Closed-form integration of the 1D density of states. Integrating g_1D(E) = (1/πℏ)·√(m/(2E)) from 0 to E₀ gives the total number of states per unit length up to energy E₀: N(E₀)/L = √(2mE₀)/(πℏ) — the momentum-space bound k_F = √(2mE₀)/ℏ…
Closed-form integration of the 1D density of states. Integrating g_1D(E) = (1/πℏ)·√(m/(2E)) from 0 to E₀ gives the total number of states per unit length up to energy E₀: N(E₀)/L = √(2mE₀)/(πℏ) — the momentum-space bound k_F = √(2mE₀)/ℏ…