Effect of Hamiltonian on Trial Wavefunction (II)

The Hamiltonian Operator for the Harmonic Oscillator is:

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Consider a trial wave function (see Dykstra's 'Physical Chemistry, A Modern Introduction', page 252):
_trial = ₀ + ₁
where each of these tow contributors is an appropriate eigenfunction of the Hamiltonian. Is this combination an eigenfunction of the Hamiltonian? Please answer `yes' or `no'.
Answer =
Query, is the above answer correct?