Author Archives: anuj
Why are the observable operators in QM required to be Hermitian?
Overview In quantum mechanics, observables (like position, momentum, energy) are represented by operators. Requiring those operators to be Hermitian (more precisely, self-adjoint) is not arbitrary—it follows from a few fundamental…
The unconnected manifold versus the affine connected manifold – describe what these are. Densities, Derivatives, Integrals, Invariant Derivatives in particular
This is a deep and beautiful distinction—one that sits right at the boundary between pure geometry and physics-ready geometry. Let’s build it cleanly and intuitively, then connect it to densities,…
Using the expansion as a Universal Clock
The Universe Has a Clock — And It’s Hidden in Its Expansion We often say the Big Bang has a “past.” But that statement quietly assumes something profound: there exists…
Wave nature and speed of proton (particle)
High-Speed Protons and de Broglie Waves If a proton is moving at high speed, does it affect its de Broglie wave nature? Answer: Yes. A proton always has a wave…
Invariant Subspaces — Definition & Examples
Invariant Subspaces — Definition & Examples Invariant Subspaces An invariant subspace is a subspace that a given linear transformation cannot “move you out of.” Definition Let V be a vector…
Mean Square Fluctuations of Energy (monoatomic gas)
Monoatomic Ideal Gas — Energy Fluctuations & Velocity Probabilities Monoatomic Perfect Gas (N particles) Below are (A) the mean-square fluctuation in energy in the canonical ensemble and (B) the probability…
The Universal Wave Function
<!doctype html> The World (Universal) Wave Function The world wave function, also called the universal wave function, is the quantum-mechanical wave function that describes the state of the entire universe…
Partial-Sum Probabilities vs. Bell Correlations
Partial-Sum Probabilities vs. Bell Correlations — Illustrated Partial-Sum Probabilities ⇄ Successive Measurements in Bell Experiments Figure: Left — running-partial-sum Markov transitions; Right — Bell measurement settings and entangled state. (Generated…
Randomy generated numbers – Probability of Running Sums
Draws are independent and uniformly chosen from the set \(\{1,2,\dots,k\}\) (with unlimited repetition). Let \(p(n)\) denote the probability that at some time the running sum equals exactly \(n\). 1. Recurrence…
Perfect Gas with Dipole Moments – what can it model?
Can a Perfect Gas of Dipolar Molecules Model Stars or Planets? Short answer: No — a “perfect gas with N molecules each having a dipole moment p” is not…