Consider a quantum system with just three linearly independent states. suppose the hamiltonian, in matrix form, is (1-) 0 0 where vo is a constant and e is some small number, e 1 write down the eigenvectors and eigenvalues of the unperturbed hamiltonian, (e a) b) solve for the exact eigenvalues of h. expand each of them as a power series in e up to second order. c) use first and second order non-degenerate perturbation theory to find the approximate eigenvalue for the state that grows out of the non-degenerate eigenvector of h. compare the exact result from part b). d) use degenerate perturbation theory to find the first-order correction to the wo initially degenerate eigenvalues. compare the exact results.

The frequency of vibrations of a vibrating violin string is given by f = 1 2l t ρ where l is the length of the string, t is its tension, and ρ is its linear density.† (a) find the rate of change of the frequency with respect to the following. (i) the length (when t and ρ are constant) (ii) the tension (when l and ρ are constant) (iii) the linear density (when l and t are constant) (b) the pitch of a note (how high or low the note sounds) is determined by the frequency f. (the higher the frequency, the higher the pitch.) use the signs of the derivatives in part (a) to determine what happens to the pitch of a note for the following. (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates df dl 0 and l is ⇒ f is ⇒ (ii) when the tension is increased by turning a tuning peg df dt 0 and t is ⇒ f is ⇒ (iii) when the linear density is increased by switching to another string df dρ 0 and ρ is ⇒ f is ⇒