At some initial time, t = 0, a particle trapped in a one-dimensional. harmonic oscillator V(x) = (1/2)mwr is described by the wavefunction (x, t = 0) = v(a) + B 44(x) + =18(2), where 1(x), 4(x), and 5(x) are normalised eigenstates and a, 3 > 0 are constants. (a) (i) By recalling the expression for the energy of the nth eigenstate of the harmonic oscillator, identify the energy that each of the above eigenstates corresponds to. (ii) State the corresponding probabilities of measuring each of these energy values. (b) (i) Write down the corresponding expression for the wavefunction at a subsequent time t. (ii) What is the highest frequency of oscillation exhibited by V(x, t)|? (Hint: This answer does not require you to work out the full ex- pression for (x, t), but can be directly stated, with appropriate rationale). (c) (i) Obtain an expression relating a and 3 such that (x, t) is nor- malised. (ii) If the expectation value of the energy takes the value (19/6)w, deduce the values of a and B.