BCMB/CHEM 8190
PROBLEM SET 3

1) Write down a Hamiltonian for a pair of spin 1/2 spins in a static B0 field using the z component spin operators Iz1, Iz2, their coupling constant J12, and their chemical shieldings, sigma 1 and 2. Make an engergy level diagram showing the positions of levels for the four simple product spin functions. Show the allowed one quantum transitions and calculate the energy differences associated with them. Sketch and label the corresponding lines in a frequency resolved spectrum.

2) Write down the operator for the total z magnetization for a two spin 1/2 system in terms of the single spin operators (call the spins 1 and 2).  Evaluate the magnetization (expectation value) associated with the alpha/alpha and alpha/beta spin states.  What is the expectation value for the total I² operator for these two states?  What is the expectation value for the z magnetization of spin 1 in these two states?

3) Derive a matrix representation for the Ix operator using the eigenfunctions for a spin 1 nucleus in a static magnetic field as a basis set. Use matrix methods to find the relative intensities of 1 - 0 and 1 - -1 transitions associated with this operator. You will find the general expression for the effect of raising and lowering operators on spin functions useful in doing this:  I(+)(psi(I,m)) = ((I(I+1)-m(m+1))**1/2) (psi(I,m+1)),  I(-)(psi(I,m) = ((I(I+1)-m(m-1))**1/2) (psi(I,m-1)).

4) Use density matrix methods to examine magnetization for a pair of spin 1/2 nuclei in a magnetic field Bo along the z axis.
a). Show the elements of an equilibrium density matrix (sigma) in the simple product basis set basis set.
b). Calculate equilibrium z magnetization using matrix methods.
c). Find the  frequency at which the 1,2 element oscillates.
d). At what frequency does the  1, 4 element oscillate?