The concept of a wave packet is universally introduced in introductory QM classics in order to "localize" a particle in free space (i.e. $V(x) = 0$ in the 1-Dimensional Schrodinger Eq.). Many textbooks use the example of the Gaussian wave packet most often written as $$\psi(x,t=0) = e^{-ax^2} \cdot e^{\imath k_{o} x} \tag{1} $$ as the initial wave packet at time $t=0$ with $a$ and $k_{o}$ being constants. I have omitted the normalization constant for brevity. However, if Equation (1) is placed into the Schrodinger Equation for a free particle $$-\frac{\hbar^2}{2m} \frac{d^2}{dx^2}\psi = E \psi$$ a real and imaginary part is obtained, after reducing and dividing by $e^{-ax^2} \cdot e^{-\imath k_{o}x}$ on both sides of the equation, with the resulting equations $$-\frac{\hbar^2}{2m} (4a^2x + 2a - k_{o}^2) = E \tag{2}$$ and $$\imath \cdot \frac{\hbar^2}{2m}(4axk_{o})=0. \tag{3}$$ In order for the Eq. 3 - the imaginary part - to be satisfied for all $x$, either $a=0$ or $k_{o}=0$. If it is the former, then there is no Gaussian envelope to shape the term $e^{\imath k_{o}x}$. On the other hand, if $k_{o}=0$, and keeping in mind that $a>0$ for all $x$, Eq.(2) mandates that $E<0$ for $x>0$ and $E<0$ for $x>0$ which seems odd. Furthermore, as $x \rightarrow \infty$, $E \rightarrow -\infty$ and the reverse must hold for $x \rightarrow -\infty$ with $E \rightarrow \infty.$
Furthermore, what potential $V(x)$ would give rise to such an initial wave packet? If $V(x)$ is placed back into the Schrodinger Eq., it arises within the real part of the resulting equations with Eq. 2 now given as $$-\frac{\hbar^2}{2m} (4a^2 x + 2a - k_{o}^2) + V(x) = E. \tag{4}$$ In order to keep $E$ finite, then $V(x)$ would have to approach $\infty$ for $x \rightarrow \infty$ and $V(x)$ approaching $- \infty$ for $x \rightarrow -\infty.$
So, in summary, how can this type of wave packet exist to begin with since it appears that it cannot come about in free space and comes about only for a specific potential. Yet, it is used often and under different variation as the beginning wave packet for multiple homework problems with almost any potential. Even if one where to say that the particle is localized prior to being placed into free space or into a given potential, how did it come to be localized in the first place outside potentials given by Eq. 4?