don't see how to "apply the group relation ##R^{ij}R^{ik}=\delta^{jk}## to both sides" of the determinant equality. The LHS of the determinant relation is a sum of ##N!## terms, each of which is a product whose factors...