Finding the wave function squared given an integral wave function

rjop455

1. The problem statement, all variables and given/known data

I need to calculate |ψ(x,t)|² and find how the wave packet moves in time.


2. Relevant equations

I am given these three equations:

(1) ψ(x,0)=∫dp A(p) Exp[ipx/hbar] where A(p) = C Exp[-a(p-p0)/(hbar² )]
Integrate from negative infinity to positive infinity

At a later time the wave function changes to:

(2) ψ(x,t)=∫dp A(p) Exp[ipx/hbar-ip²t/(2m*hbar)]
Integrate from negative infinity to positive infinity

3. The attempt at a solution

My first step was to normalize the wave function by finding the value of C. I did this by solving:
∫|A(p)|² dp =1
Integrate from negative infinity to positive infinity

I found C²=√(2a/(∏*hbar²))

Once I normalized it, I solved integral (2). I then solved for |ψ(x,t)|²; The answer I got is extremely long and depends on t. I was told that having more than a certain amount of linear combinations for a solution cause the wave function to be dependent on time. Is that correct or am I missing the point entirely? Any help would be greatly appreciated!