A state is $\psi(x,t)=\psi_2(x,t)+\psi_4(x,t)$, where $\psi_2$ and $\psi_4$ are the second and fourth normalized harmonic-oscillator wave functions and $\omega$ is the oscillator angular frequency. The state at $t=0$ is orthogonal to the state at time
Orthogonality of the energy eigenstates eliminates cross terms in the overlap: $\langle\psi(0)|\psi(t)\rangle=e^{-iE_2t/\hbar}+e^{-iE_4t/\hbar}$. Normalizing the sum would multiply this by $1/2$ without changing its zeros. Since $E_4-E_2=2\hbar\omega$, the overlap vanishes when $1+e^{-i2\omega t}=0$. The earliest positive solution is $t=\pi/(2\omega)$, option A.
More One dimensional potentials: step potential, finite rectangular well, tunnelling from a potential barrier, particle in 1,2,3-dimensional box, particle in single and double delta function potentials, 1,2,3 dimensional harmonic oscillator: concept of degeneracy questions
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