These vectors are just a column of numbers representing the coefficients of the four pure spin states: cl, c2, c3 and c4. They do not describe the whole ensemble of spins, just one Ha - Hb pair.

The Hamiltonian is an operator that represents the energies of all of the interactions in the system. In quantum mechanics, the classical energy terms are replaced by the analogous operators to generate the Hamiltonian. For NMR, the classical energy can be written as

E/h = [-Y (Bo ■ Ia) - Y (Bo ■ Ib)] + J [/ a ■ I b] (in units of Hz)

Where Bo is a vector representing the external magnetic field, J is the coupling constant, Ia and Ib are the vectors representing the nuclear magnets of Ha and Hb and the products are vector dot products. The first part represents the energy of interaction of the spins with the Bo field and the second part represents the energy of interaction of the spins with each other. Since the Bo vector has only a z component, we can write the vector products as

ab ab y y where Bo is now just the magnitude of the Bo vector. Only the z component of the nuclear magnet contributes to its interaction with the Bo field. Now we replace the classical components Ix, Iy, and Iz of the nuclear magnet's vector with the corresponding quantum mechanical operators:

h = —a iz - vbib] + J [ixib+iy ib+iaib] = Hz + Hj

Here we also account for the slightly different Beff fields experienced by Ha and Hb leading to their individual Larmor frequencies va and vb. The first part, called the Zeeman Hamiltonian, again represents the interaction of the spins with the Bo field. The second part, called the scalar coupling Hamiltonian, represents the energy of interaction of Ha with Hb, independent of the Bo field. The J-coupling interaction is isotropic: it happens in all possible directions of space depending on the relative orientation of the two nuclear magnets.

There are many other terms to the Hamiltonian but for spin-1/2 nuclei in liquids they can all be ignored. The dipole-dipole (dipolar or direct coupling) Hamiltonian is important in solids and partially oriented liquids, and the quadrupolar Hamiltonian is important for spins greater than 1/2. The dipolar interaction contains a multiplier of

where © is the angle between the Ha - Hb vector and the Bo field (z axis). This factor averages to zero for random isotropic (equal in all directions in space) molecular tumbling, and if the motion is rapid the dipolar term can be ignored. This is what defines NMR in liquids: rapid isotopic reorientation of the molecules.

From our study of product operators and the density matrix we know what these operators Ix, Iy and Iz do and we know how to represent them in matrix form. So we can write out the Hamiltonian matrix for the Ha, Hb system:

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