Author Archives: anuj - Page 2
plane wave schrodinger wave solution – Transformation under a Lorentz Transform
Key Result - The phase can be Lorentz-transformed, but the Schrödinger dispersion relation is not Lorentz invariant. Start with the free-particle Schrödinger plane wave:ψ(x,t)=Aei(kx−ωt)\psi(x,t)=A e^{i(kx-\omega t)}ψ(x,t)=Aei(kx−ωt) Using de Broglie relations,p=ℏk,E=ℏωp=\hbar…
Lorentz Invariance of Scalar Fields
This is one of the foundational derivations in relativistic quantum field theory: showing that the Klein–Gordon scalar field transforms consistently under Lorentz transformations and that the theory is Lorentz invariant.…
Treating a bipartite Hamiltonian relativistically
For a relativistic bipartite system, you usually do not start with a simple Hamiltonian likeH=HA⊗IB+IA⊗HB+HintH = H_A \otimes I_B + I_A \otimes H_B + H_{\text{int}}H=HA⊗IB+IA⊗HB+Hint unless you are in a…
Why there are no macroscopic Spinor Fields?
1. What Is a Spinor Field? A spinor field is a field describing spin-12\frac1221 particles: electrons quarks neutrinos The Dirac field is the standard example:ψ(x)\psi(x)ψ(x) Unlike scalar or vector fields,…
What is the difference between δ and ∂ in this derivation
For the derivation of the Least Action Principle in classical field theory , what is the difference between δ and ∂? This is one of the most important conceptual distinctions…
The principle of stationary action derivation
Start with the actionS=∫d4x L(ϕ,∂μϕ)S=\int d^4x \, \mathcal{L}(\phi,\partial_\mu \phi)S=∫d4xL(ϕ,∂μϕ) The physical field configuration is the one for which a small variation of the field,ϕ(x)→ϕ(x)+δϕ(x)\phi(x)\rightarrow \phi(x)+\delta\phi(x)ϕ(x)→ϕ(x)+δϕ(x) does not change the action to…
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…