Foundations

Clocks at different radii measure different light frequencies.

Homogeneity and isotropy reduce gravitational expansion to two balances.

The ideal one-electron atom has discrete bound energies.

Quantum labels specify orbitals and spin states.

A bound nucleus has less rest mass than its separated nucleons.

A fixed decay probability per time gives a population timescale.

Nuclear rearrangement can release or require energy.

Normalize a scattering rate by incident flux.

Quarks and leptons form three generations.

Gauge fields mediate three interactions; the Higgs is a scalar sector.

A balance rule must name the interaction where it applies.

A periodic crystal potential organizes energies by wavevector and band.

Band occupancy and doping control mobile carriers.

Indistinguishable particles have different equilibrium state populations.

A closed quantum system evolves by iħ d|ψ⟩/dt=H(t)|ψ⟩.

For a time-independent Hamiltonian, an energy eigenstate evolves by a phase exp(−iEt/ħ).

With zero potential inside 0<x<L and impenetrable walls, normalized eigenfunctions are √(2/L) sin(nπx/L) and energies are n²π²ħ²/(2mL²), n=1,2,… .

A real one-dimensional potential conserves probability current.

For H=p²/(2m)+mω²x²/2 on the real line, normalizable states have E_n=ħω(n+½), n=0,1,… .

For the canonical oscillator, a=√(mω/(2ħ))x+ip/√(2mħω) and its adjoint satisfy [a,a†]=1.

The Heisenberg observable A_H=U†A_SU obeys dA_H/dt=(i/ħ)[H_H,A_H]+(∂A_S/∂t)_H.

Quantum angular momentum obeys [J_i,J_j]=iħ ε_ijk J_k.

Independent commuting angular momenta combine as J=J₁+J₂.

For an isolated nondegenerate eigenvalue of H₀ and a small Hermitian perturbation V, the leading energy corrections are ⟨n|V|n⟩ and Σ_{k≠n}|⟨k|V|n⟩|²/(E_n⁰−E_k⁰).

Inside a degenerate eigenspace of H₀, first-order energy shifts and appropriate zeroth-order states come from diagonalizing the projected Hermitian perturbation PVP.

Starting in eigenstate i of fixed H₀, a weak perturbation V(t) gives first-order amplitude c_f(t)=−(i/ħ)∫_{t₀}^t V_fi(t′)exp[i(E_f−E_i)(t′−t₀)/ħ]dt′ for f≠i.

In the long-wavelength electric-dipole approximation, atomic transition strength depends on the dipole matrix element.

For a self-adjoint Hamiltonian bounded below, every normalized trial state with a finite energy expectation satisfies ⟨ψ_trial|H|ψ_trial⟩≥E₀.

For a simple compressible equilibrium system, U(S,V,{N_i}) has differential dU=T dS−p dV+Σ_i μ_i dN_i.

The Legendre transforms H=U+pV and F=U−TS change the natural state variables: dH=T dS+V dp and dF=−S dT−p dV at fixed composition.