= Finite-well confinement in a semiconductor heterostructure
{title2=$z_0=a\sqrt{2m^*V_0}/\hbar$}
Consider an original one-dimensional <effective mass> model with $V(x)=0$ for $|x|<a$ and $V(x)=V_0>0$ outside. Take the same positive <effective mass> $m^*$ everywhere, an abrupt interface, and no interactions. The <wavefunction> represents a transverse envelope; free motion parallel to the layer is outside this calculation. A <double heterostructure laser> need not be thin enough for this quantization to dominate its operation.
Set $s=x/a$, $\epsilon=\hbar^2/(2m^*a^2)$, $z_0^2=V_0/\epsilon$, $z^2=E/\epsilon$ and $w=\sqrt{z_0^2-z^2}$. The time-independent <Schrödinger equation> becomes
$$
-\frac{d^2\phi}{ds^2}+u(s)\phi=z^2\phi,\qquad
u(s)=\begin{cases}0&|s|<1,\\z_0^2&|s|>1.\end{cases}
$$
Here $u(s)$ denotes the dimensionless potential; a <bound state> has $0<z<z_0$. Matching the <wavefunction> and its <derivative> at $s=1$ gives
$$
z\tan z=w\quad\text{(even)},\qquad
-z\cot z=w\quad\text{(odd)}.
$$
For the even <bound state>, the interior solution is $A\cos(zs)$ and the exterior solution is $A\cos z\,e^{-w(|s|-1)}$. For the odd <bound state>, replace $\cos(zs)$ by $\sin(zs)$ and $\cos z$ by $\operatorname{sgn}(s)\sin z$. The physical <wavefunction> is $\psi(x)=\phi(x/a)/\sqrt a$.
<Wavefunction normalization> includes both exterior tails, not just the active layer. Direct integration gives
$$
A_{\rm even}^{-2}=1+\frac{\sin(2z)}{2z}+\frac{\cos^2 z}{w},\qquad
A_{\rm odd}^{-2}=1-\frac{\sin(2z)}{2z}+\frac{\sin^2 z}{w}.
$$
The first two terms are the interior integral; the last is the total exterior integral. Thus the interior <probability> is $P_{\rm in}=A^2[1\pm\sin(2z)/(2z)]$, with the upper sign for even <parity>.
Choose $z_0=2$. There is one even root in $(0,\pi/2)$ and one odd root in $(\pi/2,2)$; higher branches start beyond $z_0$. Each left-hand side increases on its stated interval while $\sqrt{4-z^2}$ decreases, proving uniqueness there. Numerically,
$$
\begin{array}{c|ccc}
\text{parity}&z&E/V_0&P_{\rm in}\\\hline
\text{even}&1.029867&0.265156&0.902317\\
\text{odd}&1.895494&0.898225&0.451648
\end{array}
$$
These are calculated values of the chosen <finite square well>. The excited <bound state> lies closer to the barrier and has a longer exterior tail: being bound does not mean having unit <probability> inside the well.
\Image[/semiconductor-quantum-well-bound-state-confinement.png]
{title=Two normalized <bound states> in a chosen <semiconductor quantum well>}
{description=Original dimensionless calculation at $z_0=2$. Left: <energy levels> and signed normalized envelope <wavefunctions>, vertically offset for display. Right: <probability densities> and integrated interior <probabilities>. Shading marks $|x|<a$; exterior tails remain part of the <wavefunction normalization>.}
{source=semiconductor-quantum-well-bound-state-confinement.py}
This model demonstrates electronic confinement, not optical <refractive index> confinement or a complete <laser>. Unequal <effective masses>, <band bending>, interface disorder and interactions require a more detailed model. In particular, continuity of the ordinary <derivative> above uses the chosen constant <effective mass> assumption.
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