Finite Square-Well Equations

Formula

For

V(x)=\begin{cases} -V_0,&|x|\lt a,\\ 0,&|x|\ge a, \end{cases} \qquad -V_0\lt E\lt0,

define

q=\frac{\sqrt{2m(E+V_0)}}{\hbar}, \qquad \kappa=\frac{\sqrt{-2mE}}{\hbar}.

The matching equations are

q\tan(qa)=\kappa \quad\text{(even)}, \qquad -q\cot(qa)=\kappa \quad\text{(odd)}.

With $z=qa$ and $z_0=a\sqrt{2mV_0}/\hbar$ ,

z\tan z=\sqrt{z_0^2-z^2}, \qquad -z\cot z=\sqrt{z_0^2-z^2},

on their respective parity intervals. The number of normalizable bound states is

N_{\mathrm b}=\left\lceil\frac{2z_0}{\pi}\right\rceil, \qquad z_0\gt0.

Assumptions and Conventions

The full width is $2a$ and the exterior energy zero is fixed at $V=0$ .
A convention with $V=0$ inside uses $E'=E+V_0$ instead.
Constant mass and a finite jump imply continuity of $\psi$ and $\psi'$ .
Bound states require $0\lt z\lt z_0$ and $\kappa\gt0$ .

Symbols

Symbol	Meaning
$a$	well half-width
$V_0$	positive well depth
$q$	interior oscillation wave number
$\kappa$	exterior decay constant
$z_0$	dimensionless well strength

Validity and Warnings

Do not set $\psi(\pm a)=0$ for finite walls.
Do not count the $E=0$ , $\kappa=0$ threshold solution as a bound state.
Bracket roots within one tangent or cotangent branch; an unrestricted Newton step can jump across a pole.
The finite set of bound states is not complete without the scattering continuum.
A stationary evanescent tail carries no outward flux for a real parity eigenfunction.

Canonical Treatment

Piecewise eigenfunctions, normalization, exterior probability, state-count proof, limiting cases, numerical workflow, exercises, and references are at Finite Square Well.