Physics at the 1–100 nm scale where quantum confinement, discrete charging energies, and quantised level spacings dominate — the intermediate regime between atomic and mesoscopic physics. Quantum confinement in a 1D infinite square well…
nanophysics
Coulomb blockade: E_C = e²/(2C), single-electron charging
Coulomb blockade of single-electron transport (Kulik-Shekhter 1975; Averin-Likharev 1986; Fulton-Dolan 1987 PRL 59:109): a small metallic…
1D infinite well: E_n = π²ℏ²n²/(2mL²)
Quantum confinement in a 1D infinite square-well potential of width L: Schrödinger equation −(ℏ²/2m)·d²ψ/dx² = Eψ with boundary conditions…
1D DOS: g_1D(E) = (1/πℏ)·√(m/(2E)) (van Hove 1/√E)
Density of states per unit length and unit energy for a 1D free-electron subband: starting from the dispersion E(k) = ℏ²k²/(2m) and the 1D…
Addition energy: U_N/E_C = 2N+1 (exact integer)
Closed-form electrostatic energy for sequential charging of a Coulomb-blockaded island. The chemical potential μ(N) to add the N-th…
1D box energies: E_n·2mL² − π²ℏ²n² ≡ 0 exactly
Closed-form polynomial identity of the infinite-well eigenenergies. Setting E_n = π²ℏ²n²/(2mL²) and expanding E_n·2mL² − π²ℏ²n² yields…
1D DOS integral: ∫₀^{E₀} g_1D dE = √(2mE₀)/(πℏ)
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…