Recent content by Hardy

I have been on an magnificent journey learning about the Tschirnhaus Transform and the Bring Radical. In summary, the transform takes any general 5th order polynomial: ##w^5 + A_ww^4 + B_ww^3 + C_ww^2 + D_ww + E_w## into another polynomial of the form: ##x^5 + Dx + E## The transform maps the...

Interesting comparison of multi-pole expansion to Taylor's series. In the course of learning, it was more aligned to Fourier (or Legendre or Bessel) Series ( call it 'FLB' :) ) expansion and the orthogonal nature of these expansions. I understand that multipole expansion is different than any...

I was somewhat surprised that I only found one reference to a coherent state in a book on QFT (Quantum Field Theory by Kaku from 1993). It was on page 715 out of about 760 pages, and here it is: In the Hamiltonian formalism, the transition element between two string states is given by ##\langle...

Thank you for the inspiration to be a writer! I always think of things I could do in retirement, but as the list gets longer, now am wondering if I will have the time. Quantum means two things to me: our world is discrete and randomness is the best answer we have sometimes. QFT introduces the...

Fascinating write-up, thank you very much. I have a question. Let’s say there is certain charge configuration producing a 'coherent state' such that there is a magnetic field that is detectable and predictable. Then, some additional charge density is added into the mix such that where there...

vanhees71 wrote: "Just use an infinitesimal volume with boundary surfaces defined by the coordinate lines. That's an infinitesimal cuboid." Thank you for providing the analysis. It has several interesting aspects that I have not seen before. I have been trying to understanding how you seemed...

From what I can tell, the classification of a soft vs hard magnetic material is more of an engineered property. To put it simply, elements have various levels of being able to be magnetized and to hold onto that magnetization. Magnetization is caused by spinning electrons. Materials with...

This is from an old E&M exam question where we were asked to derive the formula for the divergence of a vector field in cylindrical coordinates using Taylor's Approximation and the fundamental definition of the divergence: ∇⋅A = Lim V→0 { ( ∫S A⋅da ) / V } The vector field, A, is defined in...

Digging deeper into the problem, here is a follow-up on what I learned. As expected, when I decreased the iteration interval, all 6 possible orders of iteration gravitated to a single path, although you could still see differences in the actual numbers if you looked out to enough digits. Here...

Here is what I am taking away: 1) What I am seeing is not necessarily an error in my analysis. 2) Found a definitive statement in a discussion in wiki on the SO3 group: "Furthermore, the rotation group is nonabelian. That is, the order in which rotations are composed makes a difference. For...

Imagine a unit sphere in space that is initially aligned with a universal xyz coodinate axis. The coordinate axis are attached to the sphere at the six fixed points (+/-1,0,0) (0,+/-1,0) and (0,0,+/-1). The sphere starts rotating around each axis. The six points are now at new points in the...