Finite Square Well — Graphical Solution

a = 0.5 nm · V₀ slider 0 – 100 eV

Graphical Solution — 1D Symmetric Well

z·tan(z) even −z·cot(z) odd √(z₀²−z²)

Parameters

V₀ 5.00 eV
z₀
Bound states
V₀ (well depth)
Mode
1D
3D Spherical Well — ℓ = 0 states

In 3D the radial equation for ℓ=0 maps to the odd-parity 1D condition:
−z·cot(z) = √(z₀²−z²)

This requires z₀ > π/2, i.e.
V₀ > eV

Potential Well & Bound States

P1: 1D small V₀ — ≥1 bound state P2: Odd state threshold P3: 3D zero bound states below threshold