Selection rules in the dipole approximation are determined by the integral . If this in...

Selection rules in the dipole approximation are determined by the integral . If this integral is nonzero, the transition will be observed in an absorption spectrum. If the integral is zero, the transition is “forbidden” 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 = -ex.

a. Calculate 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 .

The standard integral

is useful for solving this problem.

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