Finding the wave function squared given an integral wave function

[rjop455]

Homework Statement

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

Homework 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

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!

 

[mark726]

Thank you for your question. It seems like you are on the right track with your calculations. The fact that the wave function depends on time is expected, as this represents the time evolution of the wave packet. In general, the wave function will depend on both position and time, and the squared magnitude of the wave function, |ψ(x,t)|2, will give you the probability density of finding the particle at a particular position at a particular time. As time progresses, the wave packet will spread out and move in a certain direction depending on the initial conditions and the potential it is in. I would recommend plotting |ψ(x,t)|2 at different time points to visualize the movement of the wave packet. Let me know if you have any further questions or need clarification on any of the concepts. Best of luck with your calculations!