At higher energies, an oscillator will experience large-amplitude motions, so that the Hooke's law potential may break down, and one may need to consider additional terms in the potential. Consider a particle in a potential consisting of a normal spring potential V(x) = 1/2 kx2 plus a small perturbing anharmonic term proportional to x4: delta V(x) = 1/4 ax4 We can say this perturbing term is small if its effect is much less than that of the spring potential. Let's first express this condition in a dimensionless fashion based on classical mechanics. Let's stipulate that delta V(x = A)/E 1, where A is amplitude and E is the total energy of the unperturbed system. Eliminating the amplitude in terms of the total energy of the system E, express the perturbation condition solely in terms as a, k and E. Eliminating x2 in terms of the definitions of the raising and lowering operators, and . Perturbing anharmonic potential |n >. We will learn next semester that one can determine the correction delta En to leading order by computing the expectation value of the perturbing potential in the unperturbed nth state: delta En = Compute the ratio delta En/E. (Hint: in order to simplify the calculation, write delta V(x) = 1/4 ax2x2, and consider the effect odf each x2 in succession on | n > .)
