Help Understanding the answer to a Quantum Mechanic problem

[David0709]

Homework Statement

The answer is as follows: [/B]

However they said that time t=0 so I am confused how the exponent has a t in it surely it should be zero. Thanks

 

[Charles Link]

They are telling you in part (b) to write $O(t)$ in terms of the "eigenstates at t=0". $O(t)$ is not required to be at $t=0$. (the eigenstates at $t=0$, is what they are referring to).b $\\$ Additional note: In the part you labeled "Answer: Unseen" , in the 3rd line, I think those should read $u^*_m(x)$ and $u_n(x)$, without any $t$ dependence.