Selection rules in the dipole approximation are determined by the integral μ mn x = «Ï * m (Ï) μ x (Ï) Ï n (Ï)

Selection rules in the dipole approximation are determined by the integral μ^mn_x=  ˆ«Ïˆ^*_m(Ï„) μ_x(Ï„) ψ_n(Ï„) dÏ„. If this integral is nonzero, the transition will be observed in an absorption spectrum. If the integral is zero, the transition is €œforbidden€ in the dipole approximation. It actually occurs with low probability because the dipole approximation is not exact. Consider the particle in the one-dimensional box and set μ_x= ˆ’e x. 

a. Calculate μ¹² _x and μ¹⁵ _x in the dipole approximation. Can you see a pattern and discern a selection rule? You may need to evaluate a few more integrals of the type μ ^1m _x. The standard integral

is useful for solving this problem.

b. Determine the ratio μ ¹² _x / μ ¹⁴ _x. On the basis of your result, would you modify the selection rule that you determined in part (a)?

axsin - 1)7x ( 1) 2 cos (n 1)TX a co- . Xsn sin- dx = a a. ( - 1?? co (n + 1)Tx ( + 1) ( + 1) axsn- cos (n + 1)7 ( + 1)