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You can find a fully worked-out derivation in these lecture notes. The formula you are looking for is equation (404), written in terms of the Wigner (small-)$d$ matrix. The relationship to the (large- …

The physical motivation for the Laplace transform is causality. Consider the linear input-output relation $$f_{\text{output}}(t)=\int_{0}^\infty R(t-t')f_{\text{input}}(t')\,dt'.$$ Causality dictates …

Let me try to answer the question "How does this logarithmic interpretation of the Heisenberg uncertainty principle compare with the conventional understanding". There are two issues that prevent a me …

This question has been explored in the context of global positioning systems, which need to account for general relativity. The traditional Minkowski coordinates $(t,x,y,z)$ of flat space-time do not …

The key physics that governs the ionisation energies is shell formation; electrons in the same atomic shell have similar ionisation energies; the number of electrons in the $n$-th shell is $2n^2$, so you …

A somewhat more promising line of approach to arrive at the spectral properties of atoms using purely mathematical reasoning (without physics input) is described in Is there a good mathematical explanation …

A gravitational geon is a space-time configuration that is bounded (asymptotically flat at spatial infinity) and stable (held together for all times by its own gravitational attraction). No such objec …

A necessary requirement for a traveling wave $u(x,t)=f(x-ct)$ to be a "solitary wave" or "soliton" is that the two limits $\lim_{s\rightarrow\pm\infty}f(s)=\alpha_\pm$ exist. This is the condition of …

I can recommend Leonard Susskind's Theoretical Minimum: A number of years ago I became aware of the large number of physics enthusiasts out there who have no venue to learn modern physics and cosmology … So I started a series of courses on modern physics at Stanford University where I am a professor of physics. …

The "manifold picture" can be applied to physics in the context of the Brillouin zone, see for example On Brillouin Zones. …

[To expand on Wojowu's comment.] Q: "Why the description of a divergence-free field as solenoidal? I expect that this name had historical origins but its unlikely that it was so named without some lin …

Early applications of $e^{i\omega t}$ in the context of electromagnetism were understood as a mathematical device: the physical fields are real, and the complex exponential is a convenient method to i …

The Palatini formalism, a variation of a Lagrangian with respect to the connection, is examined quite rigorously in On the Palatini method of variation (1978) The Palatini formulation of general rela …

The Poincaré-von Zeipel method in celestial mechanics relies on canonical transformations of the Hamiltonian to separate fast and slow degrees of freedom in a solar system. See, for example, A note on …

One way to classify multiparty Bell inequalities is via the triple $\chi=(n, m, d)$ where $n$ parties choose from among $m$ measurements each obtaining one of $d$ outcomes. I presume you wish to stick …