The Density Matrix Representation Of Spin States

The density matrix is based on the description of all NMR spin states as linear combinations or superpositions of the basic spin states or energy levels. For a system of identical spins, there are only two spin states: a and f. At equilibrium, there is a slight excess of population in the lower energy (a) state and a slightly depleted population in the higher energy (f ) state. This can be represented as a 2 x 2 matrix with the diagonal elements corresponding to the populations of the a and f states:

where N is the total number of identical spins and e (<<1) is a very small dimensionless number. This equilibrium state is broken down into two parts: the equal populations that play no role in NMR (identity matrix) and the population differences. The identity matrix and the factor Ne/2 will be ignored from now on since they will be the same in all spin states.

Oeq = N/2
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