Obtain the matrix that represents spin squared, IS2], and show it commutes with IS,I, [Sy] and [S,I. What is the significance of this result?

Math/Physic/Economic/Statistic Problems

The operator for the square of orbital angular momenturn, 1,2, has eigenfunctions and eigenvalues:

L2 Km(0, = 1(1 + 1),0 (0,)with / = 0,1,2 …

where Yin,(0, 0) are the spherical harmonic functions.

a) Obtain the energy eigenvalues and eigenfunctions associated with the rotational motion of rigid diatomic molecules in terms of the rnoment of inertia of the molecule.

b) The presence of molecules in interstellar space canb detected using microwave spectroscopy. The detected microwave radiation is associated with the spontaneous transition of molecules frorn their first excited rotational state to their ground state. At what frequency would radiation emitted from SO molecules be detected.

You may assume the separation of the S and 0 atoms is d=1.48×10-1.rn and the atornic rnasses are 32 amu for S and 16 amu for 0, the moment of inertia of a rigid dimer is given as

In the matrix representation of intrinsic spin the cartesian components of angular momenturn are represented by the Pauli spin matrices given as:
= 10) [s,]=(° Pi) IS = (0 21)

c) Obtain the matrix that represents spin squared, IS2], and show it commutes with IS,I, [Sy] and [S,I. What is the significance of this result?

d) The general quanturn spin state of an electron is written as a vector (;). Three successive measurements of S,, Sy and S, are made and a positive answer obtained in each case. What are the spin states after each measurement?

e) A final measurement is then made of Sr, what are the possible outcomes of this measurement and discuss how the predictions differ from classical physics.