Infinite well linear combo of states

[galactic]

Homework Statement

A particle of mass m is trapped in a one-dimensional infinite square well running from x= -L/2 to L/2. The particle is in a linear combination of its ground state and first excited state such that its expectation value of momentum takes on its largest possible value at t=0.

The Attempt at a Solution

I know the process of solving PDE's clear as day, that's not the issue. The problem is that I'm tripping myself out on how to write Ψ(x,t) as a linear combination of its ground state + first excited state.

My hunch is to approach the problem like this :

Ψ(x,t)=c1Ψ1(x,t)+c2Ψ2(x,t)

where 1 and 2 represent the ground state and first state, respectively

that momentum takes on the largest possible value at t=0 is confusing and not sure what to do.

 

[Simon Bridge]

Have you tried:

$\Psi(x,t)=c_1(t)\psi_1(x)+c_2(t)\psi_2(x)$