= Solution
Let $x$ run along the wire and $y$ across it. In a separable approximation,
$$
\boxed{\psi_{nk}(x,y)=\frac{e^{ikx}}{\sqrt{L_x}}\phi_n(y),\qquad E_n(k)=E_n^\perp+\frac{\hbar^2k^2}{2m^*}.}
$$
Transverse confinement gives discrete levels $E_n^\perp$, but the long unconfined direction has a nearly continuous longitudinal <wave vector> and a \b[plane-wave factor]. The result is a family of one-dimensional subbands, not a completely discrete spectrum in both in-plane directions. For a hard-wall width $W$, $E_n^\perp=\hbar^2\pi^2n^2/(2m^*W^2)$; an electrostatic confinement potential gives different thresholds. The suppressed out-of-plane degree of freedom is already frozen into the 2DEG subband.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2017/paper-2-wire-subbands.png]
{title=Discrete transverse thresholds coexist with continuous longitudinal dispersion}
{source=paper-2-wire-subbands.py}
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