= Solution
Each <energy> component acquires its own phase:
$$
\psi(x,t)=\frac1{\sqrt S}\left[
0.2e^{-i\epsilon t/\hbar}\psi_1(x)+
0.3e^{-4i\epsilon t/\hbar}\psi_2(x)+
0.4e^{-9i\epsilon t/\hbar}\psi_3(x)+
0.843e^{-16i\epsilon t/\hbar}\psi_4(x)\right].
$$
The $1/\sqrt S$ factor normalizes the printed rounded input; it can be omitted if the coefficients are regarded as an already normalized rounded representation.
\b[The <energy> probabilities and $\langle E\rangle$ remain constant] under this time-independent <hamiltonian> because phase factors have unit modulus. <position> probabilities can nevertheless vary: interference between different <energy> components generally changes with their relative phases. A stationary <energy> distribution is not the same thing as a stationary spatial <probability distribution>.
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