[UPSC CSE 2001] Schrodinger’s Wave equation and applications (Q226, (5+15)M)

(i) Write down the matrix representation of the energy operator of a linear oscillator. (ii) A linear oscillator is prepared in the state given by \psi(x) = 1/\sqrt{5} \{\hat{\psi}_0(x) + \sqrt{2} \hat{\psi}_1(x) + \sqrt{2} \hat{\psi}_2(x)\} Evaluate the energy of the oscillator in this state.