Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

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Quantum mechanically the electron in an electromagnetic field is described by minimal coupling. The non‑relativistic Hamiltonian (single electron) is

  H=12m(p−qA)2+qΦ+V(r)  .\boxed{ \; H = \frac{1}{2m}\big(\mathbf{p}-q\mathbf{A}\big)^2 + q\Phi + V(\mathbf r)\; } .H=2m1(p−qA)2+qΦ+V(r).
Expanding the kinetic term gives the free kinetic energy p22m\tfrac{\mathbf p^2}{2m}2mp2 plus interaction terms −qmp ⁣⋅ ⁣A-\tfrac{q}{m}\mathbf{p}\!\cdot\!\mathbf{A}−mqp⋅A and q22mA2\tfrac{q^2}{2m}\mathbf{A}^22mq2A2. In the electric‑dipole approximation (wavelength large compared with atom),
the dominant interaction for absorption/emission is

  Hint≃−d⋅E(t)  ,d=qr\boxed{ \; H_{\rm int} \simeq -\mathbf{d}\cdot\mathbf{E}(t)\; },\qquad \mathbf{d}=q\mathbf rHint≃−d⋅E(t),d=qr
(the electric dipole operator). This is the standard starting point for photon absorption turning potential into kinetic (and changing internal energy levels).

===== Time‑dependent perturbation theory gives the transition rate from initial state ∣i⟩|i\rangle∣i⟩ to final state ∣f⟩|f\rangle∣f⟩ (Fermi’s golden rule): =====

Γi→f=2πℏ ∣⟨f∣Hint∣i⟩∣2 ρ(Ef),\Gamma_{i\to f} = \frac{2\pi}{\hbar}\,|\langle f|H_{\rm int}|i\rangle|^2\,\rho(E_f),Γi→f=ℏ2π∣⟨f∣Hint∣i⟩∣2ρ(Ef),
and energy conservation enforces Ef−Ei≃hfE_f-E_i \simeq h fEf−Ei≃hf. That is your discrete photon energy step: a photon of frequency fff supplies the energy difference and the dipole matrix element supplies the transition amplitude.

===== Photon momentum is pγ=h/λ=hf/cp_\gamma = h/\lambda = hf/cpγ=h/λ=hf/c. If a bound atom absorbs a photon, the atom recoils so linear momentum is conserved: =====

patom after=patom before+pγ.\mathbf{p}_{\rm atom}^{\,\rm after} = \mathbf{p}_{\rm atom}^{\,\rm before} + \mathbf{p}_\gamma.patomafter=patombefore+pγ.
For a heavy atom the recoil velocity is tiny vrecoil≃hf/(Mc)v_{\rm recoil}\simeq hf/(Mc)vrecoil≃hf/(Mc) but it exists and enforces the frame dependence you mention (momentum is frame dependent — consistent with your “momentum gives local reference frames” idea).

===== If incident electromagnetic power per unit area (flux) is S=E×H\mathbf{S}=\mathbf{E}\times\mathbf{H}S=E×H, then power absorbed by area AAA is =====

Pabs=α S ⁣⋅ ⁣n^ AP_{\rm abs}=\alpha \, \mathbf{S}\!\cdot\!\hat n\, APabs=αS⋅n^A
( α\alphaα = absorption efficiency). Bookkeeping → stored mass: dM/dt=Pabs/c2dM/dt=P_{\rm abs}/c^2dM/dt=Pabs/c2. This is the classical statement of “photon energy → stored energy → mass”.

===== Electromagnetic energy–momentum density is =====

TEMμν=ε0(FμαFνα−14gμνFαβFαβ).T^{\mu\nu}_{\rm EM} = \varepsilon_0\Big( F^{\mu\alpha}F^\nu{}_\alpha - \tfrac14 g^{\mu\nu} F_{\alpha\beta}F^{\alpha\beta}\Big).TEMμν=ε0(FμαFνα−41gμνFαβFαβ).
This is what appears on the right of Einstein’s equations; any deposited EM energy (or surface energy) contributes to the stress–energy that curves spacetime. Two photons interacting can produce second‑order stress‑energy effects (but that’s different from claiming photons are gravitons).

===== Quantum decoherence can be modelled by coupling the system (atom + photon mode) to an environment. A Lindblad master equation is a standard phenomenological form: =====

ρ˙=−iℏ[H,ρ]+∑k(LkρLk†−12{Lk†Lk,ρ}),\dot\rho = -\tfrac{i}{\hbar}[H,\rho] + \sum_k \Big( L_k \rho L_k^\dagger - \tfrac12\{L_k^\dagger L_k,\rho\}\Big),ρ˙=−ℏi[H,ρ]+k∑(LkρLk†−21{Lk†Lk,ρ}),
where the jump operators LkL_kLk represent environmentally induced transitions (absorption, emission). When the system couples strongly to many degrees of freedom (the detector), off‑diagonal coherences in the measured basis decay rapidly — that is the physical mechanism underlying “wavefunction collapse” in operational terms. The rate scales like (coupling strength)2^22 × (environment spectral density), so stronger photon exchange → faster decoherence → “the future becomes classical” in the QAT sense.