**Physics H190, Spring
2002**

**Homework #1; due: Feb. 13**

Estimate the root-mean-square (rms) value of (

*R-R*_{e}), where*R*is the interatomic separation, and*R*_{e }is its equilibrium value, for a diatomic molecule in a low vibrational state.__Hint:__use, e.g., the approach we employed in class to estimate the characteristic values of electronic, vibrational, and rotational frequencies.Examine the explicit form of the spherical harmonics

**Y**_{JM }for J=0,1, and 2. Verify that the symmetry of these functions with respect to the rotation of the coordinate frame by p around the x-axis (so that x->x, y->-y, and z->-z) corresponds to (-1)^{J}.__Hint:__first determine the effect of the transformation on the angles J, j.Do you agree with the statements made in the last-but-one paragraph on p. 13 of J. M. Brown's book? If not, formulate and prove a related correct statement.

__Hint:__see, e.g., Sec. 86 (Symmetry of molecular terms) in the Landau and Lifshitz QM book. This problem was also discussed in class.