If a wave packet is given by:
My question is basically how do we choose the write $A(k)$ to fit the particle we are looking at, or does it not matter (as my matter as my textbook seems to imply) which seems counterintuitive?
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If you solve the Schrodinger equation for a free particle the solutions are plane waves, and any sum of plane waves is also a solution. Since any wave packet profile can be constructed by summing plane waves then your equation with any $A(k)$ is also a solution of the Schrodinger equation.
The $A(k)$ is not determined by the Schrodinger equation but rather it's a boundary condition. You choose the $A(k)$ that matches the system you're trying to describe. For example if your particle is completely delocalised the $A(k)$ is a delta function, which physically means there is a precise momentum but the position is completely unknown. The other extreme would be if you pinpoint the particle's position precisely, in which case $A(k) = 1$ so the momentum is completely uncertain.
In practice we'd probably usually choose an $A(k)$ that is Gaussian, because Gaussians are easy to work with. In that case the width of the Gaussian would correspond to the uncertainty in momentum.