The situation with the valence electrons in a metal is analogous to a quantum particle in a finite square well. Below is an illustration of a finite square well where the blue lines indicate the allowed energies of a quantum particle in the well. The number of allowed solutions depends on the depth of the well, $V_0$, and the width of the well, $W$. If the well is made narrower, the energies of the allowed states rise until the uppermost state pops out of the well. That electron state becomes unbound. Similarly, in a metal, the valence electrons become less bound to their nuclei as the screening increases until they eventually become unbound.

$E$ eV
	$x$ Å

$V_0=$ 5 [eV]
$W=$ 10 [Å]

Once an electron becomes unbound, it can lower its kinetic energy by spreading its wavefunction out over many atoms. The kinetic energy of an electron is,

$$E_{\text{kin}}=\frac{mv^2}{2}=\frac{p^2}{2m}=\frac{\hbar^2k^2}{2m}.$$

This energy is minimzed by making $k$ as small as possible.