Wavefunction with changing potential

[athrun200]

Homework Statement

See the attachment for question and solution

Homework Equations

See the hint in the question

The Attempt at a Solution

In part (c), it asks about the probability of finding the particle in ground state.

As far as I know, we need to write the wave function in terms of eigenfunction first.
i.e. ψ(x,t)=\sum c_{n} f_{n} (x) e^{-iωt}
Ground state correspond to n=0. Therefore, the probability we want is c_{0}

Also c_{0}=\int ψ^{*} f_{0}

f_{0} is provided in the hint which is \sqrt{\frac{\alpha}{\sqrt{\pi}}}e^{\frac{-\alpha^{2} x^{2}}{2}}
But I have no idea what ψ^{*} is.

Also I have no idea what is going on in the solution

 

[athrun200]

Also, I don't understand the solution for part d, can anyone explain it to me?
Especially why n must be even?