For the harmonic oscillator [V(x)=(1/2)kx2], the allowed energies are
En=(n+1/2)ℏω,(n=0,1,2,…),
where ω=k/m is the classical frequency. Now suppose the spring constant increases slightly: k→(1+ϵ)k. (Perhaps we cool the spring, so it becomes less flexible.)
(a) Find the *exact* new energies (trivial, in this case). Expand your formula as a power series in ϵ, up to second order.
(b) Now calculate the first-order perturbation in the energy, using Equation 7.9. What is H′ here? Compare your result with part (a). *Hint:* It is not necessary—in fact, it is not *permitted*—to calculate a single integral in doing this problem.